Titus Lupu
email: titus.lupu@upmc.fr

Isomorphisms of β-Dyson's Brownian motion with Brownian local time *

Keywords: Dyson's Brownian motion, Gaussian beta ensembles, Gaussian free field, isomorphism theorems, local time, permanental fields, topological expansion. MSC2020 subject classifications: Primary 15B52, 60B20, 60J55, Secondary 60G15, 81T18

We show that the Brydges-Fröhlich-Spencer-Dynkin and the Le Jan's isomorphisms between the Gaussian free fields and the occupation times of symmetric Markov processes generalize to the β-Dyson's Brownian motion. For β P t1, 2, 4u this is a consequence of the Gaussian case, however the relation holds for general β. We further raise the question whether there is an analogue of β-Dyson's Brownian motion on general electrical networks, interpolating and extrapolating the fields of eigenvalues in matrix-valued Gaussian free fields. In the case n " 2 we give a simple construction.

Introduction

There is a class of results, known as isomorphism theorems, relating the squares of Gaussian free fields (GFFs) to occupation times of symmetric Markov processes. They originate from the works in mathematical physics [START_REF] Symanzik | Euclidean quantum field theory[END_REF][START_REF] Brydges | The random walk representation of classical spin systems and correlation inequalities[END_REF]. For a review, see [START_REF] Marcus | Markov processes, Gaussian processes and local times[END_REF][START_REF] Sznitman | Topics in occupation times and Gaussian free field[END_REF]. Here in particular we will be interested in the Brydges-Fröhlich-Spencer-Dynkin isomorphism [START_REF] Brydges | The random walk representation of classical spin systems and correlation inequalities[END_REF][START_REF] Dynkin | Gaussian and non-Gaussian random fields associated with Markov processes[END_REF][START_REF] Dynkin | Local times and quantum fields[END_REF] and in the Le Jan's isomorphism [START_REF] Le Jan | Markov loops and renormalization[END_REF][START_REF] Le Jan | Markov paths, loops and fields[END_REF]. The BFS-Dynkin isomorphism involves Markovian paths with fixed ends. Le Jan's isomorphism involves a Poisson point process of Markovian loops, with an intensity parameter α " 1{2 in the case of real scalar GFFs. For vector-valued GFFs with d components, the intensity parameter is α " d{2.

We show that both Le Jan's and BFS-Dynkin isomorphisms have a generalization to β-Dyson's Brownian motion, and provide identities relating the latter to local times of one-dimensional Brownian motions. By doing so, we go beyond the Gaussian setting.

For β P t1, 2, 4u, a β-Dyson's Brownian motion is the diffusion of eigenvalues in a Brownian motion on the space of real symmetric pβ " 1q, complex Hermitian pβ " 2q, respectively quaternionic Hermitian pβ " 4q matrices. Yet, the β-Dyson's Brownian motion is defined for every β ě 0. The one-dimensional marginals of β-Dyson's Brownian motion are Gaussian beta ensembles GβE. The generalization of Le Jan's and BFS-Dynkin isomorphisms works for every β ě 0, and for β P t1, 2, 4u it follows from the Gaussian case. The intensity parameter α appearing in the Le Jan's type isomorphism is given by 2α " dpβ, nq " n `npn ´1q β 2 ,

where n is the number of "eigenvalues". In particular, α takes not only half-integer values, as in the Gaussian case, but a whole half-line of values. The BFS-Dynkin type isomorphism involves polynomials defined by a recurrence with a structure similar to that of the Schwinger-Dyson equation for GβE. These polynomials also give the symmetric moments of the β-Dyson's Brownian motion.

We further ask the question whether an analogue of GβE and β-Dyson's Brownian motion could exist on electrical networks and interpolate and extrapolate the distributions of the eigenvalues in matrix-valued GFFs. Our motivation for this is that such analogues could be related to Poisson point process of random walk loops, in particular to those of non half-integer intensity parameter. If the underlying graph is a tree, the construction of such analogues is straightforward, by taking β-Dyson's Brownian motions along each branch of the tree. However, if the graph contains cycles, this is not immediate, and one does not expect a Markov property for the obtained fields. However, in the simplest case Let pW pxqq xPR denote a two-sided Brownian motion, i.e. pW pxqq xě0 and pW p´xqq xě0 being two independent standard Brownian motions starting from 0 (W p0q " 0). Note that here x is rather a one-dimensional space variable then a time variable. The derivative dW pxq is a white noise on R. Let pφ R`p xqq xě0 denote the process p ? 2W pxqq xě0 . The covariance function of φ R`i s G R`. Let pφ K pxqq xPR be the stationary Ornstein-Uhlenbeck process with invariant measure N p0, 1{ ? 2Kq. It is a solution to the SDE dφ K pxq " ? 2dW pxq ´?2Kφ K pxqdx.

The covariance function of φ K is G K .

What follows is the BFS-Dynkin isomorphism (Theorem 2.2 in [START_REF] Brydges | The random walk representation of classical spin systems and correlation inequalities[END_REF], Theorems 6.1 and 6.2 in [START_REF] Dynkin | Gaussian and non-Gaussian random fields associated with Markov processes[END_REF], Theorem 1 in [START_REF] Dynkin | Local times and quantum fields[END_REF]) in the particular case of a 1D Brownian motion. In general, the BFS-Dynkin isomorphism relates the squares of Gaussian free fields to local times of symmetric Markov processes. Theorem 2.1 (Brydges-Fröhlich-Spencer [START_REF] Brydges | The random walk representation of classical spin systems and correlation inequalities[END_REF], Dynkin [START_REF] Dynkin | Gaussian and non-Gaussian random fields associated with Markov processes[END_REF][START_REF] Dynkin | Local times and quantum fields[END_REF]). Let F be a bounded measurable functional on CpR `q, resp. on CpRq. Let k ě 1 and x 1 , x 2 , . . . , x 2k in p0, `8q, resp. in R.

Then E " 2k ź i"1 φ R`p x i qF pφ 2 R`{ 2q ı " ÿ ptai,biuq 1ďiďk
partition in pairs of t1,2,...,2ku

ż γ1,...,γ k E " F pφ 2 R`{ 2 `Lpγ 1 q `¨¨¨`Lpγ k qq ı k ź i"1 µ xa i ,x b i R`p dγ i q,
resp.

E " 2k ź

i"1 φ K px i qF pφ where the sum runs over the p2kq!{p2 k k!q partitions in pairs, the γ i -s are Brownian paths and the Lpγ i q-s are the corresponding occupation fields x Þ Ñ L x pγ i q.

Remark 2.2. Since for x ă y, the measure µ x,y R`, resp. µ x,y K , can be decomposed as

µ x,x R`b
μx,y , resp. µ x,x K b μx,y K , Theorem 2.1 can be rewritten using only the measures of type µ x,x R`a nd μx,y , resp. µ x,x K and μx,y K . To a wide class of symmetric Markov processes one can associate in a natural way an infinite, σ-finite measure on loops [START_REF] Lawler | The Brownian loop soup[END_REF][START_REF] Lawler | Random walk loop soup[END_REF][START_REF] Lawler | Random walk: a modern introduction[END_REF][START_REF] Le Jan | Markov loops and renormalization[END_REF][START_REF] Le Jan | Markov paths, loops and fields[END_REF][START_REF] Le Jan | Permanental fields, loop soups and continuous additive functionals[END_REF][START_REF] Fitzsimmons | Markovian loop soups: permanental processes and isomorphism theorems[END_REF]. It originated from the works in mathematical physics [START_REF] Symanzik | Euclidean quantum field theory I: Equations for a scalar model[END_REF][START_REF] Symanzik | Euclidean quantum field theory I. Equations for a scalar model[END_REF][START_REF] Symanzik | Euclidean quantum field theory[END_REF][START_REF] Brydges | The random walk representation of classical spin systems and correlation inequalities[END_REF]. Here we recall it in the setting of a 1D Brownian motion, which has been studied in [START_REF] Lupu | Poisson ensembles of loops of one-dimensional diffusions[END_REF]. The range of a loop will be just a segment on the line, but it will carry a non-trivial Brownian local time process which will be of interest for us.

Given a Brownian loop γ, T pγq will denote its duration. The measures on (rooted) loops are µ loop R`p dγq :"

1 T pγq ż R`µ x,x R`p dγqdx, µ loop K pdγq " 1 T pγq ż R µ x,x K pdγqdx. (2.2)
Usually one considers unrooted loops, but this will not be important here. The 1D Brownian loop soups are the Poisson point processes, denoted L α R`, resp. L α K , of intensity αµ loop R`, resp. αµ loop K , where α ą 0 is an intensity parameter. LpL α R`q , resp.

LpL α K q, will denote the occupation field of L α R`, resp. L α K :

L x pL α R`q :" ÿ γPL α R `Lx pγq, L x pL α K q :" ÿ γPL α K L x pγq.
The following statement deals with the law of LpL α R`q , resp. LpL α K q. See Proposition 4.6, Property 4.11 and Corollary 5.5 in [START_REF] Lupu | Poisson ensembles of loops of one-dimensional diffusions[END_REF]. For the analogous statements in discrete space setting, see Corollary 5, Proposition 6, Theorem 13 in [START_REF] Le Jan | Markov loops and renormalization[END_REF] and Corollary 1, Section 4.1, Proposition 16, Section 4.2, Theorem 2, Section 5.1 in [START_REF] Le Jan | Markov paths, loops and fields[END_REF]. In general, one gets αpermanental fields (see also [START_REF] Le Jan | Permanental fields, loop soups and continuous additive functionals[END_REF][START_REF] Fitzsimmons | Markovian loop soups: permanental processes and isomorphism theorems[END_REF]). For α " 1 2 in particular, one gets square Gaussians.

We recall that given a matrix M " pM ij q 1ďi,jďk , its α-permanent is Perm α pM q :" ÿ σ permutation of t1,2,...,ku

α # cycles of σ k ź i"1 M iσpiq .
(2.3) Theorem 2.3 (Le Jan [START_REF] Le Jan | Markov loops and renormalization[END_REF][START_REF] Le Jan | Markov paths, loops and fields[END_REF], Lupu [START_REF] Lupu | Poisson ensembles of loops of one-dimensional diffusions[END_REF]). For every α ą 0 and x P R `, resp. x P R, the r.v. L x pL α R`q , resp. L x pL α K q, follows the distribution Gammapα, G R`p x, xq ´1q, resp. Gammapα, G K px, xq ´1q. Moreover, the process α Þ Ñ L x pL α R`q , resp. L x pL α K q, is a pure jump Gamma subordinator with Lévy measure with initial condition L 0 pL α R`q " 0. That is to say it is a square Bessel process of dimension 2α, reflected at level 0 for α ă 1. For x P R, x Þ Ñ L x pL α K q is a stationary solution to the SDE dL x pL α K q " 2 `Lx pL α K q ˘1 2 dW pxq ´2? 2KL x pL α K q `2αdx.

1 lą0 e ´l{G R `px,xq l dl, resp. 1 lą0 e ´l{G K px,xq l dl. Let x 1 , x 2 , . . . , x k P R `, resp. R. Then E " k ź i"1 L xi pL α R`q ı " Perm α `GR`p x i , x j q 1ďi,jďk ˘, E " k ź i"1 L xi pL α K q ı " Perm α `GK px i , x j q 1ďi,jďk ˘. For x ě 0, x Þ Ñ L x pL α R`q is a solution to the SDE dL x pL α R`q " 2 `Lx pL α R`q ˘1 2 dW pxq `2αdx, EJP 26
In particular, for α " 1 2 , one has the following identities in law between stochastic processes:

LpL α R`q p law q " 1 2 φ 2 R`, LpL α K q p law q " 1 2 φ 2 K .
(2.4)

Gaussian beta ensembles

For references on Gaussian beta ensembles, see [START_REF] Dumitriu | Matrix models for beta ensembles[END_REF][START_REF] Forrester | The Oxford Handbook of Random Matrix Theory[END_REF], [START_REF] Eynard | Random matrices[END_REF] For q ě 1, p q pλq will denote the q-th power sum polynomial p q pλq :"

n ÿ j"1 λ q j .
By convention, p 0 pλq " n.

