Stéphane Ouvry
email: stephane.ouvry@u-psud.fr

Alexios P Polychronakos
email: apolychronakos@ccny.cuny.edu

Exclusion statistics for particles with a discrete spectrum

We formulate and study the microscopic statistical mechanics of systems of particles with exclusion statistics in a discrete one-body spectrum. The statistical mechanics of these systems can be expressed in terms of effective single-level grand partition functions obeying a generalization of the standard thermodynamic exclusion statistics equation of state. We derive explicit expressions for the thermodynamic potential in terms of microscopic cluster coefficients and show that the mean occupation numbers of levels satisfy a nesting relation involving a number of adjacent levels determined by the exclusion parameter. We apply the formalism to the harmonic Calogero model and point out a relation with the Ramanujan continued fraction identity and appropriate generalizations.

Introduction

Anyons statistics [START_REF] Leinaas | Feynman Functional Integrals for Systems of Indistinguishable Particles[END_REF], and the related topics of fractional and exclusion statistics [START_REF] Haldane | Fractional statistics in arbitrary dimensions: A generalization of the Pauli principle[END_REF], are enjoying renewed popularity since the 2020 announcement of experimental confirmations [START_REF] Bartolomei | Fractional statistics in anyon collisions[END_REF] of lowest Landau level (LLL) anyon excitations with statistics 1/3 in fractional quantum Hall samples at filling 1/3. The statistics relevant to these experimental results are abelian, which is indeed the simplest and, in principle, easiest to observe among non conventional statistics, leaving aside possible more elaborate nonabelian extensions. On the other hand, from a theoretical perspective, anyon statistics remains truly fascinating because it arises at the microscopic level as the natural, and from this point of view unique, generalization of the standard Bose and Fermi statistics, and not as an ad hoc artificial construction. Exclusion statistics falls in this category because, as it will be detailed below, it can also be understood and derived in the context of anyons statistics. For all these reasons, it is of interest to study the manifestation of exclusion statistics in its microscopic setting, and this is the aim of the present work.

The concept of exclusions statistics, as introduced by Haldane [START_REF] Haldane | Fractional statistics in arbitrary dimensions: A generalization of the Pauli principle[END_REF], essentially holds only at the thermodynamic limit, or in situations where the Hilbert space consists of degenerate states as in the LLL of an external magnetic field. Its ab initio microscopic formulation (i.e., starting from a 1-body spectrum and filling it with particles obeying exclusion statistics, as can be done in the standard Bose and Fermi cases) is, in general, impossible for non integer (fractional) exclusion parameter g, since exact many-body states cannot be defined. Microscopic concepts become, in principle, accessible when the exclusion parameter g is constrained to be an integer, but are again to a large extent ambiguous as they are not invariant under Hilbert space state reparametrizations. This is to be contrasted to anyon statistics, defined in 2 dimensions in terms of a microscopic N -anyon quantum Hamiltonian [START_REF] Wilczek | Magnetic flux, angular momentum, and statistics[END_REF], with a statistical (exchange) anyon parameter α taking continuous values in [0, 2). All physical observables depend periodically on α with period 2, since α → α + 2 corresponds to a (nonsingular) phase redefinition of the wavefunction. Thus, α = 0 corresponds to bosons, α = 1 to fermions, and α = 2 again to bosons. It is nevertheless well known that Haldane/exclusion statistics and anyon statistics are intimately related when one considers anyons projected onto the LLL of an external magnetic field, the so-called LLL-anyon model [START_REF] Dasnières De Veigy | Equation of State of an Anyon gas in a Strong Magnetic Field[END_REF]. Here exclusion statistics manifests itself for a system of particles with a continuous degenerate 1-body density of states in the LLL.

Our present aim is to focus on exclusion statistics in its most general microscopic setting, i.e., for particles with a discrete 1-body spectrum 1 , 2 , . . . , q . In this situation, exclusion statistics requires a natural (and dynamically relevant) ordering of the 1-body energy eigenstates, as, for example, when a single quantum number k indexes them. This is typically the case for 1-dimensional systems, but exclusion statistics can be relevant in more general settings provided that a principal quantum number induces a natural ordering of the spectrum, as it happens for example in the case of the 2-dimensional LLL-anyon model properly regularized at long distances by a harmonic well.

In the next section we will give a summary of anyon statistics in the context of the LLL-anyon model [START_REF] Dasnières De Veigy | Equation of State of an Anyon gas in a Strong Magnetic Field[END_REF] and will review the intimate relation of Haldane exclusion statistics to LLL-anyon statistics. We will also consider the thermodynamics of the Calogero model [START_REF] Polychronakos | Nonrelativistic Bosonization and Fractional Statistics[END_REF][START_REF] Isakov | Fractional statistics in one dimension: modeling by means of 1/x 2 interaction and statistical mechanics[END_REF][START_REF]One-dimensional Statistical Mechanics for Identical Particles: the Calogero and Anyon Cases[END_REF] as yet another example of a microscopic realization of exclusion statistics. This will allow for a general definition of the thermodynamics relations governing a gas of particles with exclusion statistics and a continuous 1-body density of states.

We will then move to the situation of present interest, the statistical mechanics of a gas of particles in q discrete 1-body energy levels 1 , 2 , . . . , q in a specific ordering with an integer exclusion parameter g. Bosons correspond to g = 0, with no exclusion, while g = 1 is the Fermi case where no more than one particle per quantum state is allowed. For g-exclusion, levels can again be occupied by at most one particle, but in addition at least g -1 unoccupied levels must exist between any two occupied states.

Our main results, presented in section 3, are that the statistical mechanics of these systems can be written as a generalization of the thermodynamics relations governing LLL-anyon or Calogero particles, in terms of effective single-state grand partition functions obeying a generalization of the standard exclusion statistics equation of state. Interestingly, two distinct such effective partition functions can be defined, termed "forward" and "backward", obeying different equations of state but leading to identical statistical mechanics. We also define appropriate thermodynamic potentials and give their explicit expressions in terms of cluster coefficients that involve sums over generalized partitions of the particle number. The mean occupation number of each 1-body level is then expressed in terms of the effective grand partition functions and shown to satisfy nesting relations that involve g nearby levels. Finally, in section 4 the formalism is applied to the specific case of the harmonic Calogero system, and a relation to the Ramanujan continued fraction identity is pointed out for g = 2 and related generalizations for g > 2. We conclude with some directions for future research.

