Dan Betea
email: dan.betea@gmail.com

Jérémie Bouttier
email: jeremie.bouttier@ipht.fr

The periodic Schur process and free fermions at finite temperature

We revisit the periodic Schur process introduced by Borodin in 2007. Our contribution is threefold. First, we provide a new simpler derivation of its correlation functions via the free fermion formalism. In particular, we shall see that the process becomes determinantal by passing to the grand canonical ensemble, which gives a physical explanation to Borodin's "shift-mixing" trick. Second, we consider the edge scaling limit in the simplest nontrivial case, corresponding to a deformation of the poissonized Plancherel measure on partitions. We show that the edge behavior is described, in a certain crossover regime different from that for the bulk, by the universal finite-temperature Airy kernel, which was previously encountered by Johansson and Le Doussal et al. in other models, and whose extreme value statistics interpolates between the Tracy-Widom GUE and the Gumbel distributions. We also define and prove convergence for a stationary extension of our model. Finally, we compute the correlation functions for a variant of the periodic Schur process involving strict partitions, Schur's P and Q functions, and neutral fermions.

Introduction

The Schur process, introduced by Okounkov and Reshetikhin [START_REF] Okounkov | Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram[END_REF] but also appearing more or less implicitly in the works of Johansson [Joh02,[START_REF] Johansson | Discrete polynuclear growth and determinantal processes[END_REF][START_REF] Johansson | The arctic circle boundary and the Airy process[END_REF], is in many aspects a discrete analogue of a random matrix model such as Dyson's Brownian motion. It is therefore not surprizing that it can be analyzed by the same techniques and admits scaling limits in the same universality classes (e.g. sine processes in the bulk, Airy processes at the edge). See for instance [START_REF] Okounkov | Symmetric functions and random partitions[END_REF][START_REF] Johansson | Random matrices and determinantal processes[END_REF] and references therein.

In this paper we revisit the periodic Schur process introduced by Borodin [START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF]. One of his motivations was that it allows to study random cylindric partitions in the same way as the original Schur process allows to study random plane partitions [START_REF] Okounkov | Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram[END_REF][START_REF] Okounkov | Random skew plane partitions and the Pearcey process[END_REF]. See [START_REF] Gessel | Cylindric partitions[END_REF] for the definition of cylindric partitions and Figure 1 for an illustration.

The periodic Schur process is a measure on periodic sequences of integer partitions of the form µ p0q Ă λ p1q Ą µ p1q Ă ¨¨¨Ą µ pN ´1q Ă λ pN q Ą µ pN q " µ p0q (1.1) such that Probp λ, µq 9

u |µ p0q | N ź k"1
`sλ pkq {µ pk´1q `ρk ˘sλ pkq {µ pkq `ρḱ ˘˘.

(1.2)

Here u is a nonnegative real parameter smaller than 1, s λ{µ is a skew Schur function, and the ρ k are collections of variables or specializations-see the beginning of Section 3 for a summary of the relevant definitions. Note that the constraint u ă 1 arises because constant sequences would contribute an infinite mass otherwise. For u " 0 the measure is concentrated on sequences such that µ p0q is the empty partition, and we recover the original Schur process [START_REF] Okounkov | Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram[END_REF]. Taking u ‰ 0 brings an extra level of complexity: as shown by Borodin [Bor07, Theorem A], the point process naturally associated with p λ, µq requires a nontrivial "shift-mixing" transformation to be determinantal, and its correlation functions are given by an elliptic deformation of those for the original process. Here Figure 1: A cylindric partition represented as a lozenge tiling on the cylinder (the left/right sides are identified). The nonzero entries of the cylindric partition (shown in red) correspond to the displacements of the horizontal lozenges with respect to the "ground state". The corresponding sequence of integers partitions of the form (1.1) is obtained by reading the entries along vertical lines, namely we have µ p0q " p2, 1q, λ p1q " p3, 2, 1q, µ p1q " p2, 1q, λ p2q " p5, 1q, etc (here N " 6). Note the constraint that two consecutive partitions λ, µ must be interlaced-i.e., satisfy λ 1 ě µ 1 ě λ 2 ě µ 2 ě ¨¨¨. In the measure (1.2) this constraint is implemented by taking each ρ k to be a specialization in a single variable-see (3.5). Here the profile-in the sense of [START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF]-is the alternating sequence p1, ´1, 1, ´1, . . .q.

we will rederive this result using the free fermion formalism1 , which was also used recently to analyze the Schur process with free boundary conditions [START_REF] Betea | The free boundary Schur process and applications I[END_REF]. We believe that our derivation is more transparent. In particular the shift-mixing transformation follows from a passage to the grand canonical ensemble in the fermionic picture. For pedagogical purposes, we illustrate in Section 2 the key idea of our approach for the most elementary setting N " 0, where the measure (1.2) reduces to a single uniform2 random partition.

We also investigate the edge behavior of the periodic Schur process, the bulk case having been already analyzed by Borodin. We will concentrate on the simplest nontrivial instance [START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF]Example 3.4] where N " 1 and both ρ 1 , ρ 1 are exponential specializations. In other words, we consider two random partitions λ Ą µ such that Probpλ, µq "

1 Z u,γ u |µ| ˆγ|λ|´|µ| dimpλ{µq p|λ| ´|µ|q! ˙2 (1.3)
where u P r0, 1q, γ ą 0, Z u,γ :" e γ 2 1´u { ś ně1 p1 ´un q, and dimpλ{µq denotes the number of standard Young tableaux of skew shape λ{µ. The marginal distribution of λ interpolates between the uniform measure (γ " 0) and the poissonized Plancherel measure (u " 0) on partitions, and we call it the cylindric Plancherel measure.

We are interested in the thermodynamic limit (u Ñ 1 or γ Ñ 8) where the partition λ becomes large, and consider the behavior of the first part of λ. It is well-known that the fluctuations of λ 1 are asymptotically given by the Gumbel distribution in the uniform case [START_REF] Erdős | The distribution of the number of summands in the partitions of a positive integer[END_REF], and by the Tracy-Widom GUE distribution in the Plancherel case [START_REF] Baik | On the distribution of the length of the longest increasing subsequence of random permutations[END_REF]. We find that, in a suitably tuned thermodynamic limit, our model provides a "crossover" between these two behaviors. The interpolating distribution depends on a positive parameter α and was previously encountered by Johansson in the so-called MNS random matrix model [START_REF] Johansson | From Gumbel to Tracy-Widom[END_REF], and by Le Doussal et al. for free fermions in a confining trap [START_REF] Dean | Noninteracting fermions at finite temperature in a d-dimensional trap: Universal correlations[END_REF][START_REF] Le Doussal | Periodic Airy process and equilibrium dynamics of edge fermions in a trap[END_REF]-see also [START_REF] Liechty | Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe-Neuberger-Shapiro random matrix model[END_REF] and the discussion in Section 8. It is given explicitly by a Fredholm determinant F α psq :" detpI ´Mα q L 2 ps,8q , M α px, yq :"

ż 8 ´8
e αv 1 `eαv Aipx `vq Aipy `vqdv, s, x, y P R (1.4)

where Ai is the Airy function and α a positive parameter. Johansson proved that F α is well-defined and indeed interpolates between the Gumbel (α Ñ 0) and the Tracy-Widom GUE distributions (α Ñ 8). The limit α Ñ 8 is immediate since the "Fermi factor" e αv 1`e αv reduces to the indicator function 1 vą0 , and M α reduces to the well-known Airy kernel. The limit α Ñ 0 is more subtle as it requires a rescaling. The kernel M α has been called the finite-temperature Airy kernel [START_REF] Le Doussal | Periodic Airy process and equilibrium dynamics of edge fermions in a trap[END_REF]. Our main result may then be stated as follows.

Theorem 1.1. Consider the cylindric Plancherel measure (1.3) and let u Ñ 1 and γ Ñ 8 in such a way that γp1 ´uq 2 Ñ α 3 ą 0 (with possibly α " 8). Then we have

Prob ˜λ1 ´2L u,γ L 1{3 u,γ
ď s ¸Ñ F α psq, where L u,γ :" γ 1 ´u " ´γ α ¯3{2 .

(1.5)

A more general statement holds for the joint convergence of λ 1 , . . . , λ k for any k ě 1. Let us comment on the "physical" intuition behind Theorem 1.1. On the one hand, for u Ñ 1, λ 1 will have "thermal" fluctuations whose order of magnitude is p1 ´uq ´1. On the other hand, for γ Ñ 8, the area |λ| concentrates around the value L 2 u,γ hence L u,γ is the natural length scale, and we expect "quantum" Tracy-Widom-type fluctuations of order L 1{3 u,γ for λ 1 . The edge crossover regime corresponds to the situation where thermal and quantum fluctuations are of the same order of magnitude, and we expect the resulting distribution F α to be universal. Note that the edge crossover regime is not the same as that for the bulk, where one takes u Ñ 1 keeping γ fixed. As noted by Borodin, as soon as we take γ large then we recover in the bulk the usual discrete sine process, with the same density profile as in the case u " 0 corresponding to the Logan-Shepp-Vershik-Kerov limit shape [START_REF] Logan | A variational problem for random Young tableaux[END_REF][START_REF] Vershik | Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux[END_REF]. Therefore, there exists an intermediate regime 1 ! γ ! p1 ´uq ´2 where the bulk behaves as at zero temperature (u " 0) while the edge behavior is still purely governed by thermal fluctuations. We have a precise counterpart of Theorem 1.1 in the whole "high temperature" regime.

Theorem 1.2. For the cylindric Plancherel measure (1.3), set u " e ´r and assume that r Ñ 0 and γr 2 Ñ 0 (with γ possibly remaining finite). Then the rescaled fluctuations of λ 1 follow the Gumbel distribution. To wit:

Prob ˆrλ 1 ´ln I 0 p2γ `γrq r ď s ˙Ñ e ´e´s (1.6)

where I 0 pxq " 1 2π ş π ´π e x cos φ dφ is the modified Bessel function of the first kind and order zero. Intuitively speaking, for 0 ă γ ! p1 ´uq ´2, there is still a crossover, but at the level of the deterministic leading-order contribution to λ 1 rather than at the level of fluctuations. The deterministic term, which we do not expect to be universal, is consistent with both Borodin's density profile in the regime γ " Op1q, and with Theorem 1.1 in the intermediate regime 1 ! γ ! p1 ´uq ´2. Indeed, in the latter case we have r ´1 ln I0p2γ`γrq r " 2L u,γ ´lnp4πγr 2 q 2r `op1q, the logarithmic correction matching that in Johansson's formula for the limit α Ñ 0 of the F α distribution [Joh07, Theorem 1.3].

Interestingly, the cylindric Plancherel measure admits a stationary continuous-time periodic extension, which is the periodic analogue of the stationary process of Borodin and Olshanski [START_REF] Borodin | Stochastic dynamics related to Plancherel measure on partitions[END_REF], and which we call the cylindric Plancherel process-see Figure 2. We show that its correlation kernel converges in the edge crossover regime to the extended finite-temperature Airy kernel [START_REF] Le Doussal | Periodic Airy process and equilibrium dynamics of edge fermions in a trap[END_REF]. The cylindric Plancherel process can be thought as a certain "poissonian" limit of a measure on cylindric partitions, as described below.

Finally, we consider an extension of our approach to the shifted/strict setting. Vuletić [START_REF] Vuletić | The shifted Schur process and asymptotics of large random strict plane partitions[END_REF] defined the so-called shifted Schur process, which is a measure on sequences of strict partitions (an integer partition is

b = 0 b = β 0 λ 1 -1 λ 2 -2 λ 3 -3 λ 4 -4 λ 5 -5 λ 6 -6 • • • 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -7 7
Figure 2: A possible sample of the cylindric Plancherel process. Following the notations of Section 6, the horizontal axis represents the (imaginary) continuous time b, β is the period, and the i-th path represents the time-evolution of λ i P t0, 1, 2, . . .u which performs unit jumps up or down at random times (we rather represent λ i ´i so that the paths are nonintersecting). For i large enough (here i ě 7), λ i remains equal to 0 at all time. For any fixed time b, the law of the partition λpbq is the cylindric Plancherel measure. called strict if all its parts are distinct), and whose definition involves Schur's P and Q functions instead of the ordinary Schur functions. We naturally introduce the periodic variant of this process, which we prefer to call the periodic strict Schur process: to a sequence of the form (1.1) where each element is a strict partition, we assign a weight Probp λ, µq 9 u |µ p0q | N ź k"1 `Qλ pkq {µ pk´1q `ρk ˘Pλ pkq {µ pkq `ρḱ ˘˘(1.7) with u a real parameter smaller than 1, P λ{µ and Q λ{µ the Schur P and Q functions, and the ρ k strict specializations-see the beginning of Section 7 for definitions. Vuletić's definition is recovered in the case u " 0, where the measure is concentrated on sequences such that µ p0q is the empty partition. We compute the correlation functions of the periodic strict Schur process using again the free fermion formalism that now involves so-called neutral fermions as already observed in the u " 0 case [START_REF] Vuletić | The shifted Schur process and asymptotics of large random strict plane partitions[END_REF]. The approach is even simpler than in the non strict case, because the shift-mixing transformation becomes trivial. We find that the periodic strict Schur process is a pfaffian point process, whose kernel involves elliptic functions.

Outline. Our paper is organized as follows. In Section 2 we discuss the connection between uniform random partitions and Fermi-Dirac statistics, which illustrates the key idea of our approach in the most elementary setting. In Section 3, we recall the fundamental results of [START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF] on the correlation functions of the periodic Schur process. Our new derivation via the free fermion formalism is then given in Section 4. We then turn to the asymptotic analysis of the edge behavior in Section 5, and provide the proof of Theorem 1.1 (for Theorem 1.2, some technical estimates are left to Appendix D). The stationary cylindric Plancherel process is treated in Section 6. Section 7 is devoted to the periodic strict Schur process and the derivation Figure 3: Illustration of the correspondence between a partition and a Maya diagram. The Young diagram of the partition p4, 2, 1q appears in blue. The corresponding Maya diagram is represented as a sequence of "particles" (') and "holes" (˝). There must be as many particles on the right of the bottom corner of the diagram as holes on its left. We may lift this constraint on the Maya diagram by moving the origin (displayed at the intersection between the axes) to an arbitrary position on the horizontal axis. By cutting the Maya diagram at the origin, we obtain two strict partitions (parts correspond to particles on the right, holes on the left), whose number of parts differ by the charge (the abscissa of the bottom corner of the diagram, here equal to 2). of its correlation functions via neutral fermions. Finally Section 8 gathers some concluding remarks and discussion. A paper would not be complete without appendices: in Appendix A we recall the definition of the elliptic functions that we use; since it seems that basic facts about fermions should always be recalled in an appendix (see e.g. [Oko01, OR03, OR07, Vul09]), we abide by the rule in Appendices B and C that deal with charged and neutral fermions respectively; finally, in Appendix D we perform some further analysis of the discrete finite-temperature Bessel kernel arising in Section 5, which we use to complete the proof of Theorem 1.2 and also to give a short rederivation of the bulk limiting kernel.