A Gaussian beta ensemble GβE, with β ą ´2 n , follows the distribution

1 Z β,n |Dpλq| β e ´1 2 p2pλq n ź j"1 dλ j , (3.1)
where Z β,n is given by ([27, Formula (17.6.7)] and [11, Formula (1.2.23)])

Z β,n " p2πq n 2 n ź j"1 Γ `1 `j β 2 Γ`1 `β 2 ˘.
The brackets x¨y β,n will denote the expectation with respect to (3.1). For β " 0 one gets n i.i.d. N p0, 1q Gaussians. For β equal to 1, 2, resp. 4, one gets the eigenvalue distribution of GOE, GUE, resp. GSE random matrices [START_REF] Mehta | Random Matrices[END_REF][START_REF] Eynard | Random matrices[END_REF]. Usually the GβE are studied for β ą 0 [START_REF] Dumitriu | Matrix models for beta ensembles[END_REF], but the distribution (3.1) is well defined for all β ą ´2 n . For β P p´2 n , 0q there is an attraction between the λ j -s instead of a repulsion as for β ą 0. Moreover, as β Ñ ´2 n , λ under (3.1) converges in law to

´1 ? n ξ, 1 ? n ξ, . . . , 1 ? n ξ ¯, (3.2)
where ξ follows N p0, 1q.

Let dpβ, nq denote dpβ, nq " n `npn ´1q β 2 .

One can see dpβ, nq as a kind of pseudo-dimension. For β P t1, 2, 4u, dpβ, nq is the dimension of the corresponding space of matrices.

Let ν " pν 1 , ν 2 , . . . , ν m q, where m ě 1, and for all k P t1, 2, . . . , mu, ν k P Nzt0u. for the expressions of some moments.

Proposition 3.1 (Schwinger-Dyson equation [15, 16, 11]). For every β ą ´2{n and every ν as above with |ν| even,

xp ν pλqy β,n " β 2 ν mpνq ´1 ÿ i"1 xp pνrq r‰mpνq pλqp i´1 pλqp ν mpνq ´1´i pλqy β,n (3.3)
`´1 ´β 2 ¯pν mpνq ´1qxp pνrq r‰mpνq pλqp ν mpνq ´2pλqy β,n

`mpνq´1 ÿ k"1 ν k xp pνrq r‰k,mpνq pλqp ν k `νmpνq ´2pλqy β,n ,
where p 0 pλq " n. In particular, for q even, xp q pλqy β,n " β 2

q´1 ÿ i"1 xp i´1 pλqp q´1´i pλqy β,n `´1 ´β 2 ¯pq ´1qxp q´2 pλqy β,n ,
and for ν with ν mpνq " 1,

xp ν pλqy β,n " mpνq´1 ÿ k"1 ν k xp pνrq r‰k,mpνq pλqp ν k ´1pλqy β,n .
The recurrence (3.3) and the initial condition p 0 pλq " n determine all the moments xp ν pλqy β,n .

Proof. Note that (3.3) determines the moments xp ν pλqy β,n because on the left-hand side one has a degree |ν|, and on the right-hand side all the terms have a degree |ν| ´2. It is enough to check (3.3) for β ą 0, since both sides are analytic in β. For β ą 0, we outline the proof appearing in [START_REF] Croix | The combinatorics of the Jack parameter and the genus series for topological maps[END_REF]Lemma 4.13] and [START_REF] Eynard | Random matrices[END_REF]Section 4.1.1], so as to be selfcontained. Let us denote here ν :" pν 1 , ν 2 , . . . , ν mpνq´1 q, so that p ν pλq " p ν mpνq pλqp ν pλq.

We have that

B Bλ 1 ´λν mpνq ´1 1 p ν pλq|Dpλq| β e ´1 2 p2pλq
¯"

´λν mpνq 1 p ν pλq|Dpλq| β e ´1 2 p2pλq `β n ÿ j"2 λ ν mpνq ´1 1 λ 1 ´λj p ν pλq|Dpλq| β e ´1 2 p2pλq
`pν mpνq ´1qλ

ν mpνq ´2 1 p ν pλq|Dpλq| β e ´1 2 p2pλq `mpνq´1 ÿ k"1 ν k λ ν k `νmpνq ´2 1 p pνrq r‰k,mpνq pλq|Dpλq| β e ´1 2 p2pλq . Since ż R B Bλ 1 ´λν mpνq ´1 1 p ν pλq|Dpλq| β e ´1
K ą 0. Then xhpλqe ´1 2 Kp2pλq y β,n " 1 Z β,n ż R n hpλq|Dpλq| β e ´1 2 pK`1qp2pλq n ź j"1 dλ j " pK `1q ´n 2 Z β,n ż R n hppK `1q ´1 2 λq|DppK `1q ´1 2 λq| β e ´1 2 p2p λq n ź j"1 d λj " pK `1q ´1 2 `n`npn´1q β 2 `a˘x hp λqy β,n ,
where on the second line we used the change of variables λ " pK `1q

with initial condition λp0q " 0. The derivatives pdW j pxqq 1ďjďn are independent white noises. Since we will be interested in isomorphisms with Brownian local times, the variable x corresponds here to a one-dimensional spatial variable rather than a time variable. For every x ą 0, λpxq{ a G R`p x, xq " λpxq{ ? 2x, is distributed, up to a reordering of the λ j pxq-s, as a GβE (3.1). For β equal to 1, 2 resp. 4, pλpxqq xě0 is the diffusion of eigenvalues in a Brownian motion on the space of real symmetric, complex Hermitian, resp. quaternionic Hermitian matrices. For β ě 1, there is no collision between the λ j pxq-s, and for β P r0, 1q two consecutive λ j pxq-s can collide, but there is no collision of three or more particles [START_REF] Cépa | No multiple collisions for mutually repelling Brownian particles[END_REF]. Note that for β ą 0 and j P 2, n , pλ j pxq ´λj´1 pxqq{2 behaves near level 0 like a Bessel process of dimension β `1 reflected at level 0, and since β `1 ą 1, the complication with the principal value and the local time at zero does not occur; see [START_REF] Yor | The author thanks Guillaume Chapuy and Jérémie Bouttier for discussions and references on the beta ensembles. The author thanks Yves Le Jan and Wendelin Werner for their[END_REF]Chapter 10]. In particular, each pλ j pxqq xě0 is a semimartingale. For β " 0, pλpxq{ ? 2q xě0 is just a reordered family of n i.i.d. standard Brownian motions.

Remark 4.1. We restrict to β ě 0 because the case β ă 0 has not been considered in the literature. The problem is the extension of the process after a collision of λ j pxq-s.

The collision of three or more particles, including all the n together for β ă ´2pn´3q npn´1q , is no longer excluded. However, we believe that the β-Dyson's Brownian motion can be defined for all β ą ´2 n . This is indeed the case if n " 2. One can use the reflected Bessel processes for that. Let pρpxqq xě0 be the Bessel process of dimension β `1, reflected at level 0, satisfying away from 0 the SDE dρpxq " dW pxq `β 2ρpxq dx,

with ρp0q " 0. The reflected version is precisely defined for β ą ´1 " ´2 2 ; see [29, Section XI.1] and [START_REF] Lawler | Notes on the Bessel process[END_REF]Section 3]. Let p Ă W pxqq xě0 be a standard Brownian motion starting from 0, independent from pW pxqq xě0 Then, for n " 2, one can construct the β-Dyson's Brownian motion as

λ 1 pxq " Ă W pxq `ρpxq, λ 2 pxq " Ă W pxq ´ρpxq. (4.2)
Next are some simple properties of the β-Dyson's Brownian motion.

Proposition 4.2. The following holds.

1. The process `1 ? n p 1 pλpxqq ˘xě0 has the same law as φ R`. 2. The process p 1 2 p 2 pλpxqqq xě0 is a square Bessel process of dimension dpβ, nq starting from 0.

3. The processes pp 1 pλpxqqq xě0 and `λpxq ´1 n p 1 pλpxqq ˘xě0 are independent. 4. The process `1 2 `p2 pλpxqq ´1 n p 1 pλpxqq 2 ˘˘xě0 is a square Bessel process of dimension dpβ, nq ´1 starting from 0.

Proof. With Itô's formula, we get

dp 1 pλpxqq " ? 2 n ÿ j"1 dW j pxq, d 1 2 p 2 pλpxqq " 2 n ÿ j"1 λ j pxq ? 2 dW j pxq `dpβ, nqdx, d 1 2 ´p2 pλpxqq ´1 n p 1 pλpxqq 2 ¯" 2 n ÿ j"1 λ j pxq ´1 n p 1 pλpxqq ? 2 dW j pxq `pdpβ, nq ´1qdx, (4.3)
EJP 26 (2021), paper 126.

where the points x P R `for which λ j pxq " λ j´1 pxq for some j P 2, n can be neglected.

This gives (1), ( 2) and ( 4) since the processes

d Ă W pxq " n ÿ j"1 λ j pxq a p 2 pλpxqq dW j pxq, Ă W p0q " 0, and d | W pxq " n ÿ j"1 λ j pxq ´1 n p 1 pλpxqq b p 2 pλpxqq ´1 n p 1 pλpxqq 2 dW j pxq, | W p0q " 0,
are both standard Brownian motions. Again, one can neglect the points x P R `where p 2 pλpxqq ´1 n p 1 pλpxqq 2 " 0, which only occur for n " 2.

For (3), we have that

d ´λj pxq ´1 n p 1 pλpxqq ¯" ? 2d ´Wj pxq ´1 n p 1 pW pxqq β ÿ j 1 ‰j dx `λj pxq ´1 n p 1 pλpxqq ˘´`λ j 1 pxq ´1 n p 1 pλpxqq ˘,
where

p 1 pW pxqq " n ÿ j 1 "1 W j 1 pxq.
The Brownian motion p 1 pW q " 1 ? 2 p 1 pλq is independent from the family of Brownian motions `Wj ´1 n p 1 pW q ˘1ďjďn . Further, the measurability of `λj ´1 n p 1 pλq ˘1ďjďn with respect to `Wj ´1 n p 1 pW q ˘1ďjďn follows from the pathwise uniqueness of the solution to Corollary 4.3. The process `1 2 p 2 pλpxqq ˘xě0 is distributed as the occupation field pL x pL α R`q q xě0 of a 1D Brownian loop soup L α R`, with the correspondence 2α " dpβ, nq " n `npn ´1q β 2 .

(4.4)

Further, let L α´1 2 R`a nd r L 1 2
R`b e two independent 1D Brownian loop soups, α still given by (4.4). Then, one has the following identity in law between pairs of processes:

´1 2 ´p2 pλpxqq ´1 n p 1 pλpxqq 2 ¯, 1 2n p 1 pλpxqq 2 ¯xě0 (law) " pL x pL α´1 2 R`q , L x p r L 1 2
R`q q xě0 .

Symmetric moments of β-Dyson's Brownian motion

We will denote by x¨y

Rβ

,n the expectation with respect to the β-Dyson's Brownian motion (4.1). This section will be devoted to deriving a recursive way to express the symmetric moments

A mpνq ź k"1 p ν k pλpx k qq E Rβ ,n (4.5)
for ν be a finite family of positive integers with |ν| even and x 1 ď x 2 ď ¨¨¨ď x mpνq P R `.