LLL-anyons and exclusion thermodynamics

We start with a review of LLL-anyons in an isotropic harmonic trap of frequency ω and their intimate relation with exclusions statistics. The spectrum of the N -anyon system is

E N = (ω t -ω c ) N i=1 l i + α N (N -1) 2 + N ω t , 0 ≤ l 1 ≤ l 2 ≤ ... ≤ l N (1)
where ω c is half the cyclotron frequency, ω t = ω 2 c + ω 2 , the l i are 1-body LLL angular momentum quantum numbers in 2 dimensions, and α is understood to be in the interval [0, 2). We stress that the harmonic well is introduced as a long distance regulator to split the degeneracy of the LLL (when ω = 0 the LLL-anyon spectrum (1) reduces trivially to E N = N ω c ) and will be taken to vanish in the thermodynamic limit. When α : 0 → 1 the spectrum [START_REF] Leinaas | Feynman Functional Integrals for Systems of Indistinguishable Particles[END_REF] interpolates continuously between the harmonic LLL-Bose and LLL-Fermi spectra. Going beyond α = 1, we note that due to the presence of the magnetic field the Bose limit α → 2 -differs from the standard Bose case α = 0 because of some missing states. At α = 2 these missing states are restored thanks to excited states merging into the LLL ground state.

The LLL-anyon spectrum (1) is, in fact, identical in form to the spectrum of the 1-dimensional harmonic Calogero model with interaction strength α(α -1). The relation between the two models is well established (see [START_REF] Ouvry | On the Relation between the Anyon and the Calogero Models[END_REF] for an explicit mapping) and the exclusion statistics interpretation of their statistical mechanics is a common feature. We will come back later in this section to the well-known connection of the Calogero model to exclusion statistics in the thermodynamic limit βω → 0, and in section 4 we will examine in more detail its microscopic statistical mechanics in the dicrete case (βω = 0) using the results of section 3.

From the harmonic LLL-anyon spectrum (1) the N -body partition function Z N , grand partition function Z and cluster coefficients b n , defined as

Z = ∞ N =0 Z N z N , ln Z = ∞ n=1 b n z n
where z is the fugacity, can be calculated. To probe the effect of the statistics, the cluster coefficients b n were studied in [START_REF] Dasnières De Veigy | Equation of State of an Anyon gas in a Strong Magnetic Field[END_REF] and found to leading order in βω (that is, for β(ω t -ω c ) small) to be

b n = 1 β(ω t -ω c ) e -nβωc 1 n 2 n-1 k=1 k -nα k (2)
Taking then the thermodynamic limit βω → 0, (which in the present case amounts to 1/(nβ 2 ω 2 ) → V /λ 2 , where V is the macroscopically large 2-dimensional volumehere, area-of the system and λ the thermal wavelength), the LLL-anyon thermodynamic potential follows ln Z = N L ln y

where N L = BV /Φ 0 is the LLL degeneracy, i.e., the number of magnetic flux quanta in the volume -here, area -of the system. The function y was found to satisfy

y -ze -βωc y 1-α = 1 , (4)
so that

y = ∞ N =0 z N e -βN ωc N k=2 k -N α k
(by definition k 2 k=k 1 (. . .) = 1 when k 2 < k 1 ). Using Z = y N L , it was deduced [START_REF] Dasnières De Veigy | Equation of State of an Anyon gas in a Strong Magnetic Field[END_REF]

Z N = e -βN ωc N L N k=2 k + N L -N α -1 k
Z N is the LLL-anyon N -body partition function for N degenerate anyons at energy N ω c , thus identifying their degeneracy as

N L N k=2 k + N L -N α -1 k = N L N ! (N + N L -N α -1)! (N L -N α)! (5)
(factorials for fractional argument are defined in terms of the corresponding Γ-functions.)

Let us now allow α to take integer values beyond the interval [0, 2): the degeneracy (5) counts the number of ways to put N particles in N L degenerate quantum states in a circular configuration such that there are at least α -1 empty states between any two occupied states [START_REF] Polychronakos | Probabilities and path-integral realization of exclusion statistics[END_REF]. This is the hallmark of exclusion statistics with exclusion parameter α. In particular, α = 2 describes a Bose gas but with nontrivial α = 2 exclusion. Indeed, Haldane exclusion statistics postulates that given G single-particle states already populated by N -1 particles the number of quantum states available for an additional N th particle is G -(N -1)g (this is heuristic, and somewhat misleading [START_REF] Polychronakos | Probabilities and path-integral realization of exclusion statistics[END_REF]) where g is the exclusion parameter. g would need to be an integer for this to be meaningful, but this requirement can be dropped in the thermodynamic limit where G and N become large. Starting from the standard Bose degeneracy for N bosons in G quantum states

(N + G -1)! N !(G -1)!
Haldane encoded exclusion by replacing G by G -(N -1)g to propose the N -body exclusion degeneracy

(N + G -(N -1)g -1)! N !(G -(N -1)g -1)! ( 6
)
When g is a positive integer, this is the number of ways to put on a line N particles in G quantum states in such a way there are at least g -1 empty states in between two occupied states. This is the same as the LLL-anyon counting (5) discussed above upon setting g = α and G = N L in the Haldane counting and placing the states on a line rather than a circle, which is irrelevant in the thermodynamic limit where G = N L → ∞. So Haldane exclusion statistics is identical to LLL-anyon statistics provided that the anyonic exchange statistical parameter α is allowed to take integer values beyond [0, 2). Not surprisingly, in view of this intimate relation between LLL-anyon statistics and Haldane exclusion statistics, the LLL-anyon thermodynamic (3) and ( 4) can be directly recovered [START_REF] Wu | Statistical distribution for generalized ideal gas of fractional-statistics particles[END_REF] from the Haldane Hilbert space counting [START_REF] Polychronakos | Nonrelativistic Bosonization and Fractional Statistics[END_REF].

In conclusion, exclusion/LLL-anyon thermodynamics amounts to

ln Z = N L ln y , y -ze -βωc y 1-α = 1 (7)
Let us focus on the mean particle number N = z∂ ln Z/∂z or, equivalently, on the LLL filling factor

ν = N N L = z ∂ ln y ∂z
Using y -ze -βωc y 1-α = 1 we can obtain

y = 1 + ν 1 -α ν and therefore also ze -βωc = ν (1 + (1 -α)ν) 1-α (1 -α ν) α
and from ln Z = N L ln y we obtain the equation of state

ln Z = βP V = N L ln 1 + ν 1 -α ν
with a critical filling ν = 1/α where the pressure diverges [START_REF] Dasnières De Veigy | Equation of State of an Anyon gas in a Strong Magnetic Field[END_REF]. Interestingly, at the critical filling the N -body LLL-anyon wave function is nondegenerate

ψ = i<j (z i -z i ) α e -ωc N i=1 zi z i /2 (8)
which coincides with the Laughlin wavefunction when α = 2m + 1, m = 1, 2, . . ., encoding α-exclusion for an incompressible N -anyon state in the LLL.