Acknowledgments. The authors had illuminating conversations related on the subject of this note with many people, including I. Alevy, J. Baik This work was initiated while the authors were at the Département de mathématiques et applications, École normale supérieure, Paris, and continued during several visits D.B. paid to J.B. at the ENS de Lyon. It was finalized while the authors were visiting the Matrix Institute on the University of Melbourne campus in Creswick, Australia. We wish to thank all institutions for their hospitality.

We acknowledge financial support from the "Combinatoire à Paris" project funded by the City of Paris (D.B. and J.B.), from the Laboratoire International Franco-Québécois de Recherche en Combinatoire (J.B.), and from the Agence Nationale de la Recherche via the grants ANR 12-JS02-001-01 "Cartaplus" and ANR-14-CE25-0014 "GRAAL" (J.B.).

2 Warm-up: integer partitions and Fermi-Dirac statistics An (integer) partition λ is a nonincreasing sequence of integers λ 1 ě λ 2 ě ¨¨¨which vanishes eventually. The strictly positive λ i 's are called the parts, and their number is the length of the partition, denoted by pλq. The size of λ is |λ| :" ř iě1 λ i . A partition is called strict if all its parts are distinct. We denote by P (respectively SP) the set of all (respectively strict) partitions. In physics, arbitrary partitions describe "bosons", and strict partitions "fermions" [START_REF] Vershik | Statistical mechanics of combinatorial partitions, and their limit shapes[END_REF].

There are several classical bijections between various classes of partitions, strict or not. For instance, P is in bijection with pairs of strict partitions of the same length. Such pairs may conveniently be seen as Maya diagrams (or excited "Dirac seas"), and there is more generally a bijection between P ˆZ (charged partitions) and SP ˆSP-see Figure 3. This bijection yields a beautiful proof of the Jacobi triple product identity, see e.g. [START_REF] Corteel | Particle seas and basic hypergeometric series[END_REF] and references therein. We now recall this proof as it is at the core of our approach.

A Maya diagram is a bi-infinite binary sequence n " pn i q iPZ 1 P t0, 1u Z 1 such that n i " 0 for i " 0 and n i " 1 for i ! 0. Here Z 1 :" Z `1{2 denotes the set of half-integers. This latter choice is purely conventional and makes some formulas more symmetric. On Figure 3, 0's and 1's are represented as holes and particles respectively. To a Maya diagram n we associate its charge Cpnq and its energy Hpnq defined by Cpnq :"

ÿ ią0 n i `ÿ iă0 pn i ´1q, Hpnq :" ÿ ią0 in i `ÿ iă0 ipn i ´1q. (2.1)
Clearly, the energy is nonnegative and vanishes if and only if n " t1 iă0 u (vacuum). Given two parameters t, u, we associate to each Maya diagram a weight t Cpnq u Hpnq . The total weight, the sum over all Maya diagrams, clearly factorizes as an infinite product over i P Z 1 . On the other hand, if we consider the charged partition pλ, cq associated with n, then it is not difficult to check that

Cpnq " c, Hpnq " |λ| `c2 2 . (2.2)
Therefore, the weight can be rewritten as t c u |λ|`c 2 {2 . By summing over all pairs pλ, cq, we end up with the identity

8 ź k"0 p1 `tu k`1{2 qp1 `t´1 u k`1{2 q " ˜ÿ λPP u |λ| ¸˜ÿ cPZ t c u c 2 {2 ¸" 1 pu; uq 8 θ 3 pt; uq (2.3)
which is equivalent to the Jacobi triple product identity-see Appendix A for reminders and notations.

Let us now reflect on the probabilistic meaning of this construction. Viewing t Cpnq u Hpnq as a Boltzmann weight for n and normalizing by the partition function (2.3), we get a well-defined probability distribution for t P p0, 8q and u P r0, 1q. In the Maya diagram picture we learn that the n i are independent and that Probpn i " 1q " tu i 1 `tu i " 1 1 `t´1 u ´i .

(2.4) This is nothing but the Fermi-Dirac distribution for a system of noninteracting fermions with equally spaced energy levels (write t ´1u ´i " e βp i´µq to recognize more physical variables). In the charged partition picture, we learn that λ and c are independent, with λ a uniform random partition and c distributed as Probpcq " t c u c 2 {2 θ 3 pt; uq .

(2.5) This fact was observed by Borodin [Bor07, Corollary 2.6] and we suspect that it might appear elsewhere in the literature under various forms. If we condition on c " 0, then the n i 's are no longer independent. Therefore, passing to the "grand canonical ensemble"-as we do by letting c fluctuate-makes it much easier to study random partitions, at least for the observables we can capture in the Maya diagram picture. We argue that this is the fundamental reason why the shift-mixed periodic Schur process (to be defined below) is determinantal: fermions are only "free" in the grand canonical ensemble.

Before proceeding, let us briefly discuss the thermodynamic limit u Ñ 1 ´. From (2.4) we see that, if we rescale i " x{| ln u|, then the Maya diagram has a limiting density profile 1{p1 `t´1 e x q. It is not difficult to check that this is consistent with Vershik's limit shape for uniform random partitions [START_REF] Vershik | Statistical mechanics of combinatorial partitions, and their limit shapes[END_REF]. Note that the parameter t only induces a shift of the profile. This is because the charge c concentrates around its mean value of order 1{| ln u|.

Lemma 2.1. Let c be an integer-valued random variable distributed as in (2.5). Then, as u Ñ 1 ´, the random variable c :" pc `ln t{ ln uq a | ln u| converges weakly to the standard normal distribution.

Proof. Observe that, for any interval I, Probpc P Iq is a Riemann sum converging to 1

? 2π ş I e ´x2 {2 dx.
This shows in some sense the equivalence of the canonical and grand canonical ensembles for our fermions. Note that the equivalence with the microcanonical ensemble (partitions of fixed size) is another matter [START_REF] Vershik | Statistical mechanics of combinatorial partitions, and their limit shapes[END_REF], and that the equivalence between the canonical and grand canonical ensembles in the MNS model was established in [START_REF] Liechty | Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe-Neuberger-Shapiro random matrix model[END_REF]. We now turn to the edge behavior of the Maya diagram-see [START_REF] Okounkov | Infinite wedge and random partitions[END_REF]Section 3.4] (2.7)

In other words, the fluctuations of M are given by the Gumbel distribution. By Lemma 2.1, we deduce that λ 1 has asymptotically the same distribution (with t " 1), which is consistent with the result of Erdős and Lehner [START_REF] Erdős | The distribution of the number of summands in the partitions of a positive integer[END_REF]. Note that we could have alternatively started from the expression Probpλ 1 ă kq " pu k ; uq 8 to arrive at the same result.

Correlations functions of the periodic Schur process

In this section we recall the fundamental results of [START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF], for which we give a new proof in the next section.

Here we follow the notational conventions of [START_REF] Betea | The free boundary Schur process and applications I[END_REF] regarding Schur functions, which we briefly recall now. Given a sequence of numbers ph n pρqq nPZ such that h 0 pρq " 1 and h n pρq " 0 for n ă 0, and two partitions λ, µ, the skew Schur function of shape λ{µ specialized at ρ is given by s λ{µ pρq :" det

s λ{µ pex γ q " γ |λ|´|µ| dimpλ{µq p|λ| ´|µ|q! (3.4)
with dimpλ{µq the number of standard Young tableaux of shape λ{µ. When only α 1 " q is nonzero, we obtain the specialization in a single variable q for which s λ{µ pqq "

# q |λ|´|µ| if λ 1 ě µ 1 ě λ 2 ě µ 2 ě ¨¨¨, 0 otherwise. (3.5)
For simplicity, we will assume in the following that all specializations are nonnegative and such that their generating functions are analytic and nonzero in a disk of radius R ą 1 (this is clearly the case for the exponential specialization). In that case, the measure (1.2) can be normalized into a probability distribution which is the periodic Schur process-see [START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF] or the discussion below for more on the normalization.

Following [START_REF] Okounkov | Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram[END_REF], to any sequence of partitions λ " pλ p1q , . . . , λ pN q q, we associate the point configuration

Sp λq :" "ˆi , λ piq j ´j `1 2 ˙, 1 ď i ď N, j ě 1 * Ă t1, . . . , N u ˆZ1 . (3.6)
Observe that this definition is closely related to the Maya diagrams encountered in Section 2: the λ piq j ´j `1 2 , j ě 1, are precisely the positions of particles (1's) in the Maya diagram associated with the partition λ piq when the charge is 0. See again Figure 3.

Our aim is to study the point configuration when λ is the first marginal of the periodic Schur process. There is no loss of generality in discarding µ, and doing so simplifies the ensuing expressions. Sp λq is a discrete point process, and as such its properties are characterized by the data of the correlation functions pU q :" Prob ´U Ă Sp λq ¯(3.7) for any finite set U Ă t1, . . . , N u ˆZ1 .

Borodin observed that ρpU q is quite complicated in general (though it can be written in the form of a multiple contour integral-see Proposition 3.2 below), but we may transform it into a determinant at the price of "shift-mixing" the process. More precisely, we take an integer-valued random variable c, independent of λ and distributed according to (2.5) (with t an extra arbitrary parameter), and we define the shift-mixed correlation function as ˜ pU q :" Prob ´U Ă Sp λq `p0, cq ¯.

(3.8)

In other words, we are just moving Sp λq by a random vertical shift c. The shift-mixed process is determinantal in the sense that there exists a correlation kernel K : pt1, . . . , N u ˆZ1 q ˆpt1, . . . , N u ˆZ1 q Ñ C such that ˜ pU q " det 1ďi,jďn Kpu i ; u j q, U " tu 1 , . . . , u n u.

(3.9) See for instance [START_REF] Johansson | Random matrices and determinantal processes[END_REF] for general background on determinantal processes. Let us now state Borodin's main result in our current notations.

Theorem 3.1 ([Bor07, Theorem A]). The shift-mixed periodic Schur process is determinantal with correlation kernel given by

Kpi, k; i 1 , k 1 q " 1 p2πiq 2 ¿ |z|"r dz z k`1 ¿ |w|"r 1 dw w ´k1 `1 ¨F pi, zq F pi 1 , wq ¨κpz, wq (3.10)
where i, i 1 P t1, . . . , N u, k, k 1 P Z 1 , the radii r, r 1 satisfy

R ´1 ď r, r 1 ď R, r{r 1 P # pu, 1q if i ą i 1 , p1, u ´1q if i ď i 1 , (3.11)
F carries the dependence on the specializations ρ k as F pi, zq :"

ś 1ď ďi Hpρ ` ; zq ś iď ďN Hpρ ´ ; z ´1q ź ně1 ź 1ď ďN
Hpu n ρ ` ; zq Hpu n ρ ´ ; z ´1q (3.12) and κpz, wq :"

# ř mPZ 1 pz{wq m 1`ptu m q ´1
for |z{w| P p1, u ´1q, ´řmPZ 1 pz{wq m 1`tu m for |z{w| P pu, 1q.

(3.13)

Several remarks are now in order. On the one hand, we recover the correlation kernel of the original Schur process [OR03, Theorem 1] by taking u " 0, in which case κpz, wq simplifies into ? zw{pz ´wq. Note that Probpc " 0q " 1 so shift-mixing can be disregarded in that case. On the other hand, if we consider only trivial specializations (so that F pi, zq " 1), then Kpi, k; i 1 , k 1 q vanishes for k ‰ k 1 , and the shift-mixed process is a discrete Poisson process. This is nothing but the phenomenon discussed in Section 2.

As remarked by Borodin [Bor07, Remark 2.4], the two cases in (3.13) correspond to the expansions in two different annuli of the same meromorphic function, namely κpz, wq " c w z ¨pu; uq 2 8 θ u p w z q ¨θ3 p tz w ; uq θ 3 pt; uq " i ηpuq 3 θ 3 p tz w ; uq θ 1 p w z ; uqθ 3 pt; uq

(3.14)
with pu; uq 8 the infinite Pochhammer symbol, η the Dedekind eta function, θ u the "multiplicative" theta function, and θ 1 , θ 3 the Jacobi theta functions-see Appendix A for conventions. The equality between the sum expression (3.13) and the "theta" form (3.14) is a particular case of Ramanujan's 1 Ψ 1 summation formula, and may be rederived directly using the boson-fermion correspondence as explained in Section 4.

The first theta form of κpz, wq arises naturally from the computations; in the second form we introduce the Dedekind eta function to make the intriguing modular properties more apparent. In fact, κpz, wq may be interpreted as a propagator xψpzqψ ˚pwqy for the conformal field theory of charged fermions on a torus, in which modular invariance plays a crucial role [DFMS97, Chapter 10]. Finally, Borodin observed that using an elliptic version of the Cauchy determinant due to Frobenius-see Remark 4.1 below-, we may obtain an explicit contour integral representation for pU q: Proposition 3.2 ([Bor07, Corollary 2.8]). The general n-point correlation function for the periodic Schur process is given by pU q " ηpuq 3n p2iπq 2n

¿ ¨¨¨¿ n ź "1 ˜dz z k `1 ¨dw w ´k `1 ¨F pi , z q F pi , w q ¸¨ś 1ď ămďn θ 1 pz {z m ; tqθ 1 pw {w m ; tq ś 1ď ,mďn θ 1 pz {w m ; tq (3.15)
where U " tpi 1 , k 1 q, pi 2 , k 2 q, . . . , pi n , k n qu and, upon ordering i 1 ď i 2 ď ¨¨¨ď i n , the integration is taken over nested circles minpu 1{2 , Rq

ą |z 1 | ą |w 1 | ą |z 2 | ą |w 2 | ą ¨¨¨ą |z n | ą |w n | ą maxpu ´1{2 , R ´1q.
The n-point correlation ρpU q for the shift-mixed process is given by simply multiplying the integrand above by θ 3 pt z1z2¨¨¨zn w1w2¨¨¨wn ; uq{θ 3 pt; uq.

Derivation via free fermions

This section is devoted to a proof of Theorem 3.1 and Proposition 3.2 via the machinery of free fermions. Again we make use of the same conventions and notations as in [START_REF] Betea | The free boundary Schur process and applications I[END_REF]-for convenience the relevant definitions and facts are recalled in Appendix B. Our starting point is the observation that the partition function-i.e., the sum of the unnormalized weights (1.2)-of the periodic Schur process can be represented as

Z " tr `Π0 u H Γ `pρ 1 qΓ ´pρ 1 q ¨¨¨Γ `pρ Ǹ qΓ ´pρ Ń q ˘(4.1)
where tr stands for the trace over the fermionic Fock space and Π 0 denotes the orthogonal projector over the subspace of charge 0 (as we shall sum over ordinary uncharged partitions). Similarly, for U " tpi 1 , k 1 q, . . . , pi n , k n qu Ă t1, . . . , N u ˆZ1 , if the abscissas are ordered as i 1 ď ¨¨¨ď i n , then the correlation function pU q as defined in (3.7) is equal to Z U {Z, where

Z U :" tr `Π0 u H Γ `pρ 1 q ¨¨¨Γ `pρ ì1 qψ k1 ψ k1 Γ ´pρ í1 q ¨¨¨Γ `pρ ìn qψ kn ψ kn Γ ´pρ ín q ¨¨¨Γ ´pρ Ń q ˘(4.2)
is the sum of the unnormalized weights of sequences such that U Ă Sp λq.

Let us denote by Z and ZU the quantities obtained by replacing respectively in (4.1) and (4.2) the projector Π 0 by the operator t C . Then, it is not difficult to check that ZU { Z is precisely the shift-mixed correlation function ρpU q as defined in (3.8).