This generalizes the Schwinger-Dyson equation (3.3). Note that if |ν| is odd then the moment equals 0.

EJP 26 (2021), paper 126.

We will also use in the sequel the following notation. For k ě k 1 P N, k, k 1 will denote the interval of integers k, k 1 " tk, k `1, . . . , k 1 u.

We start by some lemmas.

Lemma 4.4. Let q ě 3. Then

dp q pλpxqq " q ? 2 n ÿ j"1 λ j pxq q´1 dW j pxq `β 2 q q´2 ÿ i"2 p i´1 pλpxqqp q´1´i pλpxqqdx `2 β 2 nqp q´2 pλpxqqdx `´1 ´β 2 ¯qpq ´1qp q´2 pλpxqqdx.
Proof. By Itô's formula,

dp q pλpxqq " q ? 2 n ÿ j"1 λ j pxq q´1 dW j pxq `qpq ´1qp q´2 pλpxqqdx `βq ÿ 1ďjăj 1 ďn λ j pxq q´1 ´λj 1 pxq q´1 λ j pxq ´λj 1 pxq dx. But ÿ 1ďjăj 1 ďn λ j pxq q´1 ´λj 1 pxq q´1 λ j pxq ´λj 1 pxq " ÿ 1ďjăj 1 ďn q´2 ÿ r"0 λ j pxq r λ j 1 pxq q´2´r " ´n ´q ´1 2 ¯pq´2 pλpxqq `1 2 q´2 ÿ i"2
p i´1 pλpxqqp q´1´i pλpxqq.

Lemma 4.5. Let q, q 1 ě 1 with q `q1 ą 2. Then dxp q pλpxqq, p q 1 pλpxqqy " 2qq 1 p q`q 1 ´2pλpxqqdx.

Moreover, dxp 1 pλpxqq, p 1 pλpxqqy " 2ndx.

Proof. This is a straightforward computation.

Lemma 4.6. Let ν be a finite family of positive integers and let q ě 0. Then the process

ż x 0 p ν pλpyqq n ÿ j"1 λ j pyq q dW j pyq (4.6)
is a martingale in the filtration of the Brownian motions ppW j pxqq 1ďjďn q xě0 .

Proof. The process (4.6) is a local martingale. Its quadratic variation is given by ż x 0 p ν pλpyqq 2 p 2q pλpyqqdy.

For every y ą 0, λpyq{ ? 2y follows a fixed distribution, which is up to reordering the GβE (3.1). Thus,

A ż x 0 p ν pλpyqq 2 p 2q pλpyqqdy E Rβ ,n " xp ν pλq 2 p 2q pλqy β,n ż x 0 p2yq |ν|`q dy ă `8.
So the quadratic variation is locally bounded in L 1 . It follows that (4.6) is a true martingale.

EJP 26 (2021), paper 126.

Let ν be a finite family of positive integers. and let x 1 ď x 2 ď ¨¨¨ď x mpνq P R `. For k P 1, mpνq and x ě x k´1 , let f k pxq denote the function f k pxq :"

A k´1 ź k 1 "1 p ν k 1 pλpx k 1 qq mpνq ź k 1 "k p ν k 1 pλpxqq E Rβ ,n . (4.7)
The main idea for expressing a symmetric moment (4.5) is that for x ě x k´1 , the derivative f 1 k pxq is a linear combination of symmetric moments of degree |ν| ´2, with coefficients depending on β and n. The precise expressions for these coefficients can be deduced from Lemmas 4.4 and 4.5. Further, the moment (4.5) equals f mpνq px mpνq q, for every k P 2, mpνq , f k px k´1 q " f k´1 px k´1 q, and

f 1 px 1 q " p2x 1 q |ν|{2 xp ν pλqy β,n ,
where xp ν pλqy β,n is the moment of the GβE, given by Proposition 3.1. So given the above initial conditions, and knowing the derivatives f 1 k pxq one gets the moment (4.5). It turns out that this moment is a multivariate polynomial in px k q 1ďkďmpνq . Next we describe the recursion for this polynomial.

Let pY kk q kě1 denote a family of formal commuting polynomials variables. We will consider finite families of positive integers ν " pν 1 , ν 2 , . . . , ν mpνq q with |ν| even. The order of the ν k will matter. That is to say we distinguish between ν and pν σp1q , ν σp2q , . . . , ν σpmpνqq q for σ a permutation of 1, mpνq . We want to construct a family of formal polynomials Q ν,β,n with parameters ν, β and n, where Q ν,β,n has for variables pY kk q 1ďkďmpνq . To simplify the notations, we will drop the subscripts β, n and just write Q ν . The polynomials Q ν will appear in the expression of the symmetric moments (4.5). We will denote by cpν, β, nq the solutions to the recurrence (3.3), which for β P p´2{n, `8q are the moments xp ν pλqy β,n . By convention, cpp0q, β, nq " n and cpH, β, nq " 1. For k ě 1 and Q a polynomial, Q kÐ will denote the polynomial in the variables pY k 1 k 1 q 1ďk 1 ďk , obtained from Q by replacing each variable Y k 1 k 1 with k 1 ě k `1 by the variable Y kk . Note that Q mpνqÐ ν " Q ν and that Q 1Ð ν is an univariate polynomial in Y 11 . For Y a formal polynomial variable, deg Y will denote the partial degree in Y.

Definition 4.7. The family of polynomials pQ ν q |ν| even is defined by the following.

1. Q 1Ð ν " cpν, β, nqY |ν|{2 11 . 2. If mpνq ě 2, then for every k P 2, mpνq , B BY kk Q kÐ ν " β 2 ÿ kďk 1 ďmpνq ν k 1 ą2 νpk 1 q 2 ν k 1 ´2 ÿ i"2 Q kÐ ppνrq r‰k 1 ,i´1,ν k 1 ´1´iq (4.8) `β 2 n ÿ kďk 1 ďmpνq ν k 1 ą2 νpk 1 qQ kÐ ppνrq r‰k 1 ,ν k 1 ´2q `β 2 n 2 ÿ kďk 1 ďmpνq ν k 1 "2 Q kÐ pνrq r‰k 1 `´1 ´β 2 ¯ÿ kďk 1 ďmpνq ν k 1 ą2 ν k 1 pν k 1 ´1q 2 Q kÐ ppνrq r‰k 1 ,ν k 1 ´2q `´1 ´β 2 ¯n ÿ kďk 1 ďmpνq ν k 1 "2 Q kÐ pνrq r‰k 1 EJP 26 (2021), paper 126. `ÿ kďk 1 ăk 2 ďmpνq ν k 1 `νk 2 ą2 ν k 1 ν k 2 Q kÐ ppνrq r‰k 1 ,k 2 ,ν k 1 `νk 2 ´2q `n ÿ kďk 1 ăk 2 ďmpνq ν k 1 "ν k 2 "1 Q kÐ pνrq r‰k 1 ,k 2 .
If k " mpνq, then the last two lines of (4.8) vanish.

Note that since the polynomials Q ν,β,n are formal, one is not restricted by a specific range for β. One could take any β P C or even consider β as a formal parameter. The specific range for β will only matter when relating Q ν,β,n to the symmetric moments of the β-Dyson's Brownian motion. Proposition 4.8. Definition 4.7 uniquely defines a family of polynomials pQ ν q |ν| even .

Moreover, the following properties hold.

1. For every A monomial of Q ν and every k P 2, mpνq ,

2 ÿ kďk 1 ďmpνq deg Y k 1 k 1 A ď ÿ kďk 1 ďmpνq ν k 1 , (4.9) and 2 ÿ 1ďk 1 ďmpνq deg Y k 1 k 1 A " |ν|.
In particular, Q ν is a homogeneous polynomial of degree |ν|{2. 2. For every k P 1, mpνq and every permutation σ of k, mpνq , Q kÐ pνrq 1ďrďk´1 ,pν σprq q kďrďmpνq " Q kÐ ν .

Proof. The fact that the polynomials Q ν are well defined can be proved by induction on |ν|{2.

For |ν|{2 " 1, there are only two polynomials, Q p2q and Q p1,1q . According to the condition (1),

Q p2q " cpp2q, β, nqY 11 " dpβ, nqY 11 " ´β 2 n 2 `´1 ´β 2 ¯n¯Y 11 .
The condition (2) does not apply for Q p2q . For Q p1,1q , according to the condition (2), B BY [START_REF] Le Jan | Markov paths, loops and fields[END_REF] Q p1,1q " 0.

Q kÐ ν pY kk " 0q " Q k´1Ð ν ´`Q kÐ ν ´QkÐ ν pY kk " 0q ˘k´1Ð .
Moreover, by the condition (1), Q 1Ð ν is also uniquely determined. Thus, all the polynomials pQ kÐ ν q 1ďkďmpνq are uniquely determined, with consistency by the

Q Þ Ñ Q kÐ operations. Finally, Q ν " Q mpνqÐ ν .
The properties (1) and ( 2) again follow easily by induction on |ν|{2. Proposition 4.9. Let β ě 0. Let ν be a finite family of positive integers, with |ν| even. Let Q ν " Q ν,β,n be the polynomial given by Definition 4.7. Let x 1 ď x 2 ď ¨¨¨ď x mpνq P R `.

Then,

A mpνq ź k"1 p ν k pλpx k qq E Rβ ,n " Q ν ppY kk " 2x k q 1ďkďmpνq q.
Proof. The proof is done by induction on |ν|{2.

The case |ν|{2 " 1 corresponds to ν " p1, 1q or ν " p2q.

f 1 px 1 q " cpν, β, nqp2x 1 q |ν|{2 " Q 1Ð ν pY 11 " 2x 1 q, (4.10)
where for the second equality we applied the condition (1) in Definition 4.7. If mpνq " 1, there is nothing more to check. In the case mpνq ě 2, we need only to check that for every k P 2, mpνq and every x ą x k´1 ,

f 1 k pxq " B Bx Q kÐ ν ppY k 1 k 1 " 2x k 1 q 1ďk 1 ďk´1 , Y kk " 2xq (4.11) " 2 ´B BY kk Q kÐ ν ¯ppY k 1 k 1 " 2x k 1 q 1ďk 1 ďk´1 , Y kk " 2xq.
Indeed, given (4.10), by applying (4.11) to k " 2, we further get

f 2 px 2 q " P 2Ð ν pY 11 " 2x 1 , Y 22 " 2x 2 q,
and by successively applying (4.11) to k " 3, . . . , k " mpνq, we at the end get

f mpνq px mpνq q " Q mpνqÐ ν ppY k 1 k 1 " 2x k 1 q 1ďk 1 ďmpνq q,
which is exactly what we want. To show (4.11), we proceed as follows. Let pF x q xě0 be the filtration of the Brownian motions ppW j pxqq 1ďjďn q xě0 . Then, for x ą x k´1 , f k pxq "

A k´1 ź k 1 "1 p ν k 1 pλpx k 1 qq A mpνq ź k 1 "k p ν k 1 pλpxqq ˇˇFx k´1 E Rβ ,n E Rβ ,n
, where x¨|F x k´1 y

Rβ

,n denotes the conditional expectation. To express

A mpνq ź k 1 "k p ν k 1 pλpxqq ˇˇFx k´1 E Rβ ,n
, we apply Itô's formula to

mpνq ź k 1 "k p ν k 1 pλpxqq ´A mpνq ź k 1 "k p ν k 1 pλpx k´1 qq E Rβ ,n
.

The local martingale part is, according to Lemma 4.6, a true martingale, and thus gives a 0 conditional expectation. The bounded variation part is a linear combination of terms of form p ν pλpxqqdx, with |ν| " the exact expressions following from Lemma 4.4 and Lemma 4.5. By comparing these expressions with the recurrence (4.8), and using the induction hypothesis at the step |ν|{2 ´1, we get (4.11). At this stage we omit detailing the tedious but completely elementary computations.