The above thermodynamics readily generalizes to exclusion statistics systems with arbitrary 1-body density of states ρ( ). In the thermodynamic limit we can apply relations [START_REF] Isakov | Fractional statistics in one dimension: modeling by means of 1/x 2 interaction and statistical mechanics[END_REF] for states around energy , with the number of states N L substituted by ρ( )d , obtaining

ln Z = ∞ 0 ρ( ) ln y( ) d , y( ) -ze -β y( ) 1-α = 1 (9)
The LLL-anyon result ( 7) is recovered for ρ( ) = N L δ( -ω c ). Similarly, for the mean occupation number n( ) per level at energy and the mean particle number N

n( ) = z ∂ ln y( ) ∂z , N = ∞ 0 ρ( )n( ) d
we obtain, in view of (9),

y( ) = 1 + n( ) 1 -α n( ) ze -β = n( ) [1 + (1 -α)n( )] 1-α [1 -α n( )] α (10)
and therefore

ln Z = ∞ 0 ρ( ) ln 1 + n( ) 1 -α n( ) d (11)
A question naturally arises about the existence of other microscopic quantum models with the same kind of statistics. One known example, as already mentioned, is the Calogero model. The harmonic Calogero model on the 1-dimensional line with inversesquare 2-body interactions and a confining harmonic potential is described by the Hamiltonian

H N = - 1 2 N i= ∂ 2 ∂x 2 i -α(1 -α) i<j 1 (x i -x j ) 2 + 1 2 ω 2 N i=1 x 2 i ( 12
)
Its spectrum is given by

E N = ω N i=1 l i + α N (N -1) 2 + N 2 0 ≤ l 1 ≤ l 2 ≤ ... ≤ l N ( 13
)
where the l i are now 1-dimensional "pseudo-excitation numbers" labeling the 1-body harmonic eigenstates. This spectrum directly follows from the the LLL-anyon spectrum (1) upon letting ω c → 0, i.e. in the absence of the external magnetic field, up to a trivial N -body energy shift N ω/2. Note that ω now plays the role of a 1-dimensional longdistance regulator in the Calogero case, with the difference that the limit ω → 0 does not lead anymore to an infinitely degenerate LLL but, rather, to free 1-dimensional particles with generalized statistics manifesting through their scattering phase shift [START_REF] Polychronakos | Nonrelativistic Bosonization and Fractional Statistics[END_REF].

All the thermodynamic considerations presented in this section in the context of LLL anyons apply equally to Calogero particles. In particular, when ω c = 0 the cluster coefficients (2) in the thermodynamic limit βω → 0 lead to [START_REF] Polychronakos | Probabilities and path-integral realization of exclusion statistics[END_REF][START_REF] Ouvry | On the Relation between the Anyon and the Calogero Models[END_REF][START_REF] Wu | Statistical distribution for generalized ideal gas of fractional-statistics particles[END_REF] with density of states ρ( ) = L π √ 2 i.e., a free 1-dimensional density of states on a space of macroscopically large length L.

So the 2-dimensional LLL-anyon and the 1-dimensional Calogero models both share the same exclusion statistics/thermodynamics [START_REF] Polychronakos | Probabilities and path-integral realization of exclusion statistics[END_REF][START_REF] Ouvry | On the Relation between the Anyon and the Calogero Models[END_REF][START_REF] Wu | Statistical distribution for generalized ideal gas of fractional-statistics particles[END_REF]. This is of course not surprising: as already stressed above one can show [START_REF] Ouvry | On the Relation between the Anyon and the Calogero Models[END_REF] that the 1-dimensional Calogero model is a particular projection of the 2-dimensional anyon model with same exclusion statistics and a free 1-body density of states on the line stemming in the LLL from the dimensional reduction lim ωc→0 N L δ( -ω c ) = L π √ 2 induced by the vanishing magnetic field. All said and done, [START_REF] Polychronakos | Probabilities and path-integral realization of exclusion statistics[END_REF][START_REF] Ouvry | On the Relation between the Anyon and the Calogero Models[END_REF][START_REF] Wu | Statistical distribution for generalized ideal gas of fractional-statistics particles[END_REF] can be viewed in full generality as the defining thermodynamic relations for particles with exclusion statistics α and a continuous 1-body density of states ρ( ). However, specific dynamical systems manifesting these thermodynamics, where exclusion statistics is microscopically realized in terms of N -body quantum Hamiltonians, are limited to the two cases above -the LLL-anyon and the Calogero models. In the next section we consider in general such systems, defined through a set of 1-body energy levels and an exclusion rule in filling them for many-body states, and derive their exact microscopic statistical mechanics, rather than their thermodynamics.

Exclusion statistics for a discrete 1-body spectrum

We now turn to exclusion statistics for a discrete density of states, that is for a 1-body spectrum (k), k = 1, . . . , q. As stressed in the Introduction, we assume a natural ordering of levels (1), (2), . . . , (q) in terms of the principal quantum number k in (k), which is relevant to the definition of exclusion. The Boltzmann factor for the energy level (k) is

s(k) = e -β (k)
We call s(k) the spectral function. Our focus is to derive relations analogous to [START_REF] Polychronakos | Probabilities and path-integral realization of exclusion statistics[END_REF][START_REF] Ouvry | On the Relation between the Anyon and the Calogero Models[END_REF][START_REF] Wu | Statistical distribution for generalized ideal gas of fractional-statistics particles[END_REF] for particles with exclusion statistics in the above discrete spectrum.

g = 2

As a warmup, we start with the simplest case beyond Fermi statistics, i.e., g = 2 exclusion.

The N -body partition function is, by definition,

Z N = q-2N +2 k 1 =1 k 1 k 2 =1 • • • k N -1 k N =1 s(k 1 + 2N -2)s(k 2 + 2N -4) • • • s(k N -1 + 2)s(k N )
where the cumulative +2 shifts in the arguments of the spectral function enforce g = 2 exclusion: adjacent 1-body levels k and k + 1 cannot be populated. The grand partition function follows as

Z 1,q = 1 + (q+1)/2 N =1 Z N z N
(the indices 1, q refer to the first and last levels in the spectrum). The cluster expansion of the grand potential is

ln Z 1,q = ∞ n=1 b n z n with the cluster coefficients b n calculated to be b n = (-1) n-1 l 1 ,l 2 ,...,l j composition of n j≤q c 2 (l 1 , l 2 , . . . , l j ) q-j+1 k=1 s l j (k + j -1) • • • s l 2 (k + 1)s l 1 (k) (14)
In ( 14), the sum is over all compositions (i.e., ordered partitions) of the integer n, with the number of parts j of a given composition, by definition smaller than or equal to n, also constrained to be smaller than or equal to q, the number of available 1-body quantum states. The combinatorial coefficients c 2 (l 1 , l 2 , . . . , l j ) are [START_REF] Ouvry | The algebraic area of closed lattice random walks[END_REF] c 2 (l 1 , l 2 , . . . , l j ) =

l 1 +l 2 l 1 l 1 + l 2 l 2 l 2 +l 3 l 2 l 2 + l 3 • • • l j-1 l j-1 +l j l j-1 l j-1 + l j .
By rearranging the sums in ( 14), the cluster coefficient can also take the alternative form

b n = (-1) n-1 q k=1 l 1 ,l 2 ,...,l j composition of n j≤q-k+1 c 2 (l 1 , l 2 , . . . , l j )s(k + j -1) l j • • • s(k + 1) l 2 s(k) l 1 .
A useful observation [START_REF] Ouvry | The algebraic area of closed lattice random walks[END_REF][START_REF] Ouvry | Hamiltonian and exclusion statistics approach to discrete forward-moving paths[END_REF] is that Z 1,q can be expressed as the secular determinant of the off-diagonal (q + 1) × (q + 1) matrix