Eliminating the Γ-operators. We now rewrite the quantities Z, Z U , Z, ZU following the strategy of [START_REF] Okounkov | Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram[END_REF], which was already adapted to the periodic setting in [START_REF] Bouttier | From Aztec diamonds to pyramids: steep tilings[END_REF] (in the case of Z). We call this method Γ-elimination: in a nutshell if we consider the trace of a product of operators involving some Γ ˘'s and other operators which they quasi-commute with (such as u H , ψpzq, ψ ˚pwq), then the quasi-commutation relations can be exploited to remove the Γ's from the product, up to a multiplication by the corresponding scalar factors.

Let us illustrate this method on the easiest case of Z. The basic idea is to move each of the Γ `pρ ì q to the right, using the quasi-commutation relations (B.8) with the other operators and the cyclic property of the trace (recall also that Γ-operators commute with C hence Π 0 ). For instance, performing this operation on each Γ `pρ ì q in (4.1) until it goes back into the same place, we get

Z " ź 1ďiďjďN Hpρ ì ; ρ j q ź 1ďjăiďN
Hpuρ ì ; ρ j q ˆtr `Π0 u H Γ `puρ 1 qΓ ´pρ 1 q ¨¨¨Γ `puρ Ǹ qΓ ´pρ Ń q ˘.

(4.3) (note that the specializations ρ ì have all been multiplied by u in the trace on the right). By iterating this procedure m times, and noting that the process converges for m Ñ 8 because Γ `pu m ρq Ñ 1 for m Ñ 8, we may rederive Borodin's expression for the partition function [Bor07, Proposition 1.1]-see [BCC17, proof of Theorem 12] for more details.

For the correlation functions, it is useful to first rewrite

Z U " " z k1 1 w ´k1 1 ¨¨¨z kn n w ´kn n ı tr `Π0 u H Γ `pρ 1 q ¨¨¨Γ `pρ ì1 qψpz 1 qψ ˚pw 1 qΓ ´pρ í1 q ¨¨Γ `pρ
ìn qψpz n qψ ˚pw n qΓ ´pρ ín q ¨¨¨Γ ´pρ Ń q ˘(4.4)

where rz k1 1 w ´k1 1 ¨¨¨z kn n w ´kn n s denotes coefficient extraction in the multivariate Laurent series on the right. We may now apply the same strategy to the trace in (4.4), and we will pick extra factors of the form Hpu m ρ ì ; z q{Hpu m ρ ì ; w q from the quasi-commutations between Γ `'s with ψpz qψ ˚pw q's. Note that the factors Hpu m ρ ì ; ρ j q coming from quasi-commutations between Γ `'s and Γ ´'s will eventually get cancelled when we divide by Z to get ρpU q. Furthermore, after we have "eliminated" the Γ `'s, we also need to get rid of the Γ ´'s by moving them similarly but to the left. This will produce factors of the form Hpu m ρ í ; w ´1 q{Hpu m ρ í ; z ´1 q. We end up with the expression

ρpU q " " z k1 1 w ´k1 1 ¨¨¨z kn n w ´kn n ı n ź "1
F pi , z q F pi , w q ¨xψpz 1 qψ ˚pw 1 q ¨¨¨ψpz n qψ ˚pw n qy p0q u (4.5)

where xOy p0q u :" trpΠ 0 u H Oq{ trpΠ 0 u H q denotes the "canonical" expectation value of the operator O, and where F is as in Theorem 3.1: the F pi , z q in the numerator (respectively the F pi , w q in the denominator) contains all the factors arising from the quasi-commutations of Γ's with ψpz q (respectively ψ ˚pw q) in (4.4).

All this reasoning remains valid if we consider the shift-mixed correlation function ρpU q, which then admits the expression obtained by replacing in (4.5) the canonical expectation value x¨y p0q u by the grand canonical one x¨y u,t :" trpt C u H ¨q{ trpt C u H q.

Determinantal structure of ρpU q. At this stage we may employ Wick's lemma in finite temperature (Lemma B.1) in its "determinantal" form to conclude that the shift-mixed correlation function can be rewritten as

ρpU q " " z k1 1 w ´k1 1 ¨¨¨z kn n w ´kn n ı n ź "1 F pi , z q F pi , w q ¨det 1ď ,mďn xT pψpz i q, ψ ˚pw im qqy u,t " det 1ď ,mďn Kpi , k ; i m , k m q, Kpi, k; i 1 ; k 1 q :" " z k w ´k1 ı F pi, zq F pi 1 , wq ¨#xψpzqψ ˚pwqy u,t i ď i 1 ´xψ ˚pwqψpzqy u,t i ą i 1 . (4.6)
This shows that the shift-mixed process is indeed determinantal. We may rewrite Kpi, k; i 1 ; k 1 q in the form (3.10) as follows. From the discussion of Section 2 and particularly (2.4)-or alternatively from (B.14)we see that

xψ k ψ k y u,t " tu k 1 `tu k " 1 1 `t´1 u ´k , xψ k ψ k y u,t " 1 1 `tu k , k P Z 1 (4.7)
while the expectation value of any other product of two fermionic operators vanishes. Passing to the generating series we get where we pass to the second line by Γ-elimination, and to the final form by using (2.3) with t Ñ tz{w. Upon dividing by the normalization tr `tC u H ˘" θ 3 pt; uq{pu; uq 8 we arrive at (3.14).

xψpzqψ ˚pwqy u,t " ÿ mPZ 1 pz{wq m 1 `ptu m q ´1 , xψ ˚pwqψpzqy u,t " ÿ mPZ 1 pz{wq m 1 `tu m . ( 4
Remarkably, the same trick allows to evaluate the canonical and grand canonical expectation value of the product of any number of fermionic generating series. We find xψpz 1 qψ ˚pw 1 q ¨¨¨ψpz n qψ ˚pw n qy p0q u " c w 1 ¨¨¨w n z 1 ¨¨¨z n pu; uq 2n 8 ś 1ďiăjďn θ u pz j {z i , w j {w i q ś 1ďiďjďn θ u pw j {z i q ś něiąjě1 θ u pz i {w j q .

(4.11) and xψpz 1 qψ ˚pw 1 q ¨¨¨ψpz n qψ ˚pw n qy u,t " θ 3 ˆt z 1 ¨¨¨z n w 1 ¨¨¨w n ; u ˙xψpz 1 qψ ˚pw 1 q ¨¨¨ψpz n qψ ˚pw n qy p0q u θ 3 pt; uq . (4.12)

Note that we should have u ă |w j {z i | ă 1 (respectively 1 ă |w j {z i | ă u ´1) for i ď j (respectively for i ą j) in order for the expectation values to be well-defined. By plugging these expressions in (4.5) and its shift-mixed counterpart, and writing the coefficient extractions as contour integrals, we obtain Proposition 3.2.

Remark 4.1. In fact, our computation yields a fermionic proof of the Frobenius elliptic determinant identity [START_REF] Frobenius | Ueber die elliptischen Functionen zweiter Art[END_REF]: on the one hand, using Wick's lemma, the correlator xψpz 1 qψ ˚pw 1 q ¨¨¨ψpz n qψ ˚pw n qy u,t may be written as the determinant det 1ďi,jďn κpz i , w j q whose entries can be put in the theta form (3.14); on the other hand we obtain a product of theta functions through (4.12) and (4.11).

Edge behavior of the cylindric Plancherel measure

The purpose of this section is to establish Theorems 1.1 and 1.2. Recall that we want to study the asymptotic distribution of λ 1 in the cylindric Plancherel measure, which is the λ-marginal of the measure (1.3). By (3.4), this measure is a periodic Schur process of period N " 1 with ρ 1 " ex γ the exponential specialization. We start of course by applying Theorem 3.1, which entails that the (one-dimensional) shift-mixed point configuration

Spλq `c " " λ j ´j `1 2 `c, j ě 1 * (5.1)
is a determinantal point process with correlation kernel

Kpa, bq " 1 p2πiq 2 ¿ |z|"u ´1{4 dz z a`1 ¿ |w|"u 1{4 dw w ´b`1
¨eLpz´z ´1q e Lpw´w ´1 q ¨κpz, wq (5.2)

" ÿ PZ 1 J a` p2LqJ b` p2Lq 1 `t´1 u (5.3)
where a, b P Z 1 , L " γ{p1 ´uq (this is the L u,γ from Theorem 1.1, but we drop the indices to lighten the notations) and J n denotes the Bessel function of the first kind. The choice of the integration radii is somewhat arbitrary but will prove convenient for the forthcoming analysis. We pass to the second form by using the definition (3.13) of κ and the Laurent series expansion e Lpz´z ´1q " ř nPZ J n p2Lqz n . Note that, for u Ñ 0 `, the factor 1{p1 `t´1 u q becomes the indicator function of Z 1 `and Kpa, bq becomes the celebrated Bessel kernel of Borodin-Okounkov-Olshanski [START_REF] Borodin | Asymptotics of Plancherel measures for symmetric groups[END_REF] and Johansson [START_REF] Johansson | Discrete orthogonal polynomial ensembles and the Plancherel measure[END_REF]. As such and in analogy to Johansson's "finite-temperature GUE kernel" [START_REF] Johansson | From Gumbel to Tracy-Widom[END_REF], we call the kernel Kpa, bq the discrete finite-temperature Bessel kernel.

Elementary properties of the discrete finite-temperature Bessel kernel. We note first that K is symmetric and positive semi-definite. This latter property is most evident from the Bessel representation: for any complex numbers z 1 , . . . , z n we have

n ÿ i,j"1 z i z j Kpa i , a j q " n ÿ i,j"1 ÿ PZ 1 z i z j J ai` p2LqJ aj ` p2Lq 1 `t´1 u " ÿ PZ 1 1 1 `t´1 u ˇˇˇˇn ÿ i"1 z i J ai` p2Sq ˇˇˇˇ2 ě 0. (5.4)
Second, for any m P Z 1 , K is trace-class on 2 ptm, m `1, . . . uq: here it is simpler to use the contour integral representation which immediately implies, by crudely bounding the integrand, that Kpa, bq " Opu ´pa`bq{4 q for a, b Ñ `8, hence we have ř iěm |Kpi, iq| ă 8. It follows that the Fredholm determinant detpI ´Kq 2 ptm,m`1,... uq is well-defined, and is equal to the probability that λ 1 `c ă m in the shift-mixed cylindric Plancherel measure. The parameter t governing the distribution (2.5) of c does not play a significant role in the asymptotic analysis and we will set it to 1 from now on.

Length scales and asymptotics. Before entering into the precise asymptotic analysis of K, it is useful to have a qualitative discussion of length scales. From the expression Z u,γ " e

K ´Y τ r ] , Y τ r ]¯" 1 2π ż π ´π dφ 1 `eτ´2γ cos φ ": ρpτ q, τ P R.
(5.7)

In fact he obtained more generally the bulk limiting kernel-see Proposition D.3. For τ Ñ 8 we have ρpτ q " I 0 p2γqe ´τ and we expect to find the "edge" (rightmost particle) at a position τe`Op1q r with ρpτ e q " r, i.e. τ e " ln I0p2γq r . This provides some intuition regarding the spatial rescaling in Theorem 1.2. For γ Ñ 8, the limiting density ρ tends to that of the Logan-Shepp-Vershik-Kerov limit shape, namely

lim γÑ8 ρpγxq " $ ' & ' % 0 if x ě 2, arccospxq if ´2 ă x ă 2, 1 if x ď ´2.
(5.8)

and we now expect to find the edge around the point 2 γ r " 2L, consistently with Theorem 1.1. Of course, understanding fluctuations requires more precise asymptotics, which we provide now.

To establish Theorem 1.1, we shall show that K converges to the finite-temperature Airy kernel M α in the edge crossover regime, as stated in the following.

Proposition 5.1. For t " 1 and in the edge crossover regime

$ & % L Ñ 8 u Ñ 1 Ĺ1{3 p1 ´uq Ñ α ą 0 $ & % a " t2L `xL 1{3 u b " t2L `yL 1{3 u m " t2L `sL 1{3 u ,
(5.9)

we have (5.14)

L 1{3 Kpa, bq Ñ M α px, yq (
The proposition says essentially that, near the edge in the high temperature regime, the shift-mixed determinantal process degenerates to a Poisson point process with exponentially decreasing intensity. See also the discussion in [START_REF] Johansson | From Gumbel to Tracy-Widom[END_REF]. The proof is given in Appendix D and is based a slightly nonstandard analysis of the contour integral representation for K, which may have an interest on its own. Proof that Proposition 5.1 implies Theorem 1.1. Observe that, to establish the convergence in distribution (1.5), we may replace λ 1 by the maximum of shift-mixed point configuration at t " 1, λ 1 `c´1{2. Indeed, the difference between the two rescaled random variables is pc ´1{2q{L 1{3 which, by Lemma 2.1, converges to zero in probability for t " 1. (And taking t ‰ 1 fixed simply amounts to a deterministic shift.)

Since the shift-mixed point process is determinantal, we are left with the task of proving the convergence of Fredholm determinants detpI ´Kq 2 ptm,m`1,... uq Ñ detpI ´Mα q L 2 ps,8q " F α psq

(5.15) in the scaling regime (5.9). This is done by standard arguments, which we recall for convenience. Set Kpx, yq :" L 1{3 Kpa, bq with a, b as in (5.9): by (5.10), Kpx, yq converges pointwise to M α px, yq, while we have detpI ´Kq 2 ptm,m`1,... uq "

8 ÿ n"0 p´1q n n! ż ps 1 ,8q n det 1ďi,jďn
Kpx i , x j qdx 1 ¨¨¨dx n .

(5.16) with s 1 " pm ´2Lq{L 1{3 Ñ s. The integrand converges pointwise to det 1ďi,jďn M α px i , x j q, and Hadamard's inequality states that its modulus is bounded by ś 8 i"1 Kpx i , x i q (recall that K is positive semi-definite), so we may conclude using dominated convergence and the convergence of traces (5.11) that detpI ´Kq 2 ptm,m`1,... uq Ñ 8 ÿ n"0

p´1q n n! ż ps,8q n det 1ďi,jďn
M α px i , x j qdx 1 ¨¨¨dx n " detpI ´Mα q L 2 ps,8q (5.17) as desired.

Remark since det 1ďi,jďn Kpx i , x j q Ñ e ´px1`¨¨¨`xnq almost everywhere. Note that, in the regime (5.12), we have 2Lr " 2γ `γr `op1q so the deterministic leading-order term M is essentially the same as that in Theorem 1.2.