´mpνq ÿ k 1 "k ν k 1 ¯´2, EJP 26

More general formal polynomials

In previous Section 4.2, we defined recursively a family of formal polynomials Q ν " Q ν,β,n (Definition 4.7), which encode the symmetric moments of the β-Dyson's Brownian motion (Proposition 4.9). However, these polynomials are insufficient both for the generalization of the BFS-Dynkin isomorphism (forthcoming Proposition 4.14) and for expressing the symmetric moments of the stationary version of the β-Dyson's Brownian motion (forthcoming Proposition 4.22). Therefore we introduce an other family of formal polynomials P ν " P ν,β,n , with P ν constructed out of Q ν in a straightforward way which we describe next.

On top of the formal commuting polynomial variables pY kk q kě1 appearing in the polynomials Q ν , we also consider the family of the formal commuting variables p q Y k´1 k q kě2 , also commuting with the first one. A polynomial P ν will have for variables pY kk q 1ďkďmpνq and p q Y k´1 k q 2ďkďmpνq . Definition 4.10. Given ν a finite family of positive integers with |ν| even, let P ν be the polynomial in the variables pY kk q 1ďkďmpνq , p q Y k´1 k q 2ďkďmpνq defined by the following.

1. P ν ppY kk q 1ďkďmpνq , p q Y k´1 k " 1q 2ďkďmpνq q " Q ν ppY kk q 1ďkďmpνq q.

2. For every A monomial of P ν and every k P 2, mpνq ,

deg q Y k´1 k A `2 ÿ kďk 1 ďmpνq deg Y k 1 k 1 A " ÿ kďk 1 ďmpνq ν k 1 .
(4.12)

The property (4.9) ensures that P ν " P ν,β,n is well defined. As for Q ν,β,n , P ν,β,n is defined for every β P C.

A mpνq ź k"1 p ν k pλpx k qq E Rβ ,n " P ν ppY kk " 2x k q 1ďkďmpνq , p q Y k´1 k " 1q 2ďkďmpνq q " P ν ppY kk " G R`p x k , x k qq 1ďkďmpνq , p q Y k´1 k " G R`p x k´1 , x k q{G R`p x k´1 , x k´1 qq 2ďkďmpνq q.
Next are the expressions for Q p1,1,...,1q , P p1,1,...,1q , Q p2,2,...,2q and P p2,2,...,2q .

Proposition 4.12. Let m P Nzt0u. Let M " pM kk 1 q 1ďk,k 1 ďm be the formal symmetric matrix with entries given by

M kk " Y kk , for k ă k 1 , M kk 1 " M k 1 k " Y kk ź k`1ďrďk 1 q Y r´1 r . (4.13)
The following holds. For other examples of P ν , see the Appendix. As a side remark, we observe next that the value β " ´2 n plays a special role for the polynomials Q ν,β,n and P ν,β,n . In particular, P ν,β"´2 n ,n gives the moments of the stochastic processes pφ R`p xqq xě0 and pφ K pxqq xPR introduced in Section 2, which are Gaussian. This is also related to the fact that in the limit β Ñ ´2 n , the GβE converges in law to n identical Gaussians (3.2). Proposition 4.13. Let n ě 1. Let K ą 0. Let ν be a finite family of positive integers with |ν| even. Let x 1 ď ¨¨¨ď x mpνq be mpνq points in p0, `8q, resp. in R. Then Q ν,β"´2 n ,n ppY kk " 2x k q 1ďkďmpνq q "

P ν,β"´2 n ,n ppY kk " 2x k q 1ďkďmpνq , p q Y k´1 k " 1q 2ďkďmpνq q " n mpνq´|ν|{2 E " mpνq ź k"1 φ R`p x k q ν k ı , resp. P ν,β"´2 n ,n ppY kk " 1{ ? 2Kq 1ďkďmpνq , p q Y k´1 k " e ´?2Kpx k ´xk´1 q q 2ďkďmpνq q " n mpνq´|ν|{2 E " mpνq ź k"1 φ K px k q ν k ı .
That is to say, the variables Y kk are replaced by G R`p x k , x k q, resp. G K px k , x k q, and the variables q Y k´1 k by G R`p x k´1 , x k q{G R`p x k´1 , x k´1 q, resp. G K px k´1 , x k q{G K px k´1 , x k´1 q.

EJP 26 (2021), paper 126.

Proof. First, one can check that c ´ν, β " ´2 n , n ¯" n mpνq´|ν|{2 |ν|! 2 |ν|{2 p|ν|{2q! . This follows from Proposition 3.2. The key point is that

d ´β " ´2 n , n ¯" 1.
Given ν a finite family of positive integers, let k ν : 1, |ν| Þ Ñ 1, mpνq be the function such that

k ´1 ν p1q " 1, ν 1 , for k 1 P 2, mpνq , k ´1 ν pk 1 q " ν 1 `¨¨¨`ν k 1 ´1 `1, ν 1 `¨¨¨`ν k 1 . (4.15)
Further, let p r Q ν q |ν| even be the following formal polynomials:

r Q ν " n mpνq´|ν|{2 ÿ ptai,biuq 1ďiď|ν|{2 partition in pairs of 1,|ν| |ν|{2 ź i"1
Y kν paiq^kν pbiq kν paiq^kν pbiq .

To conclude, we need only to check that r Q ν " Q ν,β"´2 n ,n for all ν with |ν| even. Indeed, this immediately implies that

P ν,β"´2 n ,n " n mpνq´|ν|{2 ÿ ptai,biuq 1ďiď|ν|{2 partition in pairs of 1,|ν| |ν|{2 ź i"1 M kν paiqkν pbiq ,
where the M kk 1 are given by (4.13), and thus n ´mpνq`|ν|{2 P ν,β"´2 n ,n corresponds to the Wick's rule. So by evaluating in Y kk " G R`p x k , x k q and q Y k´1 k " G R`p x k´1 , x k q{ G R`p x k´1 , x k´1 q, resp. Y kk " G K px k , x k q and q Y k´1 k " G K px k´1 , x k q{G K px k´1 , x k´1 q, one gets the moments of φ R`, resp. φ K .

The identity r Q ν " Q ν,β"´2 n ,n can be checked by induction over |ν|{2 by following Definition 4.7. From (4.14) follows that the r Q ν satisfy the condition (1) in Definition 4.7.

One can further check the recurrence (4.8), and this amounts to counting the pairs in k ´1 ν p k, mpνq q.

BFS-Dynkin isomorphism for β-Dyson's Brownian motion

We will denote by Υ a generic finite family of continuous paths on R, Υ " pγ 1 , . . . , γ J q, and JpΥq will denote the size J of the family. We will consider finite Brownian measures on Υ where JpΥq is not fixed but may take several values under the measure. Given x P R, L x pΥq will denote the sum of Brownian local times at x:

L x pΥq " JpΥq ÿ i"1 L x pγ i q.
LpΥq will denote the occupation field x Þ Ñ L x pΥq.

Given ν a finite family of positive integers with |ν| even and 0 ă x 1 ă x 2 ă ¨¨¨ă x mpνq , µ ν,x1,...,x mpνq R`p dΥq (also depending on β and n) will be the measure on finite families of continuous paths obtained by substituting in the polynomial P ν " P ν,β,n for each variable Y kk the measure µ x k ,x k R`, and for each variable q Y k´1 k the measure μx k´1 ,x k R`; see Section EJP 26 (2021), paper 126.

2. Since we will deal with the functional LpΥq under µ ν,x1,...,x mpνq R`p dΥq, the order of the Brownian measures in a product will not matter. For instance, for ν " p2, 1, 1q (see Appendix),

P p2,1,1q " ´β 2 n 3 `´1 ´β 2 ¯n2 ¯Y11 Y 22 q Y 23 `2nY 2 11 q Y 2 12 q Y 23 , and µ p2,1,1q,x1,x2,x3 R`" ´β 2 n 3 `´1 ´β 2 ¯n2 ¯µx1,x1 R`b µ x2,x2 R`b μx2,x3 R2 nµ x1,x1 R`b µ x1,x1 R`b μx1,x2 R`b μx1,x2 R`b μx2,x3 R`.
Note that depending on values of n and β, a measure µ

F pp pxqq xě0 q " exp ´´ż R` pxqχpxqdx ¯,
where χ is a continuous non-negative function with compact support in p0, `8q. For such F , the value returned by the right-hand side of (4.16) is well understood and is related to the local times of Brownian motions with a killing rate given by χ. In order to deal with the left-hand side of (4.16), one interprets exp ´´1 We start by some intermediate lemmas. Recall that pF x q xě0 denotes the filtration of the Brownian motions ppW j pxqq 1ďjďn q xě0 in (4.1). Consider χ a continuous non-negative function with compact support in p0, `8q. Let u χÓ denote the unique solution to λ j pyqdW j pyq.

Then pM χ pxqq xě0 is a martingale with respect to the filtration pF x q xě0 and for all x ě 0, By further performing the change of variable given by ψ χ , one gets (4.1).

D
In the sequel pG R`,χ px, yqq x,yě0 will denote the Green's function of

A mpνq ź k"1 p ν k p λpx k qq E Rβ ,n " P ν ppY kk " G R`,χ px k , x k qq 1ďkďmpνq , p q Y k´1 k " u χÓ px k q{u χÓ px k´1 qq 2ďkďmpνq q.
Proof. From Lemma 4.19 and Proposition 4.9 it follows that

A mpνq ź k"1 p ν k p λpx k qq E Rβ ,n " ´mpνq ź k"1 u χÓ px k q ν k ¯Qν ppY kk " 2ψ χ px k qq 1ďkďmpνq q. " ´mpνq ź k"1
u χÓ px k q ν k ¯Pν ppY kk " 2ψ χ px k qq 1ďkďmpνq , p q Y k´1 k " 1q 2ďkďmpνq q.

EJP 26 (2021), paper 126.

Further, let A be a monomial of P ν . One has to check that ´mpνq ź

k"1 u χÓ px k q ν k ¯AppY kk " 2ψ χ px k qq 1ďkďmpνq , p q Y k´1 k " 1q 2ďkďmpνq q " AppY kk " 2ψ χ px k qu χÓ px k q 2 q 1ďkďmpνq , p q Y k´1 k " u χÓ px k q{u χÓ px k´1 qq 2ďkďmpνq q.

This amounts to counting the power for each u χÓ px k q on both sides. On the left-hand side, each u χÓ px k q appears with power ν k . The power of u χÓ px k q on the right-hand side is

2 deg Y kk A `deg Y k´1 k A ´deg Y k k`1 A.
By (4.12), this is again ν k . Finally, by (4.20),

2ψ χ px k qu χÓ px k q 2 " G R`,χ px k , x k q.
Proof of Proposition 4.14. It is enough to show (4.16) for functionals F of form

F pp pxqq xě0 q " exp ´´ż R` pxqχpxqdx ¯,
where χ is a continuous non-negative function with compact support in p0, `8q. For such a χ,

A mpνq ź k"1 p ν k pλpx k qq exp ´´1 2 ż R`p 2 pλpxqqχpxqdx ¯ERβ ,n " A exp ´´1 2 ż R`p 2 pλpxqqχpxqdx ¯ERβ ,n A mpνq ź k"1 p ν k p λpx k qq E Rβ ,n
, where λ is given by (4.19), with λp0q " 0. The symmetric moments of λ are given by Lemma 4.20. To conclude, we use that

ż γ exp ´´ż R`L z pγqχpzqdz ¯µx,x R`p dγq " G R`,χ px, xq,
and for 0 ă x ă y,

ż γ exp ´´ż R`L z pγqχpzqdz ¯μ x,y pdγq " G R`,χ px, yq G R`,χ px, xq " u χÓ pyq u χÓ pxq ; see [24, Section 3.2].