H 1,q =          0 -1 0 • • • 0 0 s(1) 0 -1 • • • 0 0 0 s(2) 0 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 • • • 0 -1 0 0 0 • • • s(q) 0          (15)
(this H 1,q is one of several equivalent choices). Specifically,

Z 1,q = det(1 q+1 + z 1/2 H 1,q )
We also define the general truncated grand partition functions

Z k,k (1 ≤ k ≤ k ≤ q) as Z k,k = det(1 k -k+2 + z 1/2 H k,k ) (16)
with

H k,k the truncated (k -k + 2) × (k -k + 2) matrix H k,k =          0 -1 0 • • • 0 0 s(k) 0 -1 • • • 0 0 0 s(k + 1) 0 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 • • • 0 -1 0 0 0 • • • s(k ) 0         
i.e., the matrix (15) for a truncated 1-body spectrum (k), (k + 1), . . . , (k ) starting at level k and ending at level k . We also define that, trivially, Z k,k = 1 when k > k .

The grand partition function Z k,k can be expressed as

Z k,k = 1 + (k -k+2)/2 N =1 Z N (k, k )z N
where Z N (k, k ) stands for the g = 2 exclusion N -body partition function for the truncated 1-body spectrum (k), (k + 1), . . . , (k ), i.e.,

Z N (k, k ) = k -2N +2 k 1 =k k 1 k 2 =k • • • k N -1 k N =k s(k 1 + 2N -2)s(k 2 + 2N -4) • • • s(k N -1 + 2)s(k N ) .
It also follows that

ln Z k,k = - ∞ n=1 (-z) n l 1 ,l 2 ,...,l j composition of n j≤k -k+1 c 2 (l 1 , l 2 , . . . , l j ) k -j+1 l=k s l j (l + j -1) • • • s l 2 (l + 1)s l 1 (l) (17)
where the number of parts j is now bounded by the number k -k +1 of available quantum states in the truncated spectrum (here, as well as in all other similar cluster expressions which will appear below).

Expanding the determinant (16) in terms of its first row we obtain

Z 1,q = Z 2,q + zs(1)Z 3,q (18)
The recursion (18) is self explanatory and could have been written directly: because of g = 2 exclusion, the full grand partition function is the sum of a grand partition for a 1-body spectrum starting at level (2) with level (1) empty, and of zs(1) (level (1) filled) times the one starting at level (3). This is the basis for a recursion scheme: expanding the determinants Z k,q and Z 1,k , as defined in ( 16), in terms of their first and last row, respectively, yields the recursion relations

Z k,q = Z k+1,q + zs(k)Z k+2,q , Z 1,k = Z 1,k-1 + zs(k)Z 1,k-2 (19)
We now introduce "forward" and "backward" effective single-level grand partition functions y + (k) and y -(k), respectively, as

y + (k) = Z k,q Z k+1,q ; y -(k) = Z 1,k Z 1,k-1 (20)
In terms of them, the full grand partition function achieves a product form

Z 1,q = q k=1 y + (k) = q k=1 y -(k)
Moreover, the recursion relations (19) imply the nesting relations

y ± (k) - zs(k) y ± (k ± 1) = 1 (21)
where y + (0) = y -(0) = y + (q + 1) = y -(q + 1) = 1 is understood. We also note that, as a consequence of (17, 20), the cluster expansions of y + (k), y -(k) are

ln y + (k) = - ∞ n=1 (-z) n l 1 ,l 2 ,...,l j composition of n j≤q-k+1 c 2 (l 1 , l 2 , . . . , l j )s(k) l 1 s(k + 1) l 2 • • • s(k + j -1) l j ln y -(k) = - ∞ n=1 (-z) n l 1 ,l 2 ,...,l j composition of n j≤k c 2 (l 1 , l 2 , . . . , l j )s(k -j + 1) l j • • • s(k -1) l 2 s(k) l 1 (22) 10
We reach the conclusion that for a discrete 1-body spectrum the statistical mechanics of g = 2 exclusion particles amounts to

ln Z 1,q = q k=1 ln y ± (k) , y ± (k) - zs(k) y ± (k ± 1) = 1
These are indeed (9) when α = 2 but with the proviso that, because of the discreteness of the spectrum, ln Z 1,q is now a discrete sum instead of a continuous integral, and a discrete shift k → k ± 1 materializes in the argument of y + (k) or y -(k).

Turning to the mean particle number N , it is given by

N = z ∂ ln Z 1,q ∂z = q k=1 n k
where the mean occupation number n k of the energy level (k) is, by definition,

n k = s(k) ∂ ln Z 1,q ∂s(k) (23)
Using the expansion relation

Z 1,q = zs(k)Z 1,k-2 Z k+2,q + Z 1,k-1 Z k+1,q ,
(another self-explanatory identity of which both (18) and ( 19) are special cases) n k can be expressed directly in terms of truncated grand partition functions

n k = zs(k) Z 1,k-2 Z k+2,q Z 1,q ⇔ 1 -n k = Z 1,k-1 Z k+1,q Z 1,q . (24)
From this we can obtain the nesting relation for n k (see Appendix for the proof)

zs(k) = n k (1 -n k ) (1 -n k-1 -n k )(1 -n k -n k+1 ) (25)
and using (20) we can express y + (k) and

y -(k) in terms of n k y ± (k) = 1 + n k 1 -n k -n k∓1 (26)
and finally the thermodynamic potential

ln Z 1,q = q k=1 ln 1 + n k 1 -n k-1 -n k = q k=1 ln 1 + n k 1 -n k -n k+1 (27)
(25, 26, 27) are the generalization of the α = 2 LLL-anyon/Calogero models [START_REF] Ouvry | On the Relation between the Anyon and the Calogero Models[END_REF][START_REF] Wu | Statistical distribution for generalized ideal gas of fractional-statistics particles[END_REF] thermodynamics relations for the discrete spectrum at hand, i.e., with the discrete shifts k → k ± 1 in the arguments of n. (27) is the equation of state of a gas of particles with exclusion g = 2 and populating discrete energy levels (k) whose occupation numbers n k are constrained by n k + n k+1 ≤ 1.