Proof of Proposition 5.1 via the Bessel representation. We shall use Nicholson's approximation (found in [Wat22, Section 8.43] but see [BOO00, Lemma 4.4] for a recent elementary proof), as stated in [Rom15, Theorem 2.27]:

• for x P R, uniformly on compact sets,

L 1{3 J 2L`xL 1{3 p2Lq Ñ Aipxq, L Ñ 8;
(5.19)

• there exist constants c 1 , c 2 , L 0 such that for L ą L 0 and x ą 0,

L 1{3 |J 2L`xL 1{3 p2Lq| ă c 1 expp´c 2 xq.
(5.20)

In the Bessel representation (5.3), we split the sum into three parts

L 1{3 ÿ PZ 1 " L 1{3 ¨´T L 1{3 ´1 ÿ "´8 `T L 1{3 ÿ "´T L 1{3 `8 ÿ "T L 1{3 `1' ": Σ 1 `Σ2 `Σ3 . (5.21)
for some large enough T yet to be chosen. The main asymptotic contribution comes from Σ 2 which, using Nicholson's approximation (5.19), becomes a Riemann sum for the finite-temperature Airy integral:

Σ 2 « L ´1{3 T L 1{3 ÿ "´T L 1{3 ´L1{3 J 2L`xL 1{3 ` p2Lq ¯´L 1{3 J 2L`yL 1{3 ` p2Lq ¯eαL ´1{3 1 `eαL ´1{3 « L ´1{3 T L 1{3 ÿ "´T L 1{3 Aipx ` L ´1{3 q Aipy ` L ´1{3 q e αL ´1{3 1 `eαL ´1{3 « ż T ´T Aipx `vq Aipy `vq e αv 1 `eαv dv (5.22)
and the latter can be made arbitrarily close to the result by choosing T large enough. It remains to show the contributions |Σ 1 |, |Σ 3 | Ñ 0. We begin with Σ 1 . By using the same approximation and recognizing a similar Riemann sum, we have

|Σ 1 | « L ´1{3 ˇˇˇˇˇ´T L 1{3 ´1 ÿ "´8 ´L1{3 J 2L`xL 1{3 ` p2Lq ¯´L 1{3 J 2L`yL 1{3 ` p2Lq ¯eαL ´1{3 1 `eαL ´1{3 ˇˇˇˇ« L ´1{3 ˇˇˇˇˇ´T L 1{3 ´1 ÿ "´8 Aipx ` L ´1{3 q Aipy ` L ´1{3 q e αL ´1{3 1 `eαL ´1{3 ˇˇˇˇ« ˇˇˇˇż ´T ´8 Aipx `vq Aipy `vq e αv 1 `eαv dv ˇˇˇˇ(

5.23)

and the latter integral can be made arbitrarily small for appropriately large T . For instance, as the Airy function Ai is bounded, the Airy integral above is bounded by some positive constant times the integral ş ´T ´8 e αv 1`e αv dv " logp1 `expp´αT qq{α which is arbitrarily small if one chooses T large enough. We finally come to Σ 3 and for a change we shall use the asymptotics in (5.20):

|Σ 3 | « L ´1{3 ˇˇˇˇ8 ÿ "T L 1{3 `1 ´L1{3 J 2L`xL 1{3 ` p2Lq ¯´L 1{3 J 2L`yL 1{3 ` p2Lq ¯eαL ´1{3 1 `eαL ´1{3 ˇˇˇă L ´1{3 ˇˇˇˇ8 ÿ "T L 1{3 `1 ´L1{3 J 2L`xL 1{3 ` p2Lq ¯´L 1{3 J 2L`yL 1{3 ` p2Lq ¯ˇˇˇă c 2 1 L ´1{3 8 ÿ m"1 e ´c2px`y`2T `2mL ´1{3 q " c 2 1 e ´c2px`y`2T q L ´1{3 8 ÿ m"1 q m
(5.24)

where q " expp´2c 2 L ´1{3 q; for the first inequality we have used the crude bound e v {p1 `ev q ă 1; for the second we applied (5.20) (note we need at least T ě maxp´x, ´yq for this); and for the last equality we performed the change of variables ´T L 1{3 " m. The infinite series evaluates to q{p1 ´qq « L 1{3 {p2c 2 q and hence |Σ 3 | ă c 3 expp´px `y `2T qq for an appropriate positive constant c 3 , and this latter bound can be made arbitrarily small by choosing an appropriately large T . This concludes the proof of (5.10), and the proof of (5.11) is similar mutatis-mutandis.

Remark 5.4. The above argument adapts easily to the case α " 8 or more precisely to the case αL ´1{3 Ñ 8

as L Ñ 8. In equation (5.22), the term e αL ´1{3 1`e αL ´1{3 becomes the indicator of t ą 0u and by dominated convergence Σ 2 tends to the usual (here truncated) Airy kernel px, yq Þ Ñ ş T 0 Aipx `vq Aipy `vqdv while of course we still have Σ 1 , Σ 3 Ñ 0. See also Remark D.4 for the case of the bulk limit.

Proof of Proposition 5.1 via the contour integral representation. We assume here that α ă 8, see Remark 5.7 below for the case α " 8. We start with a useful lemma regarding the behavior of the propagator κpz, wq.

Lemma 5.5. Let u " e ´r and z{w " e r{2`iθ with θ P r´π, πs. Then we have κpz, wq " π r cosh πθ r `Ope ´π2 {r {rq, r Ñ 0 `(5.25)

where the big O is uniform over θ.

Proof. This immediately follows from the Poisson summation formula κpz, wq "

ÿ mPZ 1 e iθm 2 cosh rm 2 " ÿ kPZ p´1q k ż 8 ´8 e iθx´2iπkx 2 cosh rx 2 dx " ÿ kPZ p´1q k π r cosh πpθ´2πkq r .
(5.26)

Remark 5.6. Lemma 5.5 is very similar to [Bor07, Proposition 3.2], which was proved using the elliptic form (3.14) and the imaginary Jacobi transform, which is another consequence of the Poisson summation formula. It seems that our proof is much simpler.

We now rewrite the contour integral representation (5.2) in the form

L 1{3 Kpa, bq " 1 p2iπq 2 £
e LpSpzq´Spwqq κpz, wq L 1{3 dz dw z a 1 `1w ´b1 `1 (5.27)

with Spzq :" z ´z´1 ´2 ln z, a 1 :" a ´2L, b 1 :" b ´2L, and z and w are integrated over the circles of respective radii e r{4 and e ´r{4 , where we set again u " e ´r with r Ñ 0 `. Note that, in the scaling regime (5.9), we have rL 1{3 Ñ α, ra 1 Ñ αx and rb 1 Ñ αy.

Let us first record some useful properties of the "action" S: writing z " e r{4`iϕ we have

Spzq " 2 sinh ´r 4 ¯cos ϕ ´r 2 (5.28) which is maximal when ϕ " 0, i.e. when z is on the positive real axis. By the relation Spwq " ´Spw ´1q,

´ Spwq is also maximal on the positive real axis, and since Spe r{4 q " r 3 {196, the exponential factor e LpSpzq´Spwqq remains uniformly bounded. Fixing an P p0, 1{4q, we split the double integral (5.27) in two: let I be the contribution from the z and w whose arguments are both smaller than r 1´ in absolute value, and let I c be the complementary contribution. We shall show that I Ñ M α px, yq while I c Ñ 0:

• I c tends to 0: we bound the integrand as follows. Since either z or w has an argument larger than r 1´ in absolute value, we have by (5.28)

Spzq ´ Spwq ´2Spe r{4 q ď 2 sinh ´r 4 ¯`cos r 1´ ´1˘ď ´1 4 sinh ´r 4 ¯r2´2 (5.29) from which we deduce that |e LpSpzq´Spwqq | ď Ce ´α3 r ´2 {8 for some C ą 0. By Lemma 5.5, we have κpz, wq " Opr ´1q uniformly, while |z a 1 w ´b1

| " e rpa 1 `b1 q{4 remains bounded away from 0, so the integrand tends uniformly to 0 as wanted.

• I converges: we perform the change of variables z " e rζ{α , w " e rω{α (5.30)

where pζq " ´ pωq " α{4 while pζq, pωq are both smaller than αr ´ in absolute value. By a Taylor expansion of Spzq around the "monkey saddle" z " 1, we find that e LpSpzq´Spwqq " e and similarly for Aipy `vq which is given by the integral over ω. This concludes the proof of (5.10).

The proof of the convergence of traces (5.11) is entirely similar: in (5.27) the rightmost fraction should simply be replaced by dzdw pz´wqz m w ´m`1 (the factor 1 z´w is of order r ´1 " L 1{3 , and produces after the change of variables a factor 1 ζ´ω which we "decouple" by the usual integral representation 1 ζ´ω " ş 8 0 e pω´ζqv 1 dv 1 ).

Remark 5.7. In the case α " 8 the above proof may be adapted straightforwardly by integrating z and w over the circles of radii e ˘L´1{3 instead of e ˘r{4 . The change of variable (5.30) is simply replaced by z " e L ´1{3 ζ and w " e L 1{3 ω with pζq " ´ pωq " 1 and, using the Poisson summation formula as for Lemma 5.5, we see that κpz, wq may now be approximated by L 1{3 ζ´ω over the integration contour, instead of (5.32). We recover the usual zero-temperature Airy kernel as the limit.

Remark 5.8. As observed in several occasions-see e.g. [Oko02, OR03]-the asymptotic analysis of contour integral representations is particularly robust and explains why the limiting behavior should be universal. In our analysis, we simply use the fact that the action Spzq has a double critical point on the real axis, a property that is characteristic of the "edge" regime. Our proof should therefore be easily adaptable to other instances of periodic Schur processes. For example, we expect that the finite-temperature Airy kernel will be observed in cylindric partitions as the scaling limit around any generic edge point (the action now involving the dilogarithm function [START_REF] Okounkov | Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram[END_REF][START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF]).

The stationary cylindric Plancherel process

In this section we define a continuous-time periodic extension of the cylindric Plancherel measure. This process has the remarkable property of being stationary, i.e. invariant under time translation. It may be identified as the periodic analogue of the stationary process of Borodin and Olshanski [START_REF] Borodin | Stochastic dynamics related to Plancherel measure on partitions[END_REF], whose fixedtime marginal is the ordinary poissonized Plancherel measure. As we constructed our process through Fock space considerations, our presentation is somewhat different from [START_REF] Borodin | Stochastic dynamics related to Plancherel measure on partitions[END_REF].

Definition and basic properties. For β, ϑ two nonnegative parameters, let T pϑq pβq be the "transfer matrix" acting on the set P of integer partitions by T pϑq pβq λ,ν :" e ϑ 2 pe ´β ´1q ÿ µPP e ´β|µ| s λ{µ pex ϑp1´e ´β q qs ν{µ pex ϑp1´e ´β q q, λ, ν P P (6.1)

where we recall that ex denotes the exponential specialization (3.4)-we may therefore rewrite T pϑq pβq λ,ν in terms of the numbers of standard Young tableaux dimpλ{µq and dimpν{µq. What is remarkable about our definition is that the family `T pϑq pβq ˘βě0 forms a semigroup.

Proposition 6.1. For any β, β 1 , ϑ we have T pϑq pβqT pϑq pβ 1 q " T pϑq pβ `β1 q. (6.2)

Furthermore for any ϑ we have ř λPP T pϑq pβq λ,λ " Z β , where Z β :" ś kě1 p1 ´e´βk q ´1. Proof. The transfer matrix is represented in fermionic Fock space as T pϑq pβq " e ϑ 2 pe ´β ´1q Γ ´pex ϑp1´e ´β q qe ´βH Γ `pex ϑp1´e ´β q q (6.3) (we keep the same letter by a slight abuse of notation). We then leave to the reader the pleasure of checking that the commutation relations (B.8) imply the desired relation. The trace formula for Z β " trpΠ 0 T pϑq pβqq may be checked using the Γ-elimination method of Section 4.

Remark 6.2. In fact, we have T pϑq pβq " e ´βH ϑ where H ϑ :" H ´ϑpα 1 `α´1 q `ϑ2 with α ˘1 being the bosonic operators defined in (B.6). The expression (6.3) corresponds to the normal ordered form.

Definition 6.3. Fix β, ϑ two nonnegative real parameters. The (stationary) cylindric Plancherel process of period β and intensity ϑ is the random partition-valued continuous-time process λp¨q : R Ñ P which is β-periodic and whose finite-dimensional distributions are given by Probpλp0q, λpb 1 q, . . . , λpb n qq "

1 Z β n ź i"0
T pϑq pb i`1 ´bi q λpbiq,λpbi`1q (6.4) where 0 " b 0 ă b 1 ă ¨¨¨ă b n ă b n`1 " β.

By Proposition 6.1 it is clear that (6.4) defines a consistent family of finite-dimensional distributions, so the process is well-defined by the Kolmogorov extension theorem. We may see that λp¨q can be realized as a partition-valued jump process (with finitely many jumps almost surely)-see again Figure 2 for an illustration-but we shall not detail these considerations here as we are chiefly interested in the properties of the finite-dimensional distributions themselves. Note that the operator ´Hϑ mentioned in Remark 6.2, while having positive off-diagonal elements in the canonical basis, is not an intensity matrix so does not directly generate a Markov process. However, H ϑ is self-adjoint and nonnegative, hence our process might be thought as a quantum evolution in imaginary time.

The law of λp0q is the λ-marginal of the cylindric Plancherel measure (1.3) of parameters u " e ´β and γ " ϑp1 ´e´β q. This is also true for λpbq with any b P R as, from the definition and from the semigroup property, one may check easily that the process is stationary. Moreover, from (6.1) we may check that, for any times 0 ď b 1 ă ¨¨¨ă b N ď β, the tuple pλpb 1 q, . . . , λpb N qq is the λ-marginal of a periodic Schur process as defined in the introduction. Note that this requires to rewrite the weight (6.4) in the form (1.2): one sees that u is always e ´β and that the specializations ρ k will all be exponential, but their parameters have a slightly unnatural expression-which we omit here-since (1.2) breaks translation invariance due to the weight u |µ p0q | . Remark 6.4 (Infinite period/zero-temperature limit). In Fock space representation we have lim βÑ8 T pϑq pβq " e ´ϑ2 Γ ´pex ϑ q|HyxH|Γ `pex ϑ q.

(6.5)

In other words T pϑq p8q has rank one and we have the factorization T pϑq p8q λ,ν " e ´ϑ2 s λ pex ϑ qs ν pex ϑ q. From this observation, we see that the cylindric Plancherel process has well-defined limit as β Ñ 8, which is an ordinary (nonperiodic) continuous-time Schur process whose marginal at any given time is the poissonized Plancherel measure of parameter ϑ. We may identify it with the stationary process of Borodin and Olshanski [START_REF] Borodin | Stochastic dynamics related to Plancherel measure on partitions[END_REF]: this is not obvious from the definition, but it will be from the expression of the correlation functions.

Remark 6.5. In view of the characterization (3.3) for nonnegative specializations, the exponential specialization is the only one having the "infinite divisibility property" needed to define a stationary continuous-time Schur process. Therefore the cylindric Plancherel process seems to be one of a kind. In [START_REF] Borodin | Markov processes on partitions[END_REF], Borodin and Olshanski construct Markov processes preserving z-measures but, as they point out, they do not seem to belong to the class of Schur process.

Remark 6.6. The cylindric Plancherel process appears as a "poissonian" limit of a measure on cylindric partitions in the following sense: let us consider the instance of the periodic Schur process of length N given by Probp λ, µq 9 N ź k"1 ´sλ pkq {µ pk´1q pϑ q e ´ |µ pkq | s λ pkq {µ pkq pϑ q ¯. (6.6)

This weight may be rewritten in the form (1.2) but here translation invariance is manifest. Since the specializations consist of single variables, the weight is supported on sequences p λ, µq corresponding to cylindric partitions, see again Figure 1. Then, the cylindric Plancherel measure of period β and intensity ϑ appears as the limit N Ñ 8, Ñ 0 `with N Ñ β. This is because the "elementary transfer matrix" reads here Γ ´pϑ qe ´ H Γ `pϑ q " 1 ´ pH ϑ ´ϑ2 q `Op 2 q in the notation of Remark 6.2.