The stationary case

In this section we consider the stationary β-Dyson's Brownian motion on the whole line and state the analogues of Propositions 4.2, 4.9 and 4.14 for it. The proofs are omitted, as they are similar to the previous ones. As previously, n ě 2 and β ě 0. Let K ą 0. We consider the process pλpxq " pλ 1 pxq, . . . , λ n pxqqq xPR with λ 1 pxq ě ¨¨¨ě λ n pxq, satisfying the SDE dλ j pxq " ? 2dW j pxq ´?2K λ j pxq `β? 2K

ÿ j 1 ‰j dx λ j pxq ´λj 1 pxq , (4.21)
the dW j , 1 ď j ď n, being n i.i.d. white noises on R, and λ being stationary, with p2Kq 1 4 λpxq being distributed according to (3.1) (up to reordering of the λ j pxq-s).

EJP 26 (2021), paper 126.

Proposition 4.21. The following holds.

1. The process `1 ? n p 1 pλpxqq ˘xPR has the same law as φ K . 2. Let be a 1D Brownian loop soup L α K , with α given by (4.4). The process p 1 2 p 2 pλpxqqq xPR has the same law as the occupation field pL x pL α K qq xPR . 3. The processes pp 1 pλpxqqq xPR and `λpxq ´1 n p 1 pλpxqq ˘xPR are independent. K be two independent 1D Brownian loop soups, α given by (4.4).

Then, one has the following identity in law between pairs of processes:

´1 2 ´p2 pλpxqq ´1 n p 1 pλpxqq 2 ¯, 1 2n p 1 pλpxqq 2 ¯xPR (law) " pL x pL α´1 2 K q, L x p r L 1 2
K qq xPR .

We will denote by x¨y K β,n the expectation with respect to the stationary β-Dyson's Brownian motion. Given ν a finite family of positive integers with |ν| even and x 1 ă x 2 ă ¨¨¨ă x mpνq P R, µ ν,x1,...,x mpνq K pdΥq (also depending on β and n) will be the measure on finite families of continuous paths obtained by substituting in the polynomial P ν " P ν,β,n for each variable Y kk the measure µ x k ,x k K , and for each variable q Y k´1 k the measure μx k´1 ,x k K . Proposition 4.22. Let ν a finite family of positive integers with |ν| even. Let x 1 ď x 2 ď ¨¨¨ď x mpνq P R. Then,

A mpνq ź k"1 p ν k pλpx k qq E K β,n " P ν ppY kk " 1{ ? 2Kq 1ďkďmpνq , p q Y k´1 k " e
´?2Kpx k ´xk´1 q q 2ďkďmpνq q "

P ν ppY kk " G K px k , x k qq 1ďkďmpνq , p q Y k´1 k " G K px k´1 , x k q{G K px k´1 , x k´1 qq 2ďkďmpνq q.
Further, let F be a bounded measurable functional on CpRq. For x 1 ă x 2 ă ¨¨¨ă x mpνq P R,

A mpνq ź k"1 p ν k pλpx k qqF ´1 2 p 2 pλq ¯EK β,n " ż Υ A F ´1 2 p 2 pλq `LpΥq ¯EK β,n µ ν,x1,...,x mpνq K pdΥq.
5 The case of general electrical networks: a construction for n " 2 and further questions 5.1 Formal polynomials for n " 2

In this section n " 2, and β is arbitrary, considered as a formal parameter. Note that dpβ, n " 2q " β `2. In Section 4.2 we introduced the formal commuting polynomial variables pY kk q kě1 . Here we further consider the commuting variables pY kk 1 q 1ďkăk 1 , and by convention set Y kk 1 " Y k 1 k for k 1 ă k. Given ν " pν 1 , . . . , νm q with νk P N (value 0 allowed), P ν,β will be the following multivariate polynomial in the variables pY kk 1 q 1ďkďk 1 ďm : P ν,β :" Perm β`1 2 ppY f piqf pjq q 1ďi,jďν1`¨¨¨`νm q, where f is a map f : 1, ν1 `¨¨¨`ν m Ñ 1, m , such that for every k P 1, m , |f ´1pkq| " νk . Recall the expression of the α-permanents (2.3). It is clear that P ν,β does not depend on the particular choice of f . In case ν1 " ¨¨¨" νm " 0, by convention we set P ν,β " 1.

Given ν a finite family of positive integers with |ν| even, let k ν : 1, |ν| Þ Ñ 1, mpνq be the map given by (4.15). Let I ν be the following set of subsets of 1, |ν| : where | ¨| denotes the cardinal. Note that necessarily, for every I P I ν , the cardinal |I| is even. Let p P ν,β be the following multivariate polynomial in the variables pY kk 1 q 1ďkďk 1 ďmpνq :

I ν :" tI Ď 1, |ν| | @k P 1, mpνq , |k ´1 ν pkqzI| is evenu, EJP 26
p P ν,β :" ÿ IPIν 2 mpνq´|I|{2 ´ÿ ptai,biuq 1ďiď|I|{2 partition in pairs of I |I|{2 ź i"1 Y kν paiqkν pbiq ¯Pp 1 2 |k ´1 ν pkqzI|q 1ďkďmpνq ,β .
By construction, for every A monomial of p P ν,β and every k P 1, mpνq ,

2 deg Y kk A `ÿ 1ďk 1 ďmpνq k 1 ‰k deg Y kk 1 A " ν k .
(5.1) Proposition 5.1. Let ν be finite family of positive integers with |ν| even. P ν,β,n"2 is obtained from p P ν,β by replacing each variable

Y kk 1 with 1 ď k ă k 1 ď mpνq by Y kk ś k`1ďrďk 1 q Y r´1 r : P ν,β,n"2 " p P ν,β ``Y kk 1 " Y kk ź k`1ďrďk 1 q Y r´1 r ˘1ďkăk 1 ďmpνq ˘. Proof. Let be r P ν,β :" p P ν,β ``Y kk 1 " Y kk ź k`1ďrďk 1 q Y r´1 r ˘1ďkăk 1 ďmpνq ˘.
We want to show the equality r P ν,β " P ν,β,n"2 . Since a direct combinatorial proof would be a bit lengthy, we proceed differently. Let β ě 0 and let pλpxq " pλ 1 pxq, λ 2 pxqqq xě0 be the β-Dyson's Brownian motion (4.1) in the case n " 2. We use its construction through (4.2). We claim that for x 1 , x 2 , . . . , x mpνq P R `,

A mpνq ź k"1 p ν k pλpx k qq E Rβ ,n"2 " p P ν,β ``Y kk 1 " G R`p x k´1 , x k q ˘1ďkďk 1 ďmpνq ˘.
Indeed, in the expansion of ´Ă W px k q `ρpx k q ¯νk `´Ă W px k q ´ρpx k q ¯νk only enter the even powers of ρpx k q, which is how I ν appears. Then one uses that the square Bessel process pρpxqq xě0 is a pβ `1q{2-permanental field with kernel pG R`p x, yqq x,yPR`. Because of the particular form of G R`, we have that for

x 1 ď x 2 ď ¨¨¨ď x mpνq P R `, A mpνq ź k"1 p ν k pλpx k qq E Rβ ,n"2
" r P ν,β ppY kk " 2x k q 1ďkďmpνq , p q Y k´1 k " 1q 2ďkďmpνq q.

By combining with Corollary 4.11, we get that the following multivariate polynomials in the variables pY kk q 1ďkďmpνq are equal for β ě 0: r P ν,β pp q Y k´1 k " 1q 2ďkďmpνq q " P ν,β,n"2 pp q Y k´1 k " 1q 2ďkďmpνq q.

Since the coefficients of both are polynomials in β, the equality above holds for general β. To conclude the equality r P ν,β " P ν,β,n"2 , we have to deal with the variables p q Y k´1 k q 2ďkďmpνq . For this we use that both in case of P ν,β,n"2 and in case of r P ν,β , each monomial satisfies (4.12). For r P ν,β this follows from (5.1).

EJP 26 (2021), paper 126.

A construction on discrete electrical networks for n " 2

Let G " pV, Eq be an undirected connected graph, with V finite. We do not allow multiple edges or self-loops. The edges tx, yu P E are endowed with conductances Cpx, yq " Cpy, xq ą 0. There is also a non-uniformly zero killing measure pKpxqq xPV , with Kpxq ě 0. We see G as an electrical network. Let ∆ G denote the discrete Laplacian p∆ G f qpxq " ÿ y"x Cpx, yqpf pyq ´f pxqq.

Let pG G,K px, yqq x,yPV be the massive Green's function G G,K " p´∆ G `Kq ´1. The (massive) real scalar Gaussian free field (GFF) is the centered random Gaussian field on V with covariance G G,K , or equivalently with density

1 pp2πq |V | det G G,K q 1 2 exp ´´1 2 ÿ xPV Kpxqϕpxq 2 ´1 2 ÿ tx,yuPE
Cpx, yqpϕpyq ´ϕpxqq 2 ¯. (5.2) Let X t be the continuous time Markov jump process to nearest neighbors with jump rates given by the conductances. The process X t is also killed by K. Let ζ P p0, `8s be the first time X t gets killed by K. Let p G,K pt, x, yq be the transition probabilities of pX t q 0ďtăζ . Then p G,K pt, x, yq " p G,K pt, y, xq and G G,K px, yq " ż `8 0 p G,K pt, x, yqdt.

Let P t,x,y G,K be the bridge probability measure from x to y, where one conditions on t ă ζ. For x, y P V , let µ x,y G,K be the following measure on paths:

µ x,y G,K p¨q :" ż `8 0 P t,x,y G,K p¨qp G,K pt, x, yqdt.
It is the analogue of (2.1). The total mass of µ x,y G,K is G G,K px, yq, and the image of µ x,y G,K by time reversal is µ y,x G,K . Similarly, one defines the measure on (rooted) loops by µ loop G,K pdγq :"

1 T pγq ÿ xPV µ x,x G,K pdγq,
where T pγq denotes the duration of the loop γ. It is the analogue of (2.2). The measure µ loop G,K has an infinite total mass because it puts an infinite mass on trivial "loops" that stay in one vertex. For α ą 0, one considers Poisson point processes L α G,K of intensity αµ loop G,K . These are (continuous time) random walk loop soups. For details, see [START_REF] Lawler | Random walk loop soup[END_REF][START_REF] Lawler | Random walk: a modern introduction[END_REF][START_REF] Le Jan | Markov loops and renormalization[END_REF][START_REF] Le Jan | Markov paths, loops and fields[END_REF].

For a continuous time path γ on G of duration T pγq and x P V , we denote L x pγq :"

ż T pγq 0 1 γpsq"x ds. Further, L x pL α G,K q :" ÿ γPL α G,K L x pγq.
One has equality in law between pL x pL 1 2

G,K qq xPV and p 1 2 φ G,K pxq 2 q xPV , where φ G,K is the GFF distributed according to (5.2) [START_REF] Le Jan | Markov loops and renormalization[END_REF][START_REF] Le Jan | Markov paths, loops and fields[END_REF]. This is the analogue of (2.4). For general α ą 0, the occupation field pL x pL α G,K qq xPV is the α-permanental field with kernel G G,K [START_REF] Le Jan | Markov loops and renormalization[END_REF][START_REF] Le Jan | Markov paths, loops and fields[END_REF][START_REF] Le Jan | Permanental fields, loop soups and continuous additive functionals[END_REF]. In this sense it is analogous to squared Bessel processes. If pχpxqq xPV P R V is such that ´∆G `K ´χ is positive definite, then E " exp ´ÿ xPV χpxqL x pL α G,K q ¯ı " ˆdetp´∆ G `Kq detp´∆ G `K ´χq ˙α.