Note that the nesting relation (25) does not allow for finding the n k in an iterative way, starting either at k = 1 or at k = q, since already for k = 1 it involves n 1 and n 2 and similarly for k = q. On the other hand, (21) allows the calculation of y + (k) and y -(k) iteratively, starting from k = 1 for y -(k) and k = q for y + (k):

y -(1) = 1 + zs(1) , y -(2) = 1 + zs(2) 1 + zs(1)
, . . . y + (q) = 1 + zs(q) , y + (q -1) = 1 + zs(q-1) 1 + zs(q) , . . .

From these and (25) we can express n k as

n k = y + (k) -1 y + (k) + y -(k -1) -1 = y -(k) -1 y -(k) + y + (k + 1) -1

General g

For general integer exclusion parameter g the N -body partition function reads

Z N = q-gN +g k 1 =1 k 1 k 2 =1 • • • k N -1 k N =1 s(k 1 + gN -g)s(k 2 + gN -2g) • • • s(k N -1 + g)s(k N ) (28)
where the +g shift in the nested sum indices enforces g-exclusion. The grand partition function Z 1,q follows as

Z 1,q = 1 + (q+g-1)/g N =1 Z N z N
and the cluster expansion as

ln Z 1,q = - ∞ n=1 (-z) n l 1 ,l 2 ,...,l j g-composition of n j≤q c g (l 1 , l 2 , . . . , l j ) q-j+1 k=1 s l j (k + j -1) • • • s l 2 (k + 1)s l 1 (k) (29)
which is the g-generalization of ( 14). Here the sum is over all g-compositions [START_REF] Ouvry | The algebraic area of closed lattice random walks[END_REF] of the integer n which are defined as the usual compositions but where now up to g -2 successive l i can be zero. The c g (l 1 , l 2 , . . . , l j ) are given as

c g (l 1 , l 2 , . . . , l j ) = (l 1 + • • • + l g-1 -1)! l 1 ! • • • l g-1 ! j-g+1 i=1 l i + • • • + l i+g-1 -1 l i+g-1 = j-g+1 i=1 (l i + • • • + l i+g-1 -1)! j-g i=1 (l i+1 + • • • + l i+g-1 -1)! j i=1 1 l i !
As before we can express Z 1,q as the secular determinant of a (q + g -1) × (q + g -1) matrix

H 1,q =                  0 -1 0 • • • 0 0 0 • • • 0 0 0 -1 • • • 0 0 0 • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . s(1) 0 0 . . . -1 0 0 • • • 0 0 s(2) 0 . . . 0 -1 0 • • • 0 0 0 s(3) . . . 0 0 -1 • • • 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 • • • s(q -1) 0 0 • • • -1 0 0 0 • • • 0 s(q) 0 • • • 0                  ( 30
)
where there are now g -1 successive vanishing diagonals, realizing g-exclusion in the matrix representation. Then

Z 1,q = det(1 q+g-1 + z 1/g H 1,q )
Proceeding as in the g = 2 case, we define the general truncated grand partition functions 30) for the truncated 1-body spectrum (k), (k + 1), . . . , (k ) starting at level k and ending at level k , and, as before, Z k,k = 1 when k > k . The grand partition function Z k,k can be expressed as

Z k,k (1 ≤ k ≤ k ≤ q) as Z k,k = det(1 k -k+g + z 1/g H k,k ) with H k,k the (k -k + g) × (k -k + g) matrix (
Z k,k = 1 + (k -k+g)/g N =1 Z N (k, k )z N
where Z N (k, k ) stands for the truncated g-exclusion N -body partition function

Z N (k, k ) = k -gN +g k 1 =k k 1 k 2 =k • • • k N -1 k N =k s(k 1 + gN -g)s(k 2 + gN -2g) • • • s(k N -1 + g)s(k N )
We can again set up a recursion scheme by expanding the determinants Z k,q and Z 1,k in terms of their first and last row, respectively, obtaining the recursion relations

Z k,q = Z k+1,q + zs(k)Z k+g,q , Z 1,k = Z 1,k-1 + zs(k)Z 1,k-g (31)
of clear g-exclusion statistics origin. We define forward and backward effective single-level grand partition functions y + (k) and y -(k) as in the g = 2 case (20), and the full grand partition function is again expressed as their product (21). The cluster expansion for Z k,k is as in (29) but for the truncated spectrum, that is

ln Z k,k = - ∞ n=1 (-z) n l 1 ,l 2 ,...,l j g-composition of n j≤k -k+1 c g (l 1 , l 2 , . . . , l j ) k -j+1 l=k s l j (l + j -1) • • • s l 2 (l + 1)s l 1 (l) (32)
leading to expressions analogous to (22) for ln y + (k) and ln y -(k) with c 2 (l 1 , l 2 , . . . , l j ) replaced by c g (l 1 , l 2 , . . . , l j ).

The recursion relation (31) implies the nesting relations

y ± (k) - zs(k) g-1 i=1 y + (k ± i) = 1 (33)
We reach the conclusion that the statistical mechanics for g-exclusion particles in a discrete 1-body spectrum (1), ( 2), . . . , (q) amounts to

ln Z 1,q = q k=1 ln y ± (k) , y ± (k) - zs(k) g-1 i=1 y ± (k ± i) = 1
which is again basically [START_REF] Polychronakos | Probabilities and path-integral realization of exclusion statistics[END_REF] with α replaced by g, but with a discrete sum instead of a continuous integral and the discrete shifts k → k ± 1 . . . → k ± (g -1) in the argument of y + (k) and y -(k).

Likewise, because of the identity

Z 1,q = Z 1,k-1 Z k+g-1,q + k+g-2 i=k zs(i)Z 1,i-g Z i+g,q (34)
of obvious exclusion statistics origin (which can also be obtained by expanding the secular determinant with respect to its k th row), the mean occupation number n k defined in (23) becomes

n k = zs(k) Z 1,k-g Z k+g,q Z 1,q (35)
and satisfies the nesting relation (see Appendix for the proof)

zs(k) = n k g-2 j=0 1 -g-2 i=0 n k-j+i g-1 j=0 1 -g-1 i=0 n k-j+i (36)
and combining with (33) we obtain the expressions for y + (k) and y -(k)

y ± (k) = 1 + n k 1 -g-1
j=0 n k∓j leading to the expression for the thermodynamic potential

ln Z 1,q = q k=1 ln 1 + n k 1 -g-1 i=0 n k-i = q k=1 ln 1 + n k 1 -g-1 i=0 n k+i
again in close similarity to [START_REF] Ouvry | On the Relation between the Anyon and the Calogero Models[END_REF][START_REF] Wu | Statistical distribution for generalized ideal gas of fractional-statistics particles[END_REF] for α replaced by g. When the occupation numbers are such that for any given k their sum over g neighboring levels g-1 i=0 n k-i = 1, the system has reached the maximal critical filling allowed by exclusion statistics.