Correlation functions. We now turn to correlation functions, which characterize the point process

Spλq :" "ˆb , λpbq i ´i `1 2 ˙, b P r0, βs, i ě 1 * Ă r0, βs ˆZ1 . (6.7)
Following the discussion of Section 3, we shall rather consider the shift-mixed process Spλq `p0, cq, where c P Z is distributed as (2.5) with t an arbitrary positive parameter.

Proposition 6.7. The shift-mixed process Spλq `p0, cq is determinantal with correlation kernel given by

Kpb, k; b 1 , k 1 q " 1 p2iπq 2 ¿ |z|"1 `dz z k`1 ¿ |w|"1 ´dw w ´k1 `1 e ϑpz´z ´1q
e ϑpw´w ´1q κpze ´b, we ´b1 q (6.8)

where: b, b 1 P r0, βq, k, k 1 P Z 1 ; κ is as in (3.13) with u " e ´β ; and the notation 1 ˘means that we should take |z| a bit larger than |w| in the case b " b 1 , to circumvent the pole of κ at z " w.

Proof. That the shift-mixed process is determinantal is a consequence of Theorem 3.1. We may also derive the expression of Kpb, k; b 1 , k 1 q by converting the weight (6.4) in the form (1.2) and doing further manipulations, but we found it more instructive and less error-prone to rederive it directly from the free fermion formalism.

From the discussion of Section 4, the generating function Kpb, z; b 1 , wq :"

ÿ kPZ 1 ÿ k 1 PZ 1 Kpb, k; b 1 , k 1 qz k w ´k1 (6.9)
admits the Fock space representation

Kpb, z; b 1 , wq " Z ´1 β,t
ˆ#tr `tC T pϑq pbqψpzqT pϑq pb 1 ´bqψ ˚pwqT pϑq pβ ´b1 q ˘if b ď b 1 , ´tr `tC T pϑq pb 1 qψ ˚pwqT pϑq pb ´b1 qψpzqT pϑq pβ ´bq ˘if b ą b 1 . (6.10) where Z β,t " trpt C T pϑq pβqq " trpt C e ´βH q. We now plug in (6.3) and perform Γ-elimination, which yields Here, the somewhat surprising fact that the prefactor is independent of b and b 1 can be explained as follows: observe that, when performing Γ-elimination, the factor produced by the quasi-commutations with a given fermionic operator, say ψpzq, does not depend on the position of the other fermionic operator(s) in the product. Therefore this factor should be the same as for b " b 1 , and by using translation invariance we may reduce to the case b " b 1 " 0, from which we may identify the factor with the one present in (5.2) for the cylindric Plancherel measure (note that L " ϑ in our current notations). Finally, from the relations e ´bH ψpzqe bH " ψpze ´bq and e ´b1 H ψ ˚pwqe b 1 H " ψ ˚pwe ´b1 q (which result from the fact that ψ k and ψ ˚k "increase the energy" by k) and from the fermionic expression (4.9) of κ, we evaluate the remaining traces in (6.11) and get Kpb, z; b 1 , wq " e ϑpz´z ´1q e ϑpw´w ´1q κpze ´b, we ´b1 q.

(6.12)

The ratio of the arguments of κ should be in the annulus p1, e β q for b ď b 1 and in pe ´β , 1q for b ą b 1 , and this condition is ensured by taking |z| " 1 `and |w| " 1 ´. The coefficient Kpb, k; b 1 , k 1 q may then be extracted by a double contour integral.

Remark 6.8. The Bessel representation (5.3) of the correlation kernel for the cylindric Plancherel measure may be generalized as

Kpb, k; b 1 , k 1 q " # ř PZ 1 J k` p2ϑqJ k 1 ` p2ϑq e pb´b 1 q 1`t ´1 e β if b ď b 1 , ´ř PZ 1 J k` p2ϑqJ k 1 ` p2ϑq e pb´b 1 q 1`te ´β if b ą b 1 .
(6.13)

In the limit β Ñ 8, we recover the stationary version of the discrete extended Bessel kernel of [BO06b, Theorem 3.3].

Edge crossover behavior. We now study the edge crossover asymptotics of the kernel Kpb, k; b 1 , k 1 q, which we find to be described by the finite-temperature extended Airy kernel, as defined in [LDMS17, Equation (99)]. Proposition 5.1 admits the following extension. (Note that the convergence of traces need not be extended.) Proposition 6.9. For t " 1 and in the edge crossover regime

$ & % ϑ Ñ 8 β Ñ 0 θ1{3 β Ñ α $ ' ' & ' ' % b " βτ {α k " t2ϑ `xϑ 1{3 u b 1 " βτ 1 {α k 1 " t2ϑ `yϑ 1{3 u (6.14)
where we assume τ, τ 1 P r0, αq without loss of generality, we have

ϑ 1{3 Kpb, k; b 1 , k 1 q Ñ # ş 8 ´8 e pτ ´τ 1 qv 1`e ´αv Aipx `vq Aipy `vqdv if τ ď τ 1 , ´ş8 ´8 e pτ ´τ 1 qv
1`e αv Aipx `vq Aipy `vqdv if τ ą τ 1 .

(6.15)

Proof. We simply adapt the contour integral proof of Proposition 5.1. Let us highlight the few changes. First, obviously, the role of r is now played by β, and that of L by ϑ. An immediate generalization of Lemma 5.5 shows that κ " Opβ ´1q uniformly on the integration contours, and we have the estimate κpze ´b, we ´b1 q " π β sin πpζ´ω`τ 1 ´τ q α (6.16) in the vicinity of the saddle-point, where we perform the change of variables z " e βζ{α , w " e βω{α . Here, ζ and ω shall be integrated over lines parallel to the imaginary axis, with pζq ą 0 ą pωq and with pζ ´ω `τ 1 ´τ q ą 0 if τ ď τ 1 , ă 0 otherwise. In view of (6.8), all the dependence in b, b 1 being in the κ factor, we conclude that the limiting kernel is given by (5.33) where we replace the last factor by the right-hand side of (6.16). The last step is to plug in the integral representations of the cosecant and Airy functions, but for the cosecant we should be careful that it depends on the sign of τ 1 ´τ , namely π α sin πpζ´ω`τ 1 ´τ q α " # ş 8 ´8 e ´pζ´ω`τ 1 ´τ qv dv 1`e ´αv if τ ď τ 1 , ´ş8 ´8 e ´pζ´ω`τ 1 ´τ qv dv 1`e αv if τ ą τ 1 .

(6.17)

This leads to the desired expression (6.15).

Of course, some extra steps of analysis-which we omit-are needed to deduce the convergence of the rescaled process from that of the kernel. Note that Lemma 2.1 still ensures that shift-mixing is irrelevant in the considered scaling limit. We have not investigated the edge high temperature regime, namely whether Proposition 5.2 may be extended to the cylindric Plancherel process.

The periodic strict Schur process and neutral fermions

Vuletić [START_REF] Vuletić | The shifted Schur process and asymptotics of large random strict plane partitions[END_REF] defined the so-called shifted Schur process, which is a measure on sequences of strict partitions (as defined in Section 2), and whose definition involves Schur's P and Q functions instead of the ordinary Schur functions. In this section we explore the periodic variant of the shifted Schur process which we prefer to call the strict Schur process to emphasize that it deals with strict partitions.

Reminders on Schur's P and Q functions. (See [Mac95, Section III.8] for more general background.) Let SSym Ă Sym be the subalgebra of symmetric functions generated by the p n 's with n odd. This subalgebra contains the q n 's which are defined via the generating series Qpzq :"

ÿ ně0 q n px 1 , x 2 , . . . qz n " ź i 1 `zx i 1 ´zx i . (7.1)
This results from the relation Qpzq " expp ř ně1, odd 2p n z n {nq and, in fact, the q n 's with n odd form another algebraically independent family generating SSym.

For a ă b distinct nonnegative integers let Q pa,bq :" q a q b `2 ř b i"1 p´1q i q a`i q b´i and set Q pb,aq " ´Qpa,bq . For strict partitions µ, λ, write them such that λ 1 ą ¨¨¨ą λ m ą 0 and µ 1 ą ¨¨¨ą µ n ě 0 so we can assume without loss of generality that m `n is even. Then the skew Schur's Q function Q λ{µ is defined by Q λ{µ " pf M λ,µ where M λ,µ is the pm `nq ˆpm `nq antisymmetric matrix block-defined by

M λ,µ " ˆMλ N λ,µ ´Nµ,λ 0 ˙(7.2)
with pM λ q i,j " pQ pλi,λj q q and N λ,µ the m ˆn matrix pq λi´µj q. This should be viewed as an analogue of the Jacobi-Trudi formula for Schur's Q functions. We define the companion Schur's P -functions by Q λ{µ " 2 pλq´ pµq P λ{µ . We note that Q λ{µ " 0 unless µ Ă λ and likewise for P .

A strict specialization ρ is an algebra homomorphism from SSym to the complex numbers, and is completely determined by the generating function Qpρ; zq :"

ÿ ně0 q n pρqz n " exp ˜ÿ ně1, odd 2p n pρqz n n ¸. (7.3)
It is said to be nonnegative if Q λ{µ pρq ě 0 for all λ, µ. A necessary and sufficient condition for ρ to be nonnegative is [START_REF] Nazarov | Factor-representations of the infinite spin-symmetric group[END_REF][START_REF] Nazarov | Projective representations of the infinite symmetric group[END_REF] that its generating function is of the form

Qpρ; zq " e γz ź iě1 1 `αi z 1 ´αi z (7.4)
for a summable sequence of nonnegative real numbers γ, α 1 , α 2 , . . . . For two strict specializations ρ, ρ 1 , we define the specialization ρ Y ρ 1 by Qpρ Y ρ 1 ; zq :" Qpρ; zqQpρ 1 ; zq.

Recalling that SP denotes the set of strict partitions, we can finally define Qpρ; ρ 1 q by Qpρ; ρ 1 q :" ÿ λPSP Q λ pρqP λ pρ 1 q " ÿ λPSP 2 pλq P λ pρqP λ pρ 1 q.

(7.6) This is associated to the Cauchy identity for Schur's P and Q functions [Mac95, Section III.8], which amounts to Qpρ; ρ 1 q " exp ˜ÿ ně1, odd 2p n pρqp n pρ 1 q n ¸.

(7.7)

Definition of the process and the partition function. Fix N a nonnegative integer, u a nonnegative real number smaller than 1, and let ρ ȋ , 1 ď i ď N be 2N strict specializations such that z Þ Ñ Qpρ ȋ ; zq is analytic in a disk of radius R ą 1. The periodic strict Schur process is a measure on sequences of strict partitions of the form (1.1), which to a given sequence assigns the weight

W s p λ, µq :" u |µ p0q | N ź k"1
`Qλ pkq {µ pk´1q `ρk ˘Pλ pkq {µ pkq `ρḱ ˘˘.

(7.8)

The partition function Z s " Z s pu, ρ 1 , ρ 1 , . . . , ρ Ǹ , ρ Ń q is the sum of weights of all sequences.

Proposition 7.1. The partition function of the periodic strict Schur process reads

Z s " ź 1ďkď ďN Qpρ k ; ρ ´ q ź ně1
Qpu n ρ `; ρ ´qp1 `un q (7.9)

where ρ ˘:" ρ 1 Y ρ 2 Y ¨¨¨Y ρ N . (7.10)
Remark 7.2. For u " 0 we recover the partition function of the shifted Schur process [START_REF] Vuletić | The shifted Schur process and asymptotics of large random strict plane partitions[END_REF].

Correlation functions. The point configuration associated with a sample p λ, µq of the periodic strict Schur process is the set

S s p λq :" !´i , λ piq j ¯, 1 ď i ď N, 1 ď j ď pλ piq q ) Ă N ˆN˚. (7.11)
Note that, since the λ piq 's are strict, we need not shift the parts to obtain a simple point process and, as before, considering only the partitions λ p1q , . . . , λ pN q causes no loss of generality.

Fixing an auxiliary real parameter t, let c be a variable which can take the values 0 and 1, and define the shift-mixed periodic strict Schur process as the probability measure Probpc, λ, µq " 1 p1 `tqZ s ¨tc ¨Ws p λ, µq.

(7.12)

In other words, c is Bernoulli random variable of parameter t{p1 `tq independent of p λ, µq. The shiftmixed point configuration S s t p λq is the same as the point configuration, except viewed in the larger space t0, 1u ˆN ˆN˚.3 In this regard, t is a dummy parameter here more so than it was in the case of the shiftmixed periodic (non strict) Schur process, where the dependence of t and u was intertwined in the theta function θ 3 pt; uq. We nevertheless choose to keep it for keeping our exposition parallel to that of the case of the non strict process. We now state our main result.

Theorem 7.3. Let c P t0, 1u be a Bernoulli random variable of parameter t and U " tpi 1 , k 1 q, . . . , pi n , k n qu Ă t1, . . . , N u ˆN˚w ith i 1 ď ¨¨¨ď i n . The n-point shift-mixed correlation function ˜ s pU q :" Probpc ˆU Ă S s t p λqq is a pfaffian:

˜ s pU q " pf K (7.13)
where K is the 2n ˆ2n antisymmetric matrix given by

K γ,δ " $ ' & ' % 2 ´1 " z kγ w k δ ‰ F s pi γ , zqF s pi δ , wqκ s pz, wq, 1 ď γ ă δ ď n, 2 ´1 "
z kγ w ´k2n`1´δ ‰ p´1q k 2n`1´δ F s pi γ , zqF s pi δ , wqκ s pz, wq, 1 ď γ ď n ă δ ď 2n, 2 ´1 " z ´k2n`1´γ w ´k2n`1´δ ‰ p´1q k2n`1´γ `k2n`1´δ F s pi γ , zqF s pi δ , wqκ s pz, wq, n ă γ ă δ ď 2n (7.14) and

F s pi, zq " ś 1ď ďi Qpρ ` ; zq ś iď ďN Qpρ ´ ; z ´1q ź ně1 ź 1ď ďN Qpu n ρ ` ; zq Qpu n ρ ´ ; z ´1q , κ s pz, wq " θ u `w z θu `´w z ˘¨p1 `tqpu; uq 2 8 p´u; uq ´1 8 , u 1{2 ă |w| ă |z| ă u ´1{2 .
(7.15)

Remark 7.4. If u " 0 (in which case one can take t " 0) we recover the pfaffian correlations of the shifted Schur process [START_REF] Vuletić | The shifted Schur process and asymptotics of large random strict plane partitions[END_REF]. See also [START_REF] Matsumoto | Correlation functions of the shifted Schur measure[END_REF] for the case of the shifted Schur measure.

Remark 7.5. The coefficient extractions in the kernel K can be replaced by double contour integrals similar to those in Theorem 3.1.