(5.3)

EJP 26 (2021), paper 126.

See Corollary 5 in [START_REF] Le Jan | Markov loops and renormalization[END_REF] and Corollary 1, Section 4.1 in [START_REF] Le Jan | Markov paths, loops and fields[END_REF]. Now we proceed with our construction. Fix β ą ´1. Let α " 1 2 dpβ, n " 2q " β`2 2 ą 1 2 .

Let φ G,K be a GFF distributed according to (5.2), and L α´1 2 G,K an independent random walk loop soup. For x P V we set

λ 1 pxq :" 1 ? 2 φ G,K pxq `bL x pL α´1 2 G,K q, λ 2 pxq :" 1 ? 2 φ G,K pxq ´bL x pL α´1 2 G,K q,
and λ :" pλ 1 pxq, λ 2 pxqq xPV . x¨y G,K β,n"2 will denote the expectation with respect to λ. As in Section 4.4, Υ " pγ 1 , . . . , γ JpΥq q will denote a generic family of continuous time paths, this time on the graph G. For x P V , L x pΥq :"

JpΥq ÿ i"1 L x pγ i q,
and LpΥq will denote the occupation field of Υ, x Þ Ñ L x pΥq. Given ν a finite family of positive integers with |ν| even, and x 1 , x 2 , . . . , x mpνq P V , μν,β,x1,...,x mpνq G,K will denote the measure on families of |ν|{2 paths on G obtained by substituting in the polynomial p P ν,β for each variable Y kk 1 , 1 ď k ď k 1 ď mpνq, the measure µ

x k ,x k 1
G,K . The order of the paths will not matter.

Proposition 5.2. The following holds. 1. For every x P V , pλ 1 pxq{ a G G,K px, xq, λ 2 pxq{ a G G,K px, xqq is distributed, up to reordering, according to (3.1) for n " 2. (5.4)

Then the couple p ? 2λpxq{ a G G,K px, xq, ? 2ηλpyq{ a G G,K py, yqq is distributed like the β-Dyson's Brownian motion (4.1) at points 1 and η, for n " 2.

3. Let ν be finite family of positive integers with |ν| even and x 1 , x 2 , . . . , x mpνq P V .

Then

A mpνq ź

k"1 p ν k pλpx k qq E G,K β,n"2 " p P ν,β ppY kk 1 " G G,K px k , x k 1 qq 1ďkďk 1 ďmpνq q.
4. (BFS-Dynkin's isomorphism) Moreover, given F a measurable bounded function on R V , 6. Assume that β ą 0. Let φ 1 and φ 2 be two independent scalar GFFs distributed according to (5.2). L α´1 G,K be a random walk loop soup independent from pφ 1 , φ 2 q, with still α " β`2 2 . Then pλ 1 pxq, λ 2 pxqq xPV is distributed as the ordered family of eigenvalues in the matrix-valued field

A mpνq ź k"1 p ν k pλpx k qqF ´1 2 p 2 pλq ¯EG,K β,n"2 " ż Υ A F ´1 2
¨φ1 pxq b L x pL α´1 G,K q b L x pL α´1 G,K q φ 2 pxq
', x P V.

(5.7)

7. Given another killing measure r K P R V `, non uniformly zero, and λ " p λ1 , λ2 q the field obtained by using r K instead of K, G,K q{ a G G,K py, yqq is distributed as pρp1q, ρpηqq, a two-dimensional marginal of a Bessel process of dimension β `1.

The latter can be seen using the moments, that characterize the finite-dimensional marginals of the Bessel process ρ. In both cases those are pβ `1q{2-permanents, with coefficients given by the matrix (5.8).

(3) This follows by expanding

´1 ? 2 φ G,K px k q `bL x k pL α´1 2 G,K q ¯νk `´1 ? 2 φ G,K px k q ´bL x k pL α´1 2
G,K q ¯νk (5.9)

for every k P 1, mpνq . In this decomposition only the integer powers of L x k pL α´1 2 G,K q survive cancellation. The moments of pφ G,K px k qq 1ďkďmpνq give rise to the Wick part in p P ν,β (sums over partitions in pairs). The moments of pL x k pL

P V , E " r ź i"1 L yi pL α´1 2 G,K qF pLpL α´1 2 G,K qq ı " ÿ σ permutation of t1,2,...,ru ´α´1 2 ¯# cycles of σ ż γ1,...,γr E " F pLpL α´1 2 G,K q`Lpγ 1 q`¨¨¨`Lpγ r qq ı r ź i"1 µ yi,y σpiq G,K pdγ i q.
EJP 26 (2021), paper 126.

Further, by expanding (5.9) for k P 1, mpνq , we get that ś mpνq k"1 p ν k pλpx k qq is actually a polynomial in the variables pφ G,K px k qq 1ďkďmpνq and pL x k pL α´1 2 G,K qq 1ďkďmpνq , the non-integer powers of L x k pL α´1 2 G,K q cancelling out. Moreover,

1 2 p 2 pλq " 1 2 φ 2 G,K `LpL α´1 2 G,K q.
Since the fields φ G,K and LpL α´1 2

G,K q are independent, on gets (5.5) by combining the BFS-Dynkin isomorphism for φ G,K and the BFS-Dynkin isomorphism for LpL α´1 2 G,K q.

(5) Recall that for all three matrix spaces considered, β `2 is the dimension. Given pM pxqq xPV a matrix field distributed according to (5.6), M 0 pxq will denote the matrix M pxq ´1 2 TrpM pxqqI 2 , where I 2 is the 2 ˆ2 identity matrix, so that TrpM 0 pxqq " 0. Since the hyperplane of zero trace matrices is orthogonal to I 2 for the inner product pA, Bq Þ Ñ RepTrpABqq, we get that pM 0 pxqq xPV and pTrpM pxqqq xPV are independent. Moreover, p 1 ? 2 TrpM pxqqq xPV is distributed as the scalar GFF (5.2). As for pTrpM pxq 2 qq xPV , on one hand it is the sum of β `2 i.i.d. squares of scalar GFFs (5.2) corresponding to the entries of the matrices. On the other hand, TrpM pxq 2 q " TrpM 0 pxq 2 q `1 2 TrpM pxqq 2 .

So pTrpM 0 pxq 2 qq xPV is distributed as the sum of β `1

LpL α´1 G,K q `1 4 pφ 2 ´φ1 q 2 has same distribution as LpL α´1 2 G,K q. (7) The density of the GFF φ G, Ă K with respect to φ G,K is ˆdetp´∆ G `r Kq detp´∆ G `Kq ˙1 2 exp ´´1 2 ÿ xPV p r Kpxq ´Kpxqqϕpxq 2 ¯.
The density of LpL

α´1 2 G, Ă K q with respect to LpL α´1 2 G,K q is ˆdetp´∆ G `r Kq detp´∆ G `Kq ˙α´1 2 exp ´´ÿ xPV p r Kpxq ´KpxqqL x pL α´1 2 G,K q ¯,
as can be seen from the Laplace transform (5.3).

EJP 26 (2021), paper 126.

Further questions

Here we present our questions that motivated this paper. The first question is combinatorial. We would like to have the polynomials P ν,β,n given by Definition 4.7 under a more explicit form. The recurrence on polynomials (4.8) is closely related to the Schwinger-Dyson equation (3.3). Its very form suggests that the polynomials P ν,β,n might be expressible as weighted sums over maps drawn on 2D compact surfaces (not necessarily connected), where the maps associated to ν have mpνq vertices with degrees given by ν 1 , ν 2 , . . . , ν mpνq , with powers of n corresponding to the number of faces. This is indeed the case for β P t1, 2, 4u, and this corresponds to the topological expansion of matrix integrals [START_REF] Brézin | Planar diagrams[END_REF][START_REF] Itzykson | The planar approximation[END_REF][START_REF] Mulase | Duality of orthogonal and symplectic matrix integrals and quaternionic Feynman graphs[END_REF][START_REF] Lupu | Topological expansion in isomorphism theorems between matrix-valued fields and random walks[END_REF].

Question 5.3. Is there a more explicit expression for the polynomials P ν,β,n ? Can they be expressed as weighted sums over the maps on 2D surfaces (topological expansion)?

The second question is whether there is a natural generalization of Gaussian beta ensembles and β-Dyson's Brownian motion to electrical networks. For n " 2, such a generalization was given in Section 5.2.

Question 5.4. We are in the setting of an electrical network G " pV, Eq endowed with a killing measure K, as in Section 5.2. Given n ě 3 and β ą ´2 n , is there a distribution on the fields pλpxq " pλ 1 pxq, λ 2 pxq, . . . , λ n pxqqq xPV , with λ 1 pxq ą λ 2 pxq ą ¨¨¨ą λ n pxq, satisfying the following properties?

1. For β P t1, 2, 4u, λ is distributed as the fields of ordered eigenvalues in a GFF with values into n ˆn matrices, real symmetric pβ " 1q, complex Hermitian pβ " 2q, resp. quaternionic Hermitian pβ " 4q.

¯,

where φ G,K is a scalar GFF (5.2). 4. For every x P V , λpxq{ a G G,K px, xq is distributed, up to reordering, as the GβE (3.1).

5. For every x, y P V , the couple p ? 2λpxq{ a G G,K px, xq, ? 2ηλpyq{ a G G,K py, yqq, with η given by (5.4), is distributed as the β-Dyson's Brownian motion (4.1) at points 1 and η. 6. The fields p 1 pλq and λ ´1 n p 1 pλq are independent.

The field 1

? n p 1 pλq is distributed as a scalar GFF (5.2).

8. The field 1 2 `p2 pλq ´1 n p 1 pλq 2 ˘is the α ´1 2 -permanental field with kernel G G,K , where α " 1 2 dpβ, nq, and in particular is distributed as the occupation field of the continuous-time random walk loop soup L α´1 2 G,K . 9. The field 1 2 p 2 pλq is the α-permanental field with kernel G G,K , where α " with a kk 1 P N and for every k P 1, mpνq , 2a kk `ÿ 1ďk 1 ďmpνq k 1 ‰k a kk 1 " ν k , the coefficients of the linear combination being universal polynomials in β and n, not depending on the electrical network and its parameters; see also Question 5.3.

11. Given r K P R V `, non-uniformly zero, and λ " p λ1 , λ2 , . . . , λn q the field associated to the killing measure r K instead of K, the law of λ has the following density with respect to that of λ: 12. λ satisfies a BFS-Dynkin type isomorphism with continuous time random walks (already implied by ( 10)+( 11)).

If the graph G is a tree, the natural generalization λ of the β-Dyson's Brownian motion is straightforward to construct, at least for β ě 0. In absence of cycles, λ satisfies a Markov property, and along each branch of the tree one has the values of a β-Dyson's Brownian motion at different positions. On the random walk loop soup side, ( 8) and ( 9) is ensured by the covariance of the loop soups under the rewiring of graphs; see [START_REF] Le Jan | Markov paths, loops and fields[END_REF]Chapter 7]. Constructing λ on a tree for β P `´2 n , 0 ˘is a matter of constructing the corresponding β-Dyson's Brownian motion. However, if the graph G contains cycles, constructing λ is not immediate, and we have not encountered such a construction in the literature. One does not expect a Markov property, since already for β P t1, 2, 4u one has to take into account the angular part of the matrices.

Appendix: A list of moments for GβE and the corresponding formal polynomials

  (2021), paper 126.

( 4 . 1 )

 41 ; see [4, Theorem 3.1]. By combining Proposition 4.2 with Theorem 2.3, we get a first relation between the β-Dyson's Brownian motion and 1D Brownian local times. Compare it with Le Jan's isomorphism (2.4).