The harmonic Calogero model

As an illustration of the formalism developed in the previous section, let us focus on the Nbody Calogero harmonic spectrum [START_REF] Ouvry | Hamiltonian and exclusion statistics approach to discrete forward-moving paths[END_REF], here again for convenience shifted by N ω/2, not any more in the thermodynamic limit βω → 0 as in Section [START_REF] Haldane | Fractional statistics in arbitrary dimensions: A generalization of the Pauli principle[END_REF], where the harmonic well played the role of a long-distance regulator, but keeping βω physical and finite. This is an example of a exclusion-g statistics system with a discrete 1-body spectrum, arising from a microscopic N -body Hamiltonian with spectral function s(k) = x k , k = 1, 2, . . . , ∞, with x = e -βω , i.e., a harmonic linear 1-body spectrum. We could focus as well on the N -body LLL-anyon harmonic spectrum (1) with spectral function

s(k) = x k x c (37)
where x = e -β(ωt-ωc) and x c = e -βωc .

Number of states q finite

Assume, now, that the number of 1-body states is finite, that is, k = 1, 2, . . . , q. The integer q is an effective high-energy cutoff which is not actually needed for the consistency of the results; later on, the q → ∞ limit will be taken. The N -body spectrum [START_REF] Ouvry | Hamiltonian and exclusion statistics approach to discrete forward-moving paths[END_REF], with α traded for g and the l i bounded by q -(N -1)g, yields a partition function which is nothing but Z N in (28) for the spectral function s(k) = x k discussed above. The grand partition function follows as

Z 1,q = 1 + (q+g-1)/g N =1 z N x N +gN (N -1)/2 N j=1 1 -x j+q-1-(N -1)g 1 -x j
In this specific situation, where

s(k) = x k , we see that Z k,k is actually Z 1,q but with q → k -k + 1 and z → zx k-1 Z k,k = 1 + (k -k+g)/g N =1 z N x N +gN (N -1)/2 N j=1 x k-1 -x j+k -1-(N -1)g 1 -x j
From the cluster expansion (32) we obtain

ln Z k,q = - ∞ n=1 (-z) n l 1 ,l 2 ,...,l j g-composition of n j≤q-k+1 c g (l 1 , l 2 , . . . , l j ) q-j+1 l=k x nl x j i=1 (i-1)l i = - ∞ n=1 (-z) n 1 -x n l 1 ,l 2 ,...,l j g-composition of n j≤q-k+1
x nk -x n(q-j+2) c g (l 1 , l 2 , . . . , l j )

x j i=1 (i-1)l i (38)
and similarly

ln Z 1,k = - ∞ n=1 (-z) n 1 -x n l 1 ,l 2
,...,l j g-composition of n j≤k

x n -x n(k-j+2) c g (l 1 , l 2 , . . . , l j ) x j i=1 (i-1)l i which imply the cluster expansion for the forward and backward effective single-level grand partition functions

ln y + (k) = - ∞ n=1 (-zx k ) n l 1 ,l 2 ,...,l j g-composition of n j≤q-k+1 c g (l 1 , l 2 , . . . , l j ) x j i=1 (i-1)l i ln y -(k) = - ∞ n=1 (-zx k ) n l 1 ,l 2
,...,l j g-composition of n j≤k c g (l 1 , l 2 , . . . , l j ) x -j i=1 (i-1)l i (39)

The above expressions are remarkably similar, differing in the sign of the exponent and in the allowed g-compositions. Note that the expression for y -(k) does not involve q, since it looks "back" towards k = 1, and is, thus, universal.

Number of states q → ∞: the harmonic Calogero model

Clearly in the q → ∞ limit, i.e., the harmonic Calogero model, the constraint on the number of parts j ≤ q -k + 1 in (38) or (39) disappears: (38) for k = 1 is the harmonic Calogero thermodynamic potential

ln Z 1,q = - ∞ n=1 (-z) n x n 1 -x n l 1 ,l 2
,...,l j g-composition of n c g (l 1 , l 2 , . . . , l j ) x j i=1 (i-1)l i with the harmonic Calogero cluster coefficients

b n = (-1) n-1 1 1 -x n l 1 ,l 2 ,...,l j g-composition of n c g (l 1 , l 2 , . . . , l j ) x j i=1 i l i . ( 40
)
The b n in (40) encode g-exclusion statistics arising from the microscopic N -body harmonic Calogero model, i.e., from its Hamiltonian [START_REF] Ouvry | The algebraic area of closed lattice random walks[END_REF] and N -body spectrum [START_REF] Ouvry | Hamiltonian and exclusion statistics approach to discrete forward-moving paths[END_REF].

Let us consider the small βω limit to recover the cluster coefficients of Section 2. For small βω, i.e., x = e -βω 1 -βω, the cluster coefficient b n in (40) becomes b n = (-1) n-1 1 nβω l 1 ,l 2 ,...,l j g-composition of n c g (l 1 , l 2 , . . . , l j ) Using [START_REF] Ouvry | On the Relation between the Anyon and the Calogero Models[END_REF] l 1 ,l 2 ,...,l j g-composition of n c g (l 1 , l 2 , . . . , l j ) = and indeed coincides, when g is traded for α, with (2) when ω c = 0. In order to recover the full cluster coefficient (2), i.e, with ω c = 0, we have to use the spectral function (37), that is, put x = e -β(ωt-ωc) in (40) and multiply by x n c . We remark that in (40) the coefficients of the polynomial l 1 ,l 2 ,...,l j g-composition of n c g (l 1 , l 2 , . . . , l j ) x j i=1 (i-1)l i are, when multiplied by n, a g-generalization of the coefficients OEIS A227532 and the related A227543, in relation to the Ramanujan continuous fraction 1 . Indeed, denoting

H g (z, x) = Z 2,∞ (-z/x, x) = 1 + ∞ N =1 z N x gN (N -1)/2 N j=1 x x j -1 , G g (z, x) = Z 1,∞ (-z/x, x) = 1 + ∞ N =1 z N x gN (N -1)/2 N j=1 1 x j -1 , then y + (1) = Z 1,q /Z 2,q in (39) yields the q → ∞ identity ∞ n=1 z n l 1 ,l 2 ,...,l j g-composition of n c g (l 1 , l 2 , . . . , l j ) x j i=1 (i-1)l i = ln H g (z, x) G g (z, x) . (41)
When g = 1, i.e., Fermi statistics and Pauli exclusion, the c 1 (l 1 , l 2 , . . . , l j ) are non vanishing for the sole composition l 1 = n with c 1 (n) = 1/n. In this case (41) trivially reduces to ∞ n=1 z n /n = -ln(1 -z).

When g = 2, writing (21) for z → -z/x and s(k) = x k in the form

1 y + (k) = 1 1 -zx k-1 y + (k+1)
and solving it iteratively for y + (1) it yields as a solution the Ramanujan continuous fraction. Exponentiating (41), then, yields the identity exp

∞ n=1 z n l 1 ,l 2 ,...,l j composition of n c 2 (l 1 , l 2 , . . . , l j ) x j i=1 (i-1)l i = H 2 (z, x) G 2 (z, x) = 1 1 + -z 1+ -zx 1+ -zx 2 1+ -zx 3 1+ -zx 4 1+...
The LHS expresses the log of the Ramanujan fraction in terms of c 2 (l 1 , l 2 , . . . , l j ).