Proof of Theorem 7.3. We again use the formalism of free fermions, but this time in their "neutral" flavor. See Appendix C for our conventions and notations. Let us first sketch the proof of Proposition 7.1. We write Z s " tr 0 `uHs Γ s `pρ 1 qΓ s ´pρ 1 q . . . Γ s `pρ Ǹ qΓ s ´pρ Ń q ˘(7.16) where tr 0 stands for trace over the subspace N F 0 of even grading. We then proceed as in Section 4 to eliminate the Γ-operators, with the following two minor modifications. First, commuting the Γ s operators will yield Qpu n ρ ì ; ρ j q factors instead of the H factors. Second, at the end one is left with tr 0 pu Hs q " ś iě1 p1`u i q. For the shift-mixed cylindric strict Schur process, the partition function is computed similarly except one traces over all of N F " N F 0 ' N F 1 , the result being p1 `tqZ s .

We now turn to the proof of Theorem 7.3 itself, which is very similar to that of Theorem 3.1. First we notice that ˜ s pU q " 1 p1 `tqZ s tr ˆuHs t Cs Γ s `pρ 1 qΓ s ´pρ 1 q ¨¨¨Γ s `pρ ì1 q ˆ1 2 φ k1 φ k1 ˙Γs ´pρ í1 q ¨¨Γ s `pρ ìn q ˆ1 2 φ kn φ kn ˙Γs ´pρ ín q ¨¨¨Γ s `pρ Ǹ qΓ s ´pρ Ń q ˙(7.17)

where the operators 1 2 φ kγ φ kγ measure whether λ piγ q has a part of size k γ and we recall that φ k " p´1q k φ ´k for k ą 0. By commuting out the Γ-operators, we can rewrite the correlation function as ˜ s pU q " 2 ´n tr `uHs t Cs Φ k1 pi 1 qΦ k1 pi 1 q ¨¨¨Φ kn pi n qΦ kn pi n q trpu Hs t Cs q .

(7.18) with Φ k piq :" Ad `Γs

`pρ Ñ i qΓ s ´pρ Ð i q ´1˘¨φ k , Φ k piq :" Ad `Γs `pρ Ñ i qΓ s ´pρ Ð i q ´1˘¨φ k (7.19)
where Ad denotes the Lie group adjoint action AdpAq ¨B :" ABA ´1 and where

ρ Ñ i :" i ď "1 ρ ì Y ď ně1 u n ρ `, ρ Ð i :" N ď "i ρ í Y ď ně1 u n ρ ´. (7.20)
Each Φ k piq and Φ k piq is a linear combination of the φ k 's (recall that φ k " p´1q k φ ´k is just a relabelling of φ k added for convenience), a statement which follows easily if one passes to the generating field φpzq and uses the commutation between Γ s ˘and φpzq given in (C. 18). By Wick's lemma C.1, the left-hand side of equation (7.18) is a 2n ˆ2n pfaffian of a certain antisymmetric matrix K whose entries K γ,δ for γ ă δ are

$ ' ' & ' ' % 2 ´1 tr `uHs t Cs Φ kγ pi γ qΦ k δ pi δ q ˘{ trpu Hs t Cs q, 1 ď γ ă δ ď n,
2 ´1 tr ´uHs t Cs Φ kγ pi γ qΦ k2n`1´δ pi 2n`1´δ q ¯{ trpu Hs t Cs q, 1 ď γ ď n ă δ ď 2n, 2 ´1 tr ´uHs t Cs Φ k2n`1´δ pi 2n`1´δ qΦ k2n`1´γ pi 2n`1´γ q ¯{ trpu Hs t Cs q, n ă γ ă δ ď 2n.

(7.21)

Finally one checks that, after passing to the generating field φpzq via coefficient extraction, one has tr `uHs t Cs Φ kγ pi γ qΦ k δ pi δ q ˘" rz kγ w k δ s tr `uHs t Cs Γ s `pρ 1 q ¨¨¨φpzq ¨¨¨φpwq ¨¨¨Γ s ´pρ Ń q ˘, 1 ď γ ă δ ď n, tr ´uHs t Cs Φ kγ pi γ qΦ k2n`1´δ pi 2n`1´δ q ¯" p´1q k 2n`1´δ rz kγ w ´k2n`1´δ s tr `uHs t Cs Γ s `pρ 1 q ¨¨¨φpzq ¨¨¨φpwq ¨¨¨Γ s ´pρ Ń q ˘, 1 ď γ ď n ă δ ď 2n, tr ´uHs t Cs Φ k2n`1´δ pi 2n`1´δ qΦ k2n`1´γ pi 2n`1´γ q ¯" p´1q k2n`1´γ `k2n`1´δ rz ´k2n`1´γ w ´k2n`1´δ s tr `uHs t Cs Γ s `pρ 1 q ¨¨¨φpzq ¨¨¨φpwq ¨¨¨Γ s `pρ Ǹ q ˘, n ă γ ă δ ď 2n.

(7.22)

Note that the first equation above corresponds to φφ correlators, the second to φφ ˚while the third to φ ˚φc orrelators. This explains the presence of minus signs in the second and third equations as φ k " p´1q k φ k . The kernel has been arranged in this way so as to make the u " t " 0 limit correspond to the kernels of Matsumoto and Vuletić [START_REF] Matsumoto | Correlation functions of the shifted Schur measure[END_REF][START_REF] Vuletić | The shifted Schur process and asymptotics of large random strict plane partitions[END_REF]. Finally, each of the above quantities inside a coefficient extraction can be computed in exact form by Γ-elimination. At the end one is left with trpu Hs t Cs φpzqφpwqq which is given by Proposition C.3. Putting it all together, one arrives at the stated values for the entries of the correlation kernel. This concludes the proof of Theorem 7.3. We can also derive a strict analogue of Proposition 3.2. Recall that xOy :" trpu Hs t Cs Oq{ trpu Hs t Cs q. We write the n-point correlation function as follows:

˜ s pU q " 2 ´n « n ź "1 z k p´1q k w ´k ff tr `uHs t Cs Γ s `pρ 1 qΓ s ´pρ 1 q ¨¨¨Γ s `pρ ì1 qφpz 1 qφpw 1 qΓ s ´pρ í1 q ¨¨Γ s `pρ
ìn qφpz n qφpw n qΓ s ´pρ ín q ¨¨¨Γ s `pρ Ǹ qΓ s ´pρ Ń q ˘{ tr `uHs t Cs "

2 ´n « n ź "1 z k p´1q k w ´k ff n ź "1
pF pi , z qF pi , w qq ¨xφpz 1 qφpw 1 q ¨¨¨φpz n qφpw n qy (7.23)

where the second equality follows from the first after cyclically moving the Γ s operators out of the way and picking up the corresponding F factors and where brackets stand for coefficient extraction. Using the pfaffian evaluation from Proposition C.4 and replacing coefficient extraction with contour integration we arrive at the following result.

Proposition 7.6. The n-point correlation has the form where the conditions on the contours are the same as in Proposition 3.2.

˜ s pU q " p1 `

Conclusion

We conclude with a few remarks about how this work fits in the bigger picture. Let us first mention that this paper is part of a project to analyze Schur processes with "nonstandard" boundary conditions. In another paper written in collaboration with Peter Nejjar and Mirjana Vuletić, we introduced the Schur process with free boundaries [START_REF] Betea | The free boundary Schur process and applications I[END_REF]. For the edge behavior of this process, we expect to encounter pfaffian variants of the finite-temperature Airy process [START_REF] Betea | The free boundary Schur process and applications II[END_REF]. Note that this is a priori unrelated to the pfaffian process considered in Section 7, which we discuss below. As said in Remark 5.8, the finite-temperature Airy kernel should be the universal scaling limit of the periodic Schur process around a generic point at the edge of the limit shape. It is however known that, in the zero-temperature case, there exist other scaling limits (e.g., around nongeneric points such as cusps) and other specializations (e.g., those of z-measures) which lead to interesting objects like the Pearcey kernel [START_REF] Okounkov | Random skew plane partitions and the Pearcey process[END_REF], the cusp Airy kernel [START_REF] Okounkov | The birth of a random matrix[END_REF], the hypergeometric kernel [START_REF] Borodin | Distributions on partitions, point processes, and the hypergeometric kernel[END_REF], etc. It is natural to ask whether these kernels have finite-temperature variants.

We have defined in Section 6 the stationary cylindric Plancherel process through its finite-dimensional marginals, but we believe that it admits alternative nice combinatorial/probabilistic constructions via variants of the Robinson-Schensted correspondence (as used in [START_REF] Borodin | Stochastic dynamics related to Plancherel measure on partitions[END_REF] for the zero-temperature limit), Hammersley's process [START_REF] Aldous | Hammersley's interacting particle process and longest increasing subsequences[END_REF] or the polynuclear growth (PNG) model [START_REF] Prähofer | Scale invariance of the PNG droplet and the Airy process[END_REF]. We might return to this subject in another publication.

On a related aspect, it would be nice to devise a sampling algorithm-for the stationary cylindric Plancherel process or for the periodic Schur process in general-as was done in [START_REF] Borodin | Schur dynamics of the Schur processes[END_REF]BBB `18] for the zero-temperature case. In fact, we believe that the approach of [BBB `18] may be adapted straightforwardly (but we have not implemented it yet), building on ideas present in [START_REF] Langer | Enumeration of cylindric plane partitions -part II[END_REF][START_REF] Bouttier | From Aztec diamonds to pyramids: steep tilings[END_REF].

Our characterization in Section 7 of the correlation functions of the periodic strict Schur process raises the question whether they admit new (bulk or edge) scaling limits. For the edge, we expect to still get generically the finite-temperature Airy process (pfaffians reducing to determinants in the limit). Bulk limits might be more interesting and involve pfaffian variants of the kernels obtained in [START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF]. See [START_REF] Vuletić | The shifted Schur process and asymptotics of large random strict plane partitions[END_REF] for the zero-temperature case.

As said in the introduction, M α has first appeared in edge asymptotics of the Moshe-Neuberger-Shapiro (MNS) matrix model [START_REF] Johansson | From Gumbel to Tracy-Widom[END_REF], 1D free fermions in a harmonic trap [DLDMS15, DLDMS16, LW17] and nonintersecting Ornstein-Uhlenbeck processes on a cylinder [START_REF] Le Doussal | Periodic Airy process and equilibrium dynamics of edge fermions in a trap[END_REF]. All these models are different facets of the same continuous space-time finite-temperature free-fermionic model. Let us point out that the free fermions used here are slightly different from those considered in the above references. They correspond essentially to an effective theory when the number of fermions gets large.

Perhaps coincidentally, up to reparametrization, M α has also appeared in non-free-fermionic models and at the level of the KPZ equation itself, as can be seen in the works of Amir-Corwin-Quastel and Sasamoto-Spohn (see [START_REF] Amir | Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions[END_REF][START_REF] Sasamoto | The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class[END_REF], and references therein) where it appears in the scaling of the weakly asymmetric exclusion process, and in works of Calabrese-Le Doussal-Rosso, Borodin-Corwin-Ferrari and Imamura-Sasamoto [CDR10, BCF14, IS17] regarding fluctuations of the free energy of the O'Connell-Yor directed polymer. This coincidence was noticed in [START_REF] Dean | Finite-temperature free fermions and the Kardar-Parisi-Zhang equation at finite time[END_REF] and further discussed in [DLDMS16, LW17, CG18] but, to the best of our knowledge, its fundamental origin is not understood.

Finally, we note that the approach of Baik, Deift and Johansson [START_REF] Baik | On the distribution of the length of the longest increasing subsequence of random permutations[END_REF] for the edge behavior of Plancherel random partitions was based on Riemann-Hilbert techniques, while the approach that we follow here is much closer in spirit to that developed by Borodin, Okounkov and Olshanski [START_REF] Borodin | Asymptotics of Plancherel measures for symmetric groups[END_REF]. It seems an interesting technical challenge to adapt the Riemann-Hilbert approach to analyze the Fredholm determinant F α , even though its asymptotics may be studied through other techniques-see [START_REF] Corwin | Lower tail of the KPZ equation[END_REF] and references therein. We might return to this question in the future.

A Eta and theta functions

Fix q a complex parameter of modulus less than 1. The q-Pochhammer symbol of argument z and length n P N Y t8u, the multiplicative theta function and the Dedekind eta function are respectively defined as pz; qq n :" n ź k"0 p1 ´zq k q, θ q pzq :" pz; qq 8 pq{z; qq 8 , ηpqq :" q 1 24 pq; qq 8 . (A.1)

We also introduce the ("additive") Jacobi theta functions θ 3 and θ 1 :

θ 3 pz; qq :" ÿ nPZ q n 2 2 z n , θ 1 pz; qq :" 1 i ÿ nPZ p´1q n q pn`1{2q 2 2 z n`1 2 " q 1 8 z 1 2 i θ 3 p´q 1{2 z; qq. (A.2)
These conventions differ by the change q Ñ q 1{2 from those in [START_REF] Erdélyi | Higher transcendental functions[END_REF]p.355] and that for θ 1 is also slightly different from [Bor07, p.411] (the i factor makes θ 1 pz; qq real for q real and |z| " 1). The Jacobi triple product identity can then be written in the two equivalent forms θ 3 pz; qq " pq; qq 8 θ q p´q 1{2 zq, θ 1 pz; qq " iq 1 8 z ´1 2 pq; qq 8 θ q pzq (A.3)

where in the last equation we have used the relation θ q pqzq " θ q p1{zq " ´p1{zqθ q pzq.

B Basics of charged free fermions

We recall the definitions necessary for Section 4-see also [BBNV18, Section 3] and references therein. The fermionic Fock space, denoted F, may be seen as the infinite dimensional Hilbert space with orthonormal basis indexed by Maya diagrams, as defined in Section 2. We use the bra-ket notation throughout, hence denote respectively by xn| and |ny the bra and ket associated with the Maya diagram n. The charge C and energy H, as defined in (2.1), are naturally promoted as diagonal operators on F. We also defined the shift operator R, that shifts a Maya diagram one unit to the right.

Creation/annihilation operators. For k P Z 1 , the fermionic creation and annihilation operators ψ k and ψ k are defined by

ψ k |ny :" # 0 if n k " 1, p´1q ř jąk nj |n pkq y, if n k " 0, ψ k |ny :" # p´1q ř jąk nj |n pkq y, if n k " 1, 0 if n k " 0, (B.1)
where n pkq denotes the Maya diagram obtained from n by inverting the value found at position k. Because of the signs, we have the canonical anticommutation relations

tψ k , ψ ˚ u " δ k, , tψ k , ψ u " tψ k , ψ ˚ u " 0, k, P Z 1 (B.2)
where ta, bu :" ab `ba denotes the anticommutator. We also define the generating series ψpzq :"

ÿ kPZ 1 ψ k z k , ψ ˚pwq :" ÿ kPZ 1 ψ k w ´k. (B.3)
Note that the charge and energy operators can be rewritten as bilinears in the creation/annihilation operators, namely

C " ÿ kPZ 1 : ψ k ψ k :, H " ÿ kPZ 1 k : ψ k ψ k :, : ψ k ψ k : :" # ψ k ψ k , if k ą 0, ´ψk ψ k , if k ă 0. (B.4)
Here it is convenient to introduce the normal ordering : ¨: with respect to the vacuum so as to have welldefined single sums. Such bilinear quantities are essential to free field theory. A manifestation of this is Wick's lemma at finite temperature given below. where pν, cq and pµ, c 1 q are the charged partitions associated with the Maya diagrams n and m respectively, via the correspondence of Section 2. The Γ-operators commute with the charge (C) and shift (R) operators and satisfy the following quasi-commutation relations Γ ˘pρqψpzq " Hpρ; z ˘1qψpzqΓ ˘pρq, Γ ˘pρqψ ˚pwq " Hpρ; w ˘1q ´1ψ ˚pwqΓ ˘pρq, Γ `pρqΓ ´pρ 1 q " Hpρ; ρ 1 qΓ ´pρ 1 qΓ `pρq, Hpρ; ρ 1 q :" exp ˜ÿ ně1 p n pρqp n pρ 1 q n ¸,

Γ ˘pρqu H " u H Γ ˘pu ˘1ρq, Hpu ˘1ρ; zq :" Hpρ; u ˘1zq.