EJP 26 (

 26 2021), paper 126. Isomorphisms of β-Dyson's Brownian motion with Brownian local time We are ready now to express the symmetric moments (4.5).

  (2021), paper 126.

Proposition 4 .Corollary 4 . 11 .

 4411 [START_REF] Dynkin | Local times and quantum fields[END_REF] and Definition 4.10 immediately imply the following. Let β ě 0. Let ν be a finite family of positive integers, with |ν| even. Let x 1 ď x 2 ď ¨¨¨ď x mpνq P R `. Then,

  in a change of measure. Then it remains to describe the law of the stochastic process pλpxqq xě0 under the new measure, and in particular express its symmetric moments. It turns out that under the new measure, the process can still be reduced to a β-Dyson's Brownian motion through a deterministic transformation reminiscent of the scale and time changes for one-dimensional diffusions; see Lemma 4.19.

1

 1

2 d 2 dx 2

 222 ´χ on R ẁith condition 0 in 0. Then for 0 ď x ď y, G R`,χ px, yq " 2u χÓ pxqψ χ pxqu χÓ pyq.

  (2021), paper 126.

2 .

 2 Let x, y P V . Let η " G G,K px, xqG G,K py, yq G G,K px, yq 2 ě 1.

α´1 2 G 2 G

 22 ,K qq 1ďkďmpνq give rise to the permanental part in p P ν,β . (4) The GFF φ G,K satisfies the BFS-Dynkin isomorphism; see [3, Theorem 2.2], [8, Theorems 6.1, 6.2], and [9, Theorem 1]. Moreover, there is a version of BFS-Dynkin isomorphism for the occupation field LpL α´1 ,K q obtained by applying Palm's identity to Poisson point processes; see [23, Theorem 1.3] and [24, Sections 3.4, 4.3]. More precisely, for any y 1 , . . . , y r

1 2 1 p

 11 dpβ, nq, and in particular is distributed as the occupation field of the continuous-time random walk loop soup L α G,K (already implied by (6)+(7)+(8)). 10. The symmetric momentsA mpνq ź k"ν k pλpx k qq E G,K β,n are linear combination of products ź 1ďkďk 1 ďmpνq G G,K px k , x k 1 q a kk 1 ,EJP 26 (2021), paper 126.

  , Section 1.2.2], and [1, Section 4.5]. Fix n ě 2. For λ " pλ 1 , λ 2 , . . . , λ n q P R n , Dpλq will denote the Vandermonde

determinant Dpλq :" ź 1ďjăj 1 ďn pλ j 1 ´λj q.

  By convention, we set p H pλq " 1 and |H| " 0. Note that p H pλq ‰ p 0 pλq. We are interested in the expression of the moments xp ν pλqy β,n . These are 0 if |ν| is not even. For |ν| even,

Isomorphisms of β-Dyson's Brownian motion with Brownian local time Let p ν pλq denote p ν pλq :" mpνq ź k"1 p ν k pλq.

these moments are given by a recurrence known as loop equation or Schwinger-Dyson equation ([15, Lemma 4.13], [16, slide 3{15] and [11, Section 4.1.1]). See the Appendix

every β ą ´2{n, p 2 pλq{2 under GβE has for distribution Gammapdpβ, nq{2, 1q. 3. p 1 pλq and λ ´1 n p 1 pλq under GβE are independent. 4. 1 2 `p2 pλq´1 n p 1 pλq 2 ˘" 1 2 p 2 `λ´1 n p 1 pλq ˘under GβE follows a Gammappdpβ, nq´1q{2, 1q

  2 p2pλq ¯dλ 1 " 0, k xp ν k `νmpνq ´2pλqp pνrq r‰k,mpνq pλqy β,n .

	we get that By performing the change of variables λ " pK `1q	1 2 λ, we get that the expression above
	xλ equals pK `1q ´n 2 ν mpνq 1 p ν pλqy β,n " β R n Z β,n ż |DppK `1q ´1 2 λq| β e ´1 2 p2 `λ´1 n p1p λq ˘´1 n ÿ j"2 B λ ´1 F ν mpνq 1 λ 1 ´λj p ν pλq β,n `pν mpνq ´1qxλ ν mpνq 1 j"1 n 2npK`1q p1p λq 2 ź d λj	´2	p ν pλqy β,n
										`mpνq´1 ÿ k"1 " pK `1q ´1 2 dpβ,nq ν k xλ ν k `νmpνq 1 R n ´2 Z β,n ż	p pνrq r‰k,mpνq pλqy β,n . |Dp λq| β e ´1 2 p2p λq`K 2npK`1q p1p λq 2 n j"1 ź	d λj .
	Analogous relations hold for all other indices j 1 P t2, . . . , nu. By summing over j 1 P t1, 2, . . . , nu, we get Thus,
	xp ν pλqy β,n " @ e ´1 2 Kp2 `λ´1 n p1pλq ˘Dβ,n " pK `1q ´1 2 dpβ,nq @ e	K 2npK`1q p1pλq 2 D β,n " pK `1q ´1 2 pdpβ,nq´1q .
	β So we get the Laplace transform of a Gammappdpβ, nq ´1q{2, 1q r.v. ÿ 1ďjăj 1 ďn B λ ν mpνq ´1 j ´λν mpνq ´1 F j 1 λ j ´λj 1 p ν pλq β,n `pν mpνq ´1qxp ν mpνq ´2pλqp ν pλqy β,n Next is an embryonic version of the BFS-Dynkin isomorphism (Theorem (2.1)) for the
	GβE. One should imagine that the state space is reduced to one vertex, and a particle on
	it gets killed at an exponential time.	`mpνq´1 ÿ
												k"1
	Furthermore,
	ÿ 1ďjăj 1 ďn	λ ν mpνq j λ j ´λj 1 ´1 ´λν mpνq j 1	´1	"	´1 2	pν mpνq ´1qp ν mpνq ´2pλq	`1 2	ν mpνq´1 ÿ i"1	p i´1 pλqp ν mpνq ´1´i pλq.
	So we get (3.3).
	Next are some elementary properties of GβE, which follow from the form of the
	density (3.1).
	Proposition 3.2. The following holds.
	1. For every β ą ´2{n, 1 ? n p 1 pλq under GβE has for distribution N p0, 1q.
	2. For distribution.
	Proof. One can factorize the density (3.1) as
	1 Z β,n	ˇˇD ´λ	´1 n	p 1 pλq ¯ˇˇβ e ´1 2 p2 `λ´1 n p1pλq ˘n´1 ź j"1	d ´λj	´1 n	p 1 pλq ¯ˆe ´1 2n p1pλq 2	dp 1 pλq,
	where
		D ´λ	´1 n	p 1 pλq	¯" ź 1ďjăj 1 ďn	´´λ j 1	´1 n	p 1 pλq	¯´´λ	j	´1 n	p 1 pλq ¯¯" Dpλq.
	This immediately implies (3) and (1). The property (2) is implied by (4), (3) and (1). The
	property (4) can be obtained by computing a Laplace transform. Fix K ą 0. We have that
	@ e ´1 2 Kp2 `λ´1 n p1pλq	˘Dβ,n "	1 Z β,n	R n ż	|Dpλq| β e ´1 2 pK`1qp2 `λ´1 n p1pλq ˘´1 2n p1pλq 2	j"1 n ź	dλ j .
	EJP 26 (2021), paper 126. EJP 26 (2021), paper 126.

ν

Proposition 3.3. Let

  β ą ´2{n. The following holds.1. Let a ě 0. Let h : R n Ñ R be a measurable function such that x|hpλq|y β,n ă `8.Assume that h is a-homogeneous, that is to say hpsλq " s a hpλq for every s ą 0. Let F : r0, `8q Ñ R be a bounded measurable function. Let θ be a r.v. with distribution Gammappdpβ, nq `aq{2, 1q. Then xhpλqF pp 2 pλq{2qy β,n " xhpλqy β,n ErF pθqs.

					(3.4)
	2. In particular, let ν be a finite family of positive integers such that |ν| is even. Let
	T 1 , . . . , T |ν|{2 be an i.i.d. family of exponential times of mean 1, independent of the
	GβE. Then
	xp ν pλqF pp 2 pλq{2qy β,n " xp ν y β,n E	"	xF pp 2 pλq{2 `T1 `¨¨¨`T |ν|{2 qy β,n	‰	.
	Proof. (1) clearly implies (2). It is enough to check (3.4) for F of form F ptq " e ´Kt , with

Brownian motions and the occupation fields of 1D Brownian loop soups

  Section 4.3]. Let β ě 0 and n ě 2. The β-Dyson's Brownian motion is the process EJP 26 (2021), paper 126.pλpxq " pλ 1 pxq, . . . , λ n pxqqq xě0 with λ 1 pxq ě ¨¨¨ě λ n pxq, satisfying the SDE

	1 2 λ, and on the 4.1 β-Dyson's For references on β-Dyson's Brownian motion, see [10, 6, 30, 4, 5], [27, Chapter 9] third line the homogeneity. Further, pK `1q ´1 2 `n`npn´1q β 2 `a˘" Ere ´Kθ s. 4 Isomorphisms for β-Dyson's Brownian motion and [1, dλ j pxq " ? 2dW j pxq `β ÿ j 1 ‰j dx λ j pxq ´λj 1 pxq ,

  Thus, Q p1,1q contains no terms in Y 22 and Q p1,1q " Q 1Ð p1,1q . From the condition (1) we further get Q p1,1q " cpp1, 1q, β, nqY 11 " nY 11 . induction step works as follows. Assume |ν|{2 ě 2. The right hand side of (4.8) involves only families of integers ν with |ν| " |ν| ´2. According to the induction

	The hypotheses, k P 2, mpνq , Q kÐ B BY kk ν	Q kÐ ν ´QkÐ is uniquely determined for every k P 2, mpνq . Thus, for every ν pY kk " 0q is uniquely determined. On top of that,

  These are treated by Proposition 4.2, and taking into account that the one-dimensional marginals of square Bessel processes follow Gamma distributions. Now consider the induction step. Assume |ν|{2 ě 2. Recall the function f k pxq (4.7) for k P 1, mpνq . We have that

  p1,1,...,1q satisfies the Wick's rule for Gaussians: a i ^bi " minpa i , b i q and where the sums run over the m!{p2 Let ν " p2, 2, . . . , 2q, where 2 appears m times. Then Q p2,2,...,2q " 2 m Perm dpβ,nq{2 ppY k^k 1 k^k 1 q 1ďk,k 1 ďm q, P p2,2,...,2q " 2 m Perm dpβ,nq{2 pMq. The expressions for Q p1,1,...,1q and Q p2,2,...,2q are easily obtained by induction on m using Definition 4.7. Alternatively, for β ě 0, one can use that under the law of β-Dyson's Brownian motion, the process pp 1 pλpxqqq xě0 is Gaussian and the process pp 2 pλpxqqq xě0

				m{2
	Q p1,1,...,1q " n	m 2	ÿ	ź	Y ai^bi ai^bi ,
			ptai,biuq 1ďiďm{2	i"1
			partition in pairs
			of 1,m
				m{2
	P p1,1,...,1q " n	m 2	ÿ	ź	M aibi ,
			ptai,biuq 1ďiďm{2	i"1
			partition in pairs
			of 1,m
					m 2 pm{2q!q parti-
	tions in pairs.
	2.

1. Assume m is even, and let ν " p1, 1, . . . , 1q, where 1 appears m times. Then Q p1,1,...,1q EJP 26 (2021), paper 126.