For a general g, exponentiating (41) gives an alternative expression of the ratio H g (z, x)/G g (z, x) in terms of the c g (l 1 , l 2 , . . . , l j ) as a g ≥ 2-generalisation of the OEIS coefficients A227532 and A227543. Note, however, that in the case g > 2 the iterative solution of (33) for 1/y + leads to a complicated expression and does not yield any simple continuous fraction as in the g = 2 case.

Conclusions

The results in this work generalize exclusion statistics thermodynamics to the statistical mechanics of systems with discrete spectra. An interesting, and somewhat puzzling, feature of our analysis is the introduction of two distinct types of single-level effective partition functions, forward and backward y ± (k), both of which provide a complete description of the system. The physical significance of their existence is somewhat obscure: each provides an aspect of the statistics of a single level taking into account the existence of higher, resp, lower levels. By contrast, for standard noninteracting fermions or bosons, the statistics of each level is independent of the existence of other levels. y ± (k), then, encapsulate the effective interactions induced by nontrivial statistics, but in a fundamentally "chiral" way. Although a "symmetric" effective action y(k) = y + (k)y -(k) can be defined, it does not satisfy any simple nesting relations and the mean occupation number n k cannot be expressed in terms of y(k) alone, making y(k) a quantity of limited use or physical relevance. The full significance of y ± (k) and the identification of physical results stemming from their duality remain interesting open issues.

The analysis and results of this work could be applied to other systems, and in particular to the Calogero-Sutherland model of particles with an inverse sine squared two-body potential, which can be viewed as exclusion-g particles on the circle. The spectral function for this model is (setting the length of the circle to 2π) s(k) = e -βk 2 /2 , k = 0, ±1, ±2, . . . with k interpreted as the discrete momentum of exclusion particles (also referred to as "pseudomomentum"). The spectrum is symmetric and unbounded in both ends, so forward and backward effective functions y ± (k) are essentially equivalent, satisfying y + (k) = y -(-k), a relation that survives the introduction of a symmetric cutoff |k| ≤ q. We could also impose an asymmetric restriction to the spectrum, such as k ≥ 0, which actually corresponds to a symmetry-reduced Calogero-Sutherland model with inverse sine and inverse cosine squared potentials plus an additional one-body interaction with the point at the origin.

The full Calogero-Sutherland spectrum can be treated by separating in the N -body partition function the (unrestricted) sum over the smallest momentum k 1 . The sum of the remaining momenta can be cast as the grand partition function of N -1 particles with spectral function s(k) = e -β(k 1 +g+k) 2 with k ≥ 0. This makes the spectrum bounded to the left, but creates the added complication of having to sum at the end over k 1 . We defer any further discussion of the microscopic statistics of the Calogero-Sutherland model to future work.

In both the Calogero and LLL-anyon systems we can define "quasihole" excitations, corresponding to minimal "gaps" in a completely filled state, that behave as particles with exclusion statistics 1/g. We can define similar excitations in the microscopic (discrete) case, starting from a maximally filled many-body state and moving all particles to the right of a marked particle by one level, or to the left by minus one level, therefore creating a quasihole. All the standard features of quasiholes emerge in this picture: the removal of one particle creates g quasiholes, identifying them as -1/g particle each, they "move" in the energy spectrum by steps of g, therefore corresponding to an 1/g dilution of the density of states, and in a macroscopic span of K successive levels a number K of quasiholes can be"packed" together. Since the effective density of states has been decreased by a factor of 1/g, this last property identifies quasiholes as particles with 1/g exclusion statistics.

Several qualitatively new features arise, however, in the discrete setting: there are, now, g distinct "fully filled" states, related to each other through a common shift of all particles by 1, 2, . . . g -1 levels, so these states are not perturbatively connected. Further, each quasihole's energy becomes a nontrivial combination of several energy levels. The exact properties of quasiholes, and a potential reformulation of the statistical mechanics of the system in terms of them, remain fascinating objects of further study.

Exclusion statistics are related to the generating functions for the algebraic area counting of lattice walks [START_REF] Ouvry | The algebraic area of closed lattice random walks[END_REF], a connection that arises from the matrix determinant representation as exposed in section 3. The Calogero model results, in particular, are related to a set of directed walks termed Dyck or Lukasiewicz paths. This connection has been exploited to derive results for such walks [START_REF] Ouvry | Hamiltonian and exclusion statistics approach to discrete forward-moving paths[END_REF] but it can also be used in reverse: the generating functions of generalized types of walks can be considered as effective descriptions of particular exclusion-type statistics. The identification of such statistics, or of the walks corresponding to other known types of statistics, is an interesting topic for investigation.

The most interesting possible application of our results is, however, in situations of physical significance, such as fractional quantum Hall and related systems. This would require measurements that probe the microscopic quantities studied in this work, such as, e.g., the single-level mean occupation numbers n k , looking for possible signatures within recent experimental results. Also LLL-anyon statistics at maximal filling is encoded in Laughlin type wavefuctions [START_REF]One-dimensional Statistical Mechanics for Identical Particles: the Calogero and Anyon Cases[END_REF] describing bulk particles with exclusion α, while experimentalists [START_REF] Bartolomei | Fractional statistics in anyon collisions[END_REF] so far observe quasiholes with statistics 1/α. In this regard it is worth noting that unpublished data from [START_REF] Bartolomei | Fractional statistics in anyon collisions[END_REF] reveal in the strong-backscattering regime some conductance nonlinearities [14] which might hint to charge carriers with exclusion equal to 3. Clearly identifying possible measurements allowing for an experimental confirmation of bulk particles with exclusion statistics remains an important subject for further theoretical research and experimental developments.

  b n = (-1) n-1 1 βω gn n gn 2

See The On-Line Encyclopedia of Integer Sentences (OEIS), https://oeis.org/

Acknowledgments

We would like to thank Gwendal Feve for interesting conversations. We also thank the anonymous referee for useful comments and remarks that helped improve the manuscript. A.P. also acknowledges the hospitality of LPTMS, CNRS at Université Paris-Saclay (Faculté des Sciences d'Orsay), where this work was seeded.

Appendix

To derive (25), we use the recursion (19) to rewrite Z k+1,q = Z k+2,q + zs(k + 1)Z k+3,q so that (24) becomes

Then from (24) and (42) get

Likewise from (42) get

or, again using (24),

Multiplying ( 43) and (44) we obtain (25).