(B.8)

We may reconstruct the creation/annihilation operators from the Γ-operators as

ψpzq " z C´1 2 R Γ ´pzqΓ 1 ``´z ´1˘, ψ ˚pwq " R ´1w ´C`1 2 Γ 1 ´p´wqΓ ``w ´1˘, (B.9)
a result often referred to as the boson-fermion correspondence.

Wick's lemma. For u P p0, 1q and t ą 0, we denote by x¨y u,t the grand canonical expectation value defined by xOy u,t :" trpt C u H Oq{ trpt C u H q for any operator O acting on Fock space. Then we have the following "finite-temperature" version of Wick's lemma.

Lemma B.1. Let Ψ be the vector space spanned by (possibly infinite) linear combinations of the ψ k and ψ k , k P Z 1 . For ϕ 1 , . . . , ϕ 2n P Ψ , we have xϕ 1 ¨¨¨ϕ 2n y u,t " pf A (B.10)

where A is the 2n ˆ2n antisymmetric matrix defined by A i,j " xϕ i ϕ j y u,t for i ă j.

In particular, if ϕ 2i´1 (respectively ϕ 2i ) is a linear combination of the ψ k 's only (respectively the ψ k 's only) for all i " 1, . . . , n, we have

xϕ 1 ¨¨¨ϕ 2n y u,t " det 1ďi,jďn xT pϕ 2i´1 , ϕ 2j qy u,t (B. 11
)
where T is the "time-ordered product": T pϕ 2i´1 , ϕ 2j q " ϕ 2i´1 ϕ 2j for i ď j and T pϕ 2i´1 , ϕ 2j q " ´ϕ2j ϕ 2i´1 for i ą j.

The more usual Wick's lemma at zero temperature corresponds to the case u " 0, for which x¨y u,t reduces to the vacuum expectation value xH| ¨|Hy. For convenience we provide a proof of Lemma B.1, which seems basically due to Gaudin [START_REF] Gaudin | Une démonstration simplifée du théorème de Wick en mécanique statistique[END_REF].

Proof. Let us introduce the density matrix D :" t C u H { trpt C u H q so that xOy u,t " trpDOq for any operator O. By direct computation we have

Dψ k " tu k ψ k D, Dψ k " ptu k q ´1ψ k D. (B.12)
By multilinearity, it suffices to prove (B.10) when each ϕ i is equal to either ψ ki or ψ ki for some k i . The left hand side of (B.10) can be telescopically rewritten as:

trpDϕ 1 ¨¨¨ϕ 2n q " 2n ÿ i"2 p´1q i trpDϕ 2 ¨¨¨ϕ i´1 tϕ 1 , ϕ i uϕ i`1 ¨¨¨ϕ 2n q ´trpDϕ 2 ϕ 3 ¨¨¨ϕ 2n ϕ 1 q " 2n ÿ i"2 p´1q i tϕ 1 , ϕ i u trpDϕ 2 ¨¨¨ϕ i´1 ϕ i`1 ¨¨¨ϕ 2n q ´c1 trpDϕ 1 ϕ 2 ϕ 3 ¨¨¨ϕ 2n q (B.13)
where c 1 " ptu k1 q ´1 (respectively " tu k1 ) if ϕ 1 " ψ k1 (respectively " ψ k1 )-to pass to the second line we use the fact that, by the canonical anticommutation relations, tϕ 1 , ϕ i u is a scalar, and we use cyclicity and (B.12) to rewrite the rightmost trace. In particular, for n " 1, (B.13) yields tϕ 1 , ϕ 2 u " p1 `c1 qxϕ 1 ϕ 2 y u,t (B.14) which of course still holds when replacing ϕ 2 by ϕ i for any i. We deduce the recursion relation

xϕ 1 ¨¨¨ϕ 2n y u,t " 2n ÿ i"2 p´1q i xϕ 1 ϕ i y u,t xϕ 2 ¨¨¨ϕ i´1 ϕ i`1 ¨¨¨ϕ 2n y u,t . (B.15)
The proof of (B.10) is then done by induction: it is a tautology for n " 1, and assuming that it holds at rank n ´1, we have xϕ 2 ¨¨¨ϕ i´1 ϕ i`1 ¨¨¨ϕ 2n y u,t " pf A p1,iq where A p1,iq is the p2n ´2q ˆp2n ´2q submatrix of A with the first and i-th rows and columns removed. We recognize in the right-hand side of (B.15) the expansion of the pfaffian of A with respect to the first row/column, and conclude that (B.10) holds at rank n.

In the case where ϕ 2i´1 (respectively ϕ 2i ) is a linear combination of the ψ k 's only (respectively the ψ k 's only) for all i " 1, . . . , n, A i,j vanishes whenever i and j have the same parity, hence pf A " det 1ďi,jďn A 2i´1,2j which is equivalent to the stated form (B.11).

Remark B.2. The main ingredients of the proof are the facts that (i) the anti-commutator of any two elements of Ψ is a scalar, and (ii) the density matrix D is the exponential of a bilinear combination of elements of Ψ. Here we assume that D has a diagonal form, which simplifies the proof but is not necessary-see e.g. [START_REF] Blaizot | Quantum theory of finite systems[END_REF]Ch. 4 Given two strict specializations ρ, ρ 1 , as p n pρ Y ρ 1 q " p n pρq `pn pρ 1 q, we have

Γ s `pρqΓ s `pρ 1 q " Γ s `pρ Y ρ 1 q " Γ s `pρ 1 qΓ s `pρq.
(C.16)

The commutation relations (C.13) and the Cauchy identity (7.7) imply that Γ s `pρqΓ s ´pρ 1 q " Qpρ; ρ 1 qΓ s ´pρ 1 qΓ s `pρq (C.17) while Γ s ˘pρqφpzq " Qpρ; z ˘1qφpzqΓ s ˘pρq.

(C.18)

These latter relations always make sense at a formal level; at an analytic level they require that the parameter of Qpρ; ¨q be within its disk of convergence. The crucial property of these half-vertex operators is that skew Schur's P and Q functions arise as their matrix elements (see [START_REF] Matsumoto | Correlation functions of the shifted Schur measure[END_REF] or [Whe11, Sections 1.4.9 and 3. The entries of the pfaffian are given by Proposition C.3, and the left-hand side can also be written as a product of theta functions using the boson-fermion correspondence. The result follows upon cancelling all diagonal terms appearing on both sides.

Remark C.5. This pfaffian, while simple to write down, has appeared only recently in the (mathematical) literature in the works [Ros07, Remark 2.9] and [Ros08, Lemma 3.1] by Rosengren. Remarkably, using an algebraic geometric-type no-go argument, Rains (personal communication with the first author) has proved it was the most general pfaffian evaluation of the form pf iăj A i,j " ś iăj A i,j . Taking u Ñ 0 leads to a famous pfaffian evaluation of Schur (see, e.g., [START_REF] Macdonald | Symmetric functions and Hall polynomials. Oxford Mathematical Monographs[END_REF]

D On the discrete finite-temperature Bessel kernel

We provide some further analysis of the discrete finite-temperature Bessel kernel of Section 5. Our approach relies on the contour integral representation (5.2) where we set u " e ´r and t " 1.

A convenient change of variables. Set which is maximal at pφ, pζqq " p0, 0q, a property which is essential for doing saddlepoint approximations. where I D is the modified Bessel function of the first kind and order D.

The bulk limit revisited. We now give a short rederivation of Borodin's bulk limiting kernel for the cylindric Plancherel measure. where we use standard tails pruning/completion arguments to justify passing to the second line. By integrating over ν we obtain wanted expression (D.5).

We believe that this proof may be adapted-keeping the same change of variables pz, wq Ñ pφ, νq-to give a streamlined derivation of [Bor07, Theorem 3.1] regarding the bulk limiting kernels for more general periodic Schur processes.

Remark D.4. If we let γ Ñ 8 with τ {γ " x fixed in the right-hand side of (D.5), we recover the discrete sine kernel as for the (noncylindric) poissonized Plancherel measure-see the remark at the end of [START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF]Example 3.4]. In all rigor, Proposition D.3 does not apply to the situation where we let u Ñ 1 ´and γ Ñ 8 jointly as in Theorem 1.1. But it is not difficult to adapt to the finite-temperature setting the proof given in [Oko02, Section 3.2] for the convergence of the discrete Bessel kernel to the sine kernel, the key fact being that κpz, wq has a residue 1 at z " w-as mentioned in Section 4 this is a general consequence of the fermionic canonical anticommutations.

Proof of Proposition 5.2. The difficulty to circumvent is that, in the contour integral representation (5.2) of Kpa, bq, the exponential factor e Lpz´z ´1´w`w ´1q is now subdominant. Instead, we may see that the asymptotics is governed by the pole of κ at z{w " u ´1. The strategy is to deform the integration contours into two circles such that |z{w| is larger than u ´1 (but smaller than u ´2). As we cross the pole, we pick a contribution from the residue which turns out to be dominant.

More precisely, let us first establish the pointwise convergence (5.13). We use the reparametrization (D.2) and change the integration contour of ζ into the segment 3r 2 `ir´π, πs plus a small negatively oriented circle around the pole at r. Noting that the residue of κ at this pole is ´1 and using the integral representation where G :" 4L sinh r 2 . We claim that, in the edge high temperature regime (5.12), the first term T 1 in (D.8) tends to e ´xδ x,y (the Kronecker delta) while the second term T 2 tends to 0. Note that G " 2Lr `op1q " 2γ may or may not remain finite, and the forthcoming discussion is valid in both cases.

The fact that T 1 Ñ e ´x for x " y (i.e. b ´a " 0) is a straightforward consequence of the scaling (5.12). Indeed we have We now observe that f 1 pxq " arsinhpxq so f is convex and minimal at x " 0, with f pxq ě ´1 `x2 2 . Hence e ´Gf p D G q " O ´eG´D 2 2G ¯" O ˆeG´p y´xq 2 4Lr 3 ˙" ope G q, which establishes (D.10) hence that T 1 Ñ 0 for x ‰ y.

We now show that T 2 Ñ 0: using (D.3) and proceeding as for Lemma 5.5, we see that |κpζq| may be bounded over the ζ-integration contour by where we use again (D.9) and the fact that I 0 pGq " Θ ´eG ? G ¯. Upon integrating over φ and ζ, and multiplying by the prefactor r ´1, we deduce that T 2 " O `pr ? Gq 1{2 ˘Ñ 0, which completes the proof of the pointwise convergence (5.13).

The proof of the trace convergence (5.14) is entirely similar: starting from (D.8), we find that and all the above analysis of (D.8) may be adapted straightforwardly.

  , P. Biane, A. Borodin, S. Corteel, P. Di Francesco, P. Ferrari, T. Imamura, L. Hodgkinson, K. Johansson, C. Krattenthaler, G. Lambert, P. Le Doussal, S. Majumdar, G. Miermont, M. Mucciconi, P. Nejjar, E. Rains, N. Reshetikhin, T. Sasamoto, G. Schehr, M. Schlosser, M. Vuletić and M. Wheeler.

a

  5.10) and ÿ iěm Kpi, iq Ñ ż 8 s M α ps 1 , s 1 qds 1 .(5.11)Note that the right-hand side of (5.11) is finite by [Joh07, Proposition 1.1]. For good measure, we will give below two proofs of Proposition 5.1. The first proof uses the Bessel representation, and is an adaptation of the zero-temperature proofs from [BOO00, Joh01]-see Romik's book[START_REF] Romik | The Surprising Mathematics of Longest Increasing Subsequences[END_REF] for a pedagogical exposition. The second proof is based on the contour integral representation. Let us now give the analogous statement, implying Theorem 1.2, for the high temperature regime. Proposition 5.2. For t " 1 and in the edge high temperature regime " tM `xr ´1u b " tM `yr ´1u m " tM `sr ´1u , M :" r ´1 ln I 0 p2Lrq r , (5.12) we have r ´1K pa, bq Ñ # e ´x if x " y, 0 otherwise, (5.13) and ÿ iěm Kpi, iq Ñ e ´s.

  exponentially small error term. Finally, z a 1 w ´b1 Ñ e xζ´yω , which allows to conclude by dominated convergence that the L 1{3 of (5.27) and the 1{r of (5.32) are absorbed by the Jacobian of the change of variables z Ñ ζ, w Ñ ω).It is straightforward to identify the right-hand side of (5.33) as M α px, yq, using the integral representations

Kpb, z; b 1

 1 , wq " Z ´1 β,t e ϑpz´z ´1q e ϑpw´w ´1q ˆ$ & % tr ´tC e ´bH ψpzqe ´pb 1 ´bqH ψ ˚pwqe ´pβ´b 1 qH ¯if b ď b 1 , ´tr ´tC e ´b1 H ψ ˚pwqe ´pb´b 1 qH ψpzqe ´pβ´bqH ¯if b ą b 1 . (6.11)

  Section III.8]) pf iăj xi´xj xi`xj " ś iăj xi´xj xi`xj .

  , w " e ´ζ 2 `iφ (D.1) so that z{w " e ζ and ?zw " e iφ , where φ may be integrated over the interval r´π, πs and ζ over ζ 0 `ir´π, πs with ζ 0 P p0, rq. We get Kpa, bq " rm . Note that the modulus of the exponential factor e 4L cos φ sinh ζ 2 is equal to e 4L cos φ sinh ζ 0 2 cos pζq 2

Remark D. 1 .