P

where Proof. is dpβ, nq{2-permanental; see Proposition 4.2. This gives the expression of Q p1,1,...,1q and Q p2,2,...,2q for β ě 0. To extend it to general β one can use that the coefficients of the polynomials Q ν are themselves polynomials in β. The expressions for P p1,1,...,1q and P p2,2,...,2q are immediately deducible from those for Q p1,1,...,1q and Q p2,2,...,2q by following Definition 4.10.

  Let ν be a finite family of positive integers, with |ν| even and let 0 ă x 1 ă x 2 ă ¨¨¨ă x mpνq . Let F be a bounded measurable functional on CpR `q. Then In the limiting case when x k " x k´1 for some k P 2, mpνq , q Y k´1 k in P ν has to be replaced by the constant 1 instead of a measure on Brownian paths. For β P t0, 1, 2, 4u, (4.16) reduces to the Gaussian case of Theorem 2.1.Let us first outline our strategy for proving Proposition 4.14. By density arguments it is enough to show (4.16) for functionals F of form

										ν,x1,...,x mpνq R`m ay be signed.
	Next is a version of BFS-Dynkin isomorphism (Theorem (2.1)) for β-Dyson's Brownian
	motion.
	Proposition 4.14. A mpνq ź k"1 p ν k pλpx k qqF	´1 2	p 2 pλq	¯ERβ	,n	"	ż Υ	A F	´1 2	p 2 pλq `LpΥq ¯ERβ	,n	µ ν,x1,...,x mpνq
	Remark 4.16.

R`p dΥq. (4.16) Remark 4.15.

  Let D χ p`8q be the positive r.v. D χ p`8q :" u χÓ p`8q ´1 2 dpβ,nq exp

									´´1 2	0 ż `8	p 2 pλpyqqχpyqdy ¯.	(4.17)
	Then xD χ p`8qy Rβ	,n " 1. Moreover,
	Rβ D χ pxq :" xD χ p`8q|F x y	,n
	" u χÓ pxq ´1 2 dpβ,nq exp	´´1 2	ż x 0	p 2 pλpyqqχpyqdy ¯exp	´1 4	p 2 pλpxqq	u 1 χÓ pxq u χÓ pxq	¯. (4.18)
	Let	M χ pxq :"	1 ? 2	ż x 0	u 1 χÓ pyq u χÓ pyq	n ÿ j"1
							1 2	d 2 dx	u " χu

which is positive non-increasing on R `, with u χÓ p0q " 1. See [24, Section 2.1] for details. Then u χÓ p`8q " lim xÑ`8 u χÓ pxq ą 0. EJP 26 (2021), paper 126.

Lemma 4.17.

  Further consider a change of measure with density D χ p`8q (4.17) on the filtered probability space with filtration pF x q xě0 . Then λ after the change of measure and λ before the change of measure have the same law. χÓ pxq ´1 λj pxq ´uχÓ pxq ´1 λj 1 pxq dx u χÓ pxq 2 .

										`:
										ψ χ pxq "	ż x 0	dy u χÓ pyq 2 .
	Let ψ ´1 χ be the inverse diffeomorphism.
	Lemma 4.19. If λ is a solution to the SDE (4.19), then the process
										´1 u χÓ pψ ´1 χ pxqq	χ pxqq λpψ ´1 ¯xě0
	satisfies the SDE (4.1).
	Proof. The process	´1 u χÓ pxq	¯xě0 λpxq	satisfies
		d ´1 u χÓ pxq	λj pxq ¯" ? u χÓ pxq 2	dW j pxq	`β ÿ j 1 ‰j	u	1
	d	´1 4	p 2 pλpxqq	u 1 χÓ pxq u χÓ pxq	¯" dM χ pxq	`1 2	p 2 pλpxqqχpxqdx
										´1 4	p 2 pλpxqq	u 1 χÓ pxq 2 u χÓ pxq 2 dx	`1 2	dpβ, nq	u 1 χÓ pxq u χÓ pxq	dx,
	and						d	1 2	xM χ , M χ ypxq "	1 4	p 2 pλpxqq	u 1 χÓ pxq 2 u χÓ pxq 2 dx.
	the SDE
				d λj pxq "	?	2dW j pxq	`u1 χÓ pxq u χÓ pxq	λj pxqdx	`β ÿ j 1 ‰j	dx λj pxq ´λ j 1 pxq	,	(4.19)
	with initial condition λp0q " 0. Proof. The existence and uniqueness of strong solutions to (4.19) is given by [4, The-
	orem 3.1]. The rest is a consequence of Girsanov's theorem; see Theorems (1.7) and
	(1.12), Section VIII.1, in [29]. Indeed,
								dxW j pxq, M χ pxqy "	1 ? 2	u 1 χÓ pxq u χÓ pxq	λ j pxqdx.
	Thus, after the change of measure, the
										W j pxq ´1 ? 2	ż x 0	u 1 χÓ pyq u χÓ pyq	λ j pyqdy

χ pxq " exp ´Mχ pxq ´1 2 xM χ , M χ ypxq ¯. Proof. (4.17) and (4.18) follow from the properties of square Bessel processes. See Theorem (1.7), Section XI.1 in [29]. pM χ pxqq xě0 is obviously a (true) martingale, as can be seen with the quadratic variation. Further, Thus d ´Mχ pxq ´1 2 xM χ , M χ ypxq ¯" d logpD χ pxqq. Lemma 4.18. Let be p λpxq " p λ1 pxq, . . . , λn pxqqq xě0 with λ1 pxq ě ¨¨¨ě λn pxq, satisfying for j P 1, n are n i.i.d. standard Brownian motions. EJP 26 (2021), paper 126.

Let ψ χ denote the following diffeomorphism of R

  By 2 ´2u χÓ pxqψ χ pxqu χÓ pyq ¯" χpyq ´2u χÓ pxqψ χ pxqu χÓ pyq Bx 2 ´2u χÓ pxqψ χ pxqu χÓ pyq Let ν be a finite family of positive integers, with |ν| even. Let x 1 ď x 2 ď ¨¨¨ď x mpνq P R `. Then,

		1	B 2					¯,
		2
	1 2	B 2			¯" 1 2	B Bx	´2u 1 χÓ pxqψ χ pxqu χÓ pyq	u χÓ pxq `2 u χÓ pyq	"
								χpxq ´2u χÓ pxqψ χ pxqu χÓ pyq ¯`0,
	and		1 2 ´B Bx	ˇˇx	"y ´B By	ˇˇy	"x	¯´2u

χÓ pxqψ χ pxqu χÓ pyq ¯" 1.

Lemma 4.20. Let p λpxqq xě0 be the solution to (4.19) with λp0q " 0.

  pλ 1 pxq, λ 2 pxqq xPV is distributed like the ordered family of eigenvalues in a GFF with values in 2 ˆ2 real symmetric pβ " 1q, complex Hermitian pβ " 2q, resp. quaternionic Hermitian pβ " 4q matrices, with density proportional

							¯EG,K
					p 2 pλq `LpΥq	β,n"2	μν,β,x1,...,x mpνq G,K	pdΥq. (5.5)
	5. For β P t1, 2, 4u, to
	exp	´´1 2	ÿ xPV	Kpxq TrpM pxq 2 q	´1 2	ÿ tx,yuPE

Cpx, yq TrppM pyq ´M pxqq 2 q ¯. (5.6) EJP 26 (2021), paper 126.

  the density of the law of λ with respect to that of λ is This follows from Proposition 3.2 and the fact that φ G,K pxq{ a G G,K px, xq is distributed according to N p0, 1q, and L x pL

	detp´∆ G `Kq ˆdetp´∆ G `r Kq	˙β`2 2	exp	2 ´´1	xPV ÿ	p r Kpxq ´Kpxqqp 2 pλpxqq	¯.
						α´1 2 G,K q{ a G G,K px, xq according to Gamma `α	1
	2 , 1 ˘.
	(2) One uses the decomposition (4.2) of a β-Dyson's Brownian motion for n " 2. In-deed, p ? 2φ G,K pxq{ a G G,K px, xq, ? 2ηφ G,K pyq{ a ˙. G ˆ2 2 2 2η (5.8)
	Moreover, the couple p	?	2L x pL α´1 2 G,K q{ a	G G,K px, xq,	?	2ηL y pL α´1 2

Proof. (1) G,K py, yqq and pφ R`p 1q, φ R`p ηqq are two Gaussian vectors with the same distribution, with covariance matrix given by

  2. For β " 0, λ is obtained by reordering n i.i.d. scalar GFFs (5.2). 3. As β Ñ ´2 n , λ converges in law to

	´1 ? n	φ G,K ,	1 ? n	φ G,K , . . . ,	1 ? n	φ G,K

  xp 1 pλq 2 y β,n " n, P p1,1q " nY 11 q Y 12 , ¯n¯Y 11 " dpβ, nqY 11 , xp 1 pλq 4 y β,n " 3n 2 , P p1,1,1,1q " n 2 Y 11 q Y 12 Y 33 q Y 34 `2n 2 Y 11 q Y 12 Y 22 q

	xp 2 pλq 2 y β,n "			β 2 4	2 n 4 `2 β	´1	´β 2	¯n3
					`´´1	´β 2	¯2	`2 β 2	¯n2 `2´1	´β 2	¯n
		" dpβ, nqpdpβ, nq `2q,
	P p2,2q "		´β2 4	n 4 `2 β 2	´1	´β 2	¯n3 `´1	´β 2	¯2n 2	¯Y11 Y 22
					`´2	β 2	n 2 `2´1	´β 2	¯n¯Y 11 2	q Y 2 12 ,
	xp 3 pλqp 1 pλqy β,n " 3	β 2	n 2 `3´1	´β 2	¯n,
			P p3,1q "	´3 β 2	n 2 `3´1	´β 2	¯n¯Y 11 2	q Y 12 ,
			P p1,3q "	´3 β 2	n 2 `3´1	´β 2	¯n¯Y 11 q Y 12 Y 22 ,
	xp 4 pλqy β,n " 2	β 2 4	n 3 `5 β 2	´1	´β 2	¯n2 `´β 2	`3´1	´β 2	¯2¯n ,
	P p4q "	´2 β 2 4	n 3 `5 β 2	´1	´β 2	¯n2 `´β 2	`3´1	´β 2	¯2¯n¯Y 11 , 2
	xp 3 pλq 2 y β,n " 12	β 2 4	n 3 `27	β 2	´1	´β 2	¯n2 `´3	β 2	`15 ´1	´β 2	¯2¯n ,
	P p3,3q " 9 ´β2 4	n 3 `2 β 2	´1	´β 2	¯n2 `´1	´β 2	¯2n ¯Y2 11	q Y 12 Y 22
		`3´β2 4	n 3 `3 β 2	´1	´β 2	¯n2 `´β 2	`2´1	´β 2	¯2¯n¯Y 11 3	q Y 3 12 .
	xp 2 pλqy β,n "				β 2	n 2 `´1	´β 2	¯n " dpβ, nq,
	P p2q "			´β 2	n 2 `´1	´β 2
												Y 2 23	q Y 34 ,
	xp 2 pλqp 1 pλq 2 y β,n "		β 2	n 3 `´1	´β 2	¯n2 `2n,
	P p2,1,1q "		´β 2	n 3 `´1	´β 2	¯n2	¯Y11 Y 22 q Y 23 `2nY 2 11	q Y 2 12	q Y 23 ,
	P p1,2,1q "		´β 2	n 3 `´1	´β 2	¯n2 `2n ¯Y11 q Y 12 Y 22 q Y 23 ,
	P p1,1,2q "		´β 2	n 3 `´1	´β 2	¯n2	¯Y11 q Y 12 Y 33 `2nY 11 q Y 12 Y 22 q Y 2 23 ,
	EJP 26 (2021), paper 126.

https://www.imstat.org/ejp

* This work was supported by the French National Research Agency (ANR) grant within the project MALIN (ANR-16-CE93-0003).

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