 1 Applying the Poisson summation formula as in Lemma 5.5, we get κpζq " is manifest that κ is a meromorphic function of ζ in C, having a simple pole at each point of the lattice rZ `2iπZ, with residue p´1q n` at ζ " nr `2iπ . If we assume that argpζq P r´π, πs, then keeping only the term " 0 in (D.3) yields an approximation of κpζq with uniform error Ope ´π2 {r {rq.Remark D.2. By integrating over φ in (D.2), we get Kpa, bq "

  Proposition D.3 (see[START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF] Example 3.4]). Consider the limit u " e ´r Ñ 1 ´with Lr Ñ γ fixed, and assume that ra, rb Ñ τ P R with D :" b ´a fixed. ThenKpa, bq Ñ 1 2π ż π ´π e iDφ dφ 1 `eτ´2γ cos φ (D.5)Proof. We use the integral representation (D.2) for Kpa, bq, and perform the change of variable ζ " r 2 `irν with ν P r´π{r, π{rs. Then, using Lemma 5.5 and simple Taylor expansions, we get Kpa, bq " ż π ´π e iDφ dφ 4π 2 ż π{r ´π{r dν e pγ cos φ´τ {2qp1`2iνq`op1q ´π cosh πν `

I

  D pGq " 1 2π ż π ´π dφe iDφ`G cos φ (D.7)for the modified Bessel function of the first kind and integer order D, we find that r ´1K pa, bq "

  safe to replace the argument of I 0 even if G Ñ 8 since I 0 pGq " e G ? 2πG . For x ‰ y, the order D :" b ´a of the Bessel function now tends to ˘8 as py ´xqr ´1, and showing that T 1 Ñ 0 amounts to showing thatI D pGq " opI 0 pGqq " o ˆeG ? G ˙(D.10)which is done by the saddle-point method: in the integral representation (D.7), let us move the integration path to the segment r´π, πs `i arsinhp D G q. On this segment we may bound the integrand as|e iDφ e G cos φ | " e f pxq :" x arsinhpxq ´?1 `x2 . By integrating over φ we get I D pGq " O `G´1{2 exp `´Gf `D G ˘˘˘.

  e 4L cos φ sinh ζ 2 ¸(D.13)

  for the bulk behavior. It is elementary to check that

	ProbpM ă kq "	1 p´tu k ; uq 8	,	M :" maxti : n i " 1u.	(2.6)
	Recognizing a q-exponential function, and using the asymptotics pp1 ´uqz; uq 8 Ñ e ´z [GR04, p.11], we
	easily deduce that
	lim uÑ1 ´Prob ˆM ă	´lnp1 ´uq ln u	`ξ | ln u|	˙" e ´te ´ξ .

  , β 1 , α 2 , β 2 , . . . a summable collection of nonnegative parameters (the specialization ρ is then called nonnegative). In particular, when only γ is nonzero, we obtain the exponential specialization denoted ex γ , for which

	1ďi,jď pλq	h λi´i´µj `j pρq.	(3.1)
	It vanishes unless λ{µ is a skew shape (i.e. we have λ Ą µ, i.e. λ i ě µ i for all i). The "specialization" ρ is
	conveniently encoded into the generating function
	Hpρ; zq :"	ÿ	h n pρqz n .	(3.2)
		ně0
	Thoma's theorem [Tho64, AESW51] states that s λ{µ pρq is nonnegative for all λ, µ if and only if the generating
	function Hpρ; zq is of the form
	Hpρ; zq " e γz	ź iě1	1 `βi z 1 ´αi z	(3.3)
	with γ, α 1

  .8) Note that, at an analytic level, the first sum converges for |uz| ă |w| ă |z| and the second for |uw| ă |z| ă |w|, and in fact we have (by the canonical anticommutation relations, it is a general fact that fermionic propagators xψpzqψ ˚pwqy and ´xψ ˚pwqψpzqy should be two Laurent series expansions of a same meromorphic function with a pole of residue 1 at z " w). We conclude the proof of Theorem 3.1 by representing the coefficient extraction rz k w ´k1 s in (4.6) as a double contour integral. Note that, by our analyticity assumptions, F pi, ¨q is analytic and nonzero in the annulus R ´1 ă | ¨| ă R.Theta form for fermionic expectations. Interestingly, it is possible to derive the "theta" form (3.14) of κpz, wq from the boson-fermion correspondence (B.9). Assuming |z| ą |w| we may write tr `tC u H ψpzqψ ˚pwq

	κpz, wq "	#	xψpzqψ ˚pwqy u,t ´xψ ˚pwqψpzqy u,t for |z{w| P pu, 1q for |z{w| P p1, u ´1q,	(4.9)
	˘" c	w z	¨tr	˜ˆtz w	˙C u H Γ ´pzqΓ 1 ``´z	´1˘Γ 1 ´p´wqΓ ``w	´1˘"
	c	w z	¨pu; uq 2 8 θ u p w z q	¨tr	˜ˆtz w	˙C u H	¸" c	w z	¨pu; uq 2 8 θ u p w z q	w ; uq pu; uq 8 ¨θ3 p tz	(4.10)

  This shows that |λ|{Λ u,γ tends to 1 in probability. Note that Λ u,γ " L 2 for γ Ñ 8, hence L is the natural length scale for the Young diagram of λ and for the point process Spλq. For u Ñ 1 ´and fixed γ ą 0, L remains a good length scale but it is easier to work with r ´1, where u " e ´r .

									γ 2 1´u {pu; uq 8 of the partition
	function of the cylindric Plancherel measure (1.3), it is straightforward to check that
	E|λ| "	γ 2 p1 ´uq 2	´u d du	lnpu; uq 8 ,	Var|λ| "	γ 2 p1 `uq p1 ´uq 3 ´ˆu	d du	˙2 lnpu; uq 8 .	(5.5)
	From the asymptotics lnpu; uq 8 "	´π2 6p1´uq for u Ñ 1 ´, we see that
			E|λ| "	γ 2 `π2 6 p1 ´uq 2 ": Λ u,γ ,	Var|λ| ! pE|λ|q 2		(5.6)
	whenever u Ñ 1 ´or γ Ñ 8. Borodin has shown the following "bulk" asymptotics for the one-point function [Bor07, Example 3.4]:
	lim u"e ´r , rÑ0

γ fixed, t"1

  5.3. An extension of this argument shows that the joint law of the k first parts of λ converges to that of the k righmost particles in the finite-temperature Airy process. See [BOO00, Section 4.2] for the zero-temperature case.Proof that Proposition 5.2 implies Theorem 1.2. The argument of the previous proof applies mutatismutandis. Moreover, we are now able to explicitly evaluate the limiting Fredholm determinant: detpI ´Kq 2 ptm,m`1,... uq Ñ

	8 ÿ n"0	p´1q n n!	e ´ns " e

´e´s

(5.

18)

  tq n pu; uq 2n

	8 8 2 n p´u; uq n	¿	¨¨¨¿	n ź "1	¨dz dw F pi , z qF pi , w q p2πiq 2 p´1q k z k `1w ´k `1	´z w w ´´z ¨θu	θu ¯' ź
						θ u	´z zm ¯θu	´w wm ¯θu	´z wm ¯θu	zm ´w	θu
					1ď ămďn	´´z zm ¯θu	´´w wm ¯θu	´´z wm ¯θu	zm ´´w	¯(7.24)

  Γ-operators. For ρ a specialization with generating function Hpρ, ¨q as defined in (3.2), we consider the sequence pp n pρqq ně1 of power sums specialized at ρ given by If x is a variable, we denote by Γ ˘pxq (respectively Γ 1 ˘pxq) the half-vertex operators for the specialization in the single variable x (respectively its dual x), for which p n pxq " x n (respectively p n pxq " p´1q n´1 x n ). It is well-known that xn|Γ `pρq|my " xm|Γ ´pρq|ny "

		ÿ ně1	p n pρq n	z n :" ln Hpρ; zq.			(B.5)
	The half-vertex operators Γ ˘pρq are then defined by
	Γ ˘pρq :" exp	˜ÿ ně1	p n pρqα ˘n n	¸,	α n :"	kPZ 1 ÿ	ψ k´n ψ k .	(B.6)
					#	s µ{ν pρq if c " c 1 , 0 otherwise,	(B.7)

  and P4.1]. The canonical density matrix Π 0 u H does not satisfy the property (ii), which is why we need to pass to the grand canonical ensemble to have determinantal correlations.Remark C.2. We remark that above, the trace of an operator O over N F is understood as Proof of Lemma C.1. The proof of Lemma B.1 applies mutatis-mutandis except we now use the canonical anticommutation relations for neutral fermions from equation (C.3). Note we still have the simple commutationsDφ k " u k tψ k D, Dφ k " pu k tq ´1φ k D (C.11)for k ą 0 where D :" u Hs t Cs { trpu Hs t Cs q.Bosonic and half-vertex operators. We can define the so-called bosonic operators α s ˘n as follows. Fix n a positive odd integer. Then set

	tr O "			ÿ	xλ _ , c|O|λ, cy	(C.10)
				λPSP, cPt0,1u
	and we have that trpu Hs t Cs q " p1 `tq	ś iě1 p1 `ui q.
		α s n :"	1 4	kPZ ÿ p´1q k φ ´k´n φ k .	(C.12)
	We have that α s ´n is the adjoint of α s n , that α s n |Hy " 0 for n ą 0, and for n, m P 2Z `1 that they satisfy
	the following commutation relations
	rα n , α m s "	n 2	δ n,´m ,	rα ˘pρq by
	Γ s ˘pρq :" exp	˜ÿ nP2N`1	2p n pρqα s ˘n n	¸.	(C.14)
	When x is a variable, we denote by Γ s ˘pxq the half-vertex operators for the specialization in the single variable x, for which p n pxq " x n . Γ s ´pρq is the adjoint of Γ s `pρq for any real ρ, and
	Γ s `pρq|Hy " |Hy,	xH|Γ s ´pρq " xH|.	(C.15)

n , φpzqs " z n φpzq. (C.13) For ρ a (strict) specialization of the algebra SSym we define the half-vertex operators Γ s

  Hs Γ s R s |Hy " φ 0 |Hy and R s φ i " φ i R s for i ‰ 0. The following averages are useful for our purposes.Proof. Notice that the first equality is the u " t " 0 case of the second. For the second we use the boson-fermion correspondence (C.21) and Γ-elimination.Using neutral fermions, we can also derive the following pfaffian evaluation.Proof. For an operator O acting on neutral Fock space N F let us introduce, as usual, its expectation to be xOy " trpu Hs t Cs Oq{ trpu Hs t Cs q. By Wick's lemma C.1 we have xφpx 1 qφpx 2 q ¨¨¨φpx 2n´1 qφpx 2n qy " pf

	Finally, the boson-fermion correspondence in this setting reads [Kac90, Exercise 14.13]:
		φpzq " R s Γ s ´pzqΓ	s ``´z	´1˘,	(C.21)
	where R s satisfies R 2 s " 1, Proposition C.3. We have
	xH|φpzqφpwq|Hy " trpu Hs t Cs φpzqφpwqq "	z z θ u `w z ´w `w , `´w θu ˘¨p1 `tqpu; uq 2 |w| ă |z|, 8 p´u; uq ´1 8 ,	u 1{2 ă |w| ă |z| ă u ´1{2 .	(C.22)
		z
	Proposition C.4. We have
		pf	θ u ´xi xj ´´xi θu ¯"		ź	θu ´´xi θ u ´xi xj ¯.	(C.23)
		1ďiăjď2n	xj	1ďiăjď2n	xj
						2.8]):
	xλ _ , c|Γ s `pρq|µ, dy " Q µ{λ pρqδ c,d ,	xµ _ , c|Γ s ´pρq|λ, dy " P µ{λ pρqδ c,d .	(C.19)
	where λ, µ are strict partitions and c, d P Z 2 . This results from (C.18), Wick's lemma C.1 at u " 0, and the
	Jacobi-Trudi-like identity for Schur's P and Q functions we took for their definition-see [Mat05], [Whe11,
	Sections 1.4.9 and 3.2.8] for elementary proofs.
	The half-vertex operators Γ ˘pu ˘1ρq.	(C.20)

s commute with the grading operator C s and satisfy the following quasicommutation with the energy operator H s : Γ s ˘pρqu Hs " u 1ďiăjďn xφpx i qφpx j qy. (C.24)

Borodin's proof involves so-called L-ensembles. As stated in[START_REF] Borodin | Periodic Schur process and cylindric partitions[END_REF], his initial derivation was also based on the formalism of (fermionic) Fock space, but our approach is different (personal communication with Alexei Borodin).

By a slight abuse of language, the model for which Probpλq9u |λ| will be called uniform as it is the "macrocanonical" ensemble associated with uniform random partitions of fixed size[START_REF] Vershik | Statistical mechanics of combinatorial partitions, and their limit shapes[END_REF]. Note that this is not the same notion as the fermionic grand canonical ensemble that we will consider.

The fiber c ˆSs p λq-c P t0, 1u-corresponds to states |λ, cy in neutral Fock space N F in the notation of Appendix C.

C Basics of neutral free fermions

We recall the theory of neutral free fermions, useful for the study of the strict Schur process in [START_REF] Vuletić | The shifted Schur process and asymptotics of large random strict plane partitions[END_REF], following the conventions of [Whe11, Sections 1.4 and 3.8] (note they differ by a factor of 2 from the conventions of [Kac90, Exercise 14.13] and by a factor of 4 from those of [START_REF] Matsumoto | Correlation functions of the shifted Schur measure[END_REF][START_REF] Vuletić | A generalization of MacMahon's formula[END_REF]). We give a more detailed outline-neutral fermions being less often used in the literature-than the one in Appendix B on which we rely in our construction of neutral fermions.

Neutral Fock space and fermionic operators. We begin by constructing the neutral fermionic Fock space N F in analogy with how we constructed F. We start with the neutral fermionic operators, which we define in terms of the charged fermionic operators ψ k , ψ k as follows:

(notice how we have switched the indexing from Z 1 to Z). They form the neutral fermionic field φpzq "

and satisfy the following canonical anticommutation relations:

Given an ordered subset tµ 1 ą ¨¨¨ą µ r ě 0u Ă N, a basis for N F (respectively its dual N F ˚) is given by states of the form φ µ1 ¨¨¨φ µr |Hy, xH|φ μr ¨¨¨φ μ1 (C.6) which can be naturally identified with pairs |µ, cy (respectively xµ, c|) where µ is a strict partition (a partition with all its parts distinct) and c " 0 if r is even, and 1 otherwise. The number c P Z 2 can be seen as an even/odd grading N F " N F 0 ' N F 1 , based on whether a basis state comes from an even/odd number of fermions acting on the vacuum. For example, the vectors φ 3 φ 2 φ 0 |Hy ": |p3, 2q, 1y P N F 1 , φ 3 φ 2 |Hy ": |p3, 2q, 0y P N F 0 correspond to the subsets t3, 2, 0u, t3, 2u Ă N and both to the same strict partition p3, 2q. One can think of basis states in N F aforedescribed as collections of finitely many particles sitting on the N lattice. Note φ k for k ą 0 is the adjoint of φ k under the inner product xλ, c|µ, dy " 2 pλq δ λ,µ δ c,d for λ, µ strict partitions and c, d P Z 2 . We have φ ´n|Hy " φ n|Hy " 0 " xH|φ n " xH|φ ˚n|Hy for all n ą 0.

We denote xλ _ , c| " 2 ´ pλq xλ, c| so that xλ _ , c|µ, dy " δ λ,µ δ c,d .

The energy operator H s and grading operator C s (the analogue of the charge operator C) are given by (the subscript standing for strict) u Hs |λ, cy :" u |λ| |λ, cy, t Cs |λ, cy :" t c |λ, cy.

(C.7)

For k ą 0 we have 1 2 φ k φ k |λ, cy " 1 λ has a part of size k |λ, cy.

(C.8)

We also have a (finite-temperature) Wick lemma for neutral fermions similar to Lemma B.1:

Lemma C.1. Let Φ be the vector space spanned by (possibly infinite linear combinations of ) the φ k 's. For ϕ 1 , . . . , ϕ 2n P Φ, we have trpu Hs t Cs ϕ 1 ¨¨¨ϕ 2n q " pf A (C.9)

where A is the antisymmetric matrix defined by A i,j " tr `uHs t Cs ϕ i ϕ j ˘{ trpu Hs t Cs q for i ă j.