The proof of the corresponding relation for g > 2 proceeds along similar lines. Dividing the relation (34) by Z 1,q and using (35) for each term in the sum we obtain

(we also changed summation variable form i to i + k). Using recursion relation (31) to write Z k+g-1,q = Z k+g,q + zs(k + g -1)Z k+2g-1,q in the above relation, and using again (35) to express the second term in terms of n k+g-1 we obtain

Dividing ( 45) and ( 46) we obtain

The product of the above ratio for successive values of k "telescopes", and we obtain

using also (46) = Z k+1,q Z k+g,q • Z 1,q Z 1,k-g Z k+1,q = Z 1,q Z k+g,q Z 1,k-g = zs(k) n k thus proving (36).


Alexis Hermida

MD Momar Diouf

Maciej Kubala

MD, pHD Floriane Fay

MD Jacqueline Burtin

MD Pierre-Marc Lallemand

MD Otilia Buiciuc

MD Audrey Lieu

MD Mustafa Zaitouni

MD Christophe Beyls

MD Jean-Sylvain Hermida

Results and Predictive Factors After One Cryoablation for Persistent Atrial Fibrillation

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Standalone pulmonary vein isolation (PVI) as the first step of a strategy for the ablation of persistent atrial fibrillation (AF) may be an option, as the addition of linear ablation or the ablation of complex fractionated electrograms is not superior to PVI alone. [START_REF] Verma | Approaches to catheter ablation for persistent atrial fibrillation[END_REF] The proportion of patients in sinus rhythm after one procedure of cryoballoon PVI for persistent AF ranges from 50 to 72%. [START_REF] Koektuerk | Cryoballoon Ablation for Pulmonary Vein Isolation in Patients With Persistent Atrial Fibrillation: One-Year Outcome Using Second Generation Cryoballoon[END_REF][START_REF] Ciconte | Pulmonary vein isolation as index procedure for persistent atrial fibrillation: One-year clinical outcome after ablation using the second-generation cryoballoon[END_REF][START_REF] Aryana | Acute and Long-Term Outcomes of Catheter Ablation of Atrial Fibrillation Using the Second-Generation Cryoballoon versus Open-Irrigated Radiofrequency: A Multicenter Experience[END_REF][START_REF] Guhl | Efficacy of Cryoballoon Pulmonary Vein Isolation in Patients With Persistent Atrial Fibrillation[END_REF][START_REF] Tscholl | Two years outcome in patients with persistent atrial fibrillation after pulmonary vein isolation using the second-generation 28-mm cryoballoon[END_REF][START_REF] Boveda | Outcomes after cryoballoon or radiofrequency ablation for persistent atrial fibrillation: a multicentric propensity-score matched study[END_REF][START_REF] Lemes | One-year clinical outcome after pulmonary vein isolation in persistent atrial fibrillation using the second-generation 28 mm cryoballoon: a retrospective analysis[END_REF][START_REF] Straube | Cryoballoon ablation for persistent atrial fibrillation -Large single-center experience[END_REF][START_REF] Tondo | Pulmonary vein isolation cryoablation for patients with persistent and long-standing persistent atrial fibrillation: Clinical outcomes from the real-world multicenter observational project[END_REF][START_REF] Akkaya | Second-generation cryoballoon ablation for treatment of persistent atrial fibrillation: Three-year outcome and predictors of recurrence after a single procedure[END_REF][START_REF] Yalin | Safety and efficacy of persistent atrial fibrillation ablation using the second-generation cryoballoon[END_REF][START_REF] Boveda | Single-Procedure Outcomes and Quality-of-Life Improvement 12 Months Post-Cryoballoon Ablation in Persistent Atrial Fibrillation: Results From the Multicenter CRYO4PERSISTENT AF Trial[END_REF][START_REF] Sawhney | Cryoablation for persistent and longstanding 15 persistent atrial fibrillation: results from a multicentre European registry[END_REF][START_REF] Reissmann | Predictors of freedom from atrial arrhythmia recurrence after cryoballoon ablation for persistent atrial fibrillation: A multicenter study[END_REF][START_REF] Su | Cryoballoon Ablation of Pulmonary Veins for Persistent Atrial Fibrillation: Results From the Multicenter STOP Persistent AF Trial[END_REF] The interpretation of the results of these studies is limited by the short follow-up of under two years, except for one, which reached 37 months. [START_REF] Akkaya | Second-generation cryoballoon ablation for treatment of persistent atrial fibrillation: Three-year outcome and predictors of recurrence after a single procedure[END_REF] Moreover, only two studies have followed large cohorts of 609 [START_REF] Sawhney | Cryoablation for persistent and longstanding 15 persistent atrial fibrillation: results from a multicentre European registry[END_REF] and 486 patients [START_REF] Tondo | Pulmonary vein isolation cryoablation for patients with persistent and long-standing persistent atrial fibrillation: Clinical outcomes from the real-world multicenter observational project[END_REF] and provided results for patients with long-standing persistent AF. Given the recurrence rate, it would be useful to identify long-term predictors of recurrence before performing the procedure. Data concerning such predictors have been published [START_REF] Koektuerk | Cryoballoon Ablation for Pulmonary Vein Isolation in Patients With Persistent Atrial Fibrillation: One-Year Outcome Using Second Generation Cryoballoon[END_REF][START_REF] Ciconte | Pulmonary vein isolation as index procedure for persistent atrial fibrillation: One-year clinical outcome after ablation using the second-generation cryoballoon[END_REF][START_REF] Tscholl | Two years outcome in patients with persistent atrial fibrillation after pulmonary vein isolation using the second-generation 28-mm cryoballoon[END_REF][START_REF] Boveda | Outcomes after cryoballoon or radiofrequency ablation for persistent atrial fibrillation: a multicentric propensity-score matched study[END_REF][START_REF] Straube | Cryoballoon ablation for persistent atrial fibrillation -Large single-center experience[END_REF][START_REF] Akkaya | Second-generation cryoballoon ablation for treatment of persistent atrial fibrillation: Three-year outcome and predictors of recurrence after a single procedure[END_REF][START_REF] Yalin | Safety and efficacy of persistent atrial fibrillation ablation using the second-generation cryoballoon[END_REF][START_REF] Sawhney | Cryoablation for persistent and longstanding 15 persistent atrial fibrillation: results from a multicentre European registry[END_REF][START_REF] Reissmann | Predictors of freedom from atrial arrhythmia recurrence after cryoballoon ablation for persistent atrial fibrillation: A multicenter study[END_REF] but only three studies included left atrium dimension. [START_REF] Akkaya | Second-generation cryoballoon ablation for treatment of persistent atrial fibrillation: Three-year outcome and predictors of recurrence after a single procedure[END_REF][START_REF] Sawhney | Cryoablation for persistent and longstanding 15 persistent atrial fibrillation: results from a multicentre European registry[END_REF][START_REF] Reissmann | Predictors of freedom from atrial arrhythmia recurrence after cryoballoon ablation for persistent atrial fibrillation: A multicenter study[END_REF] Here, we evaluated the results and predictors of recurrence after a long follow-up of two years for a large cohort of patients who were treated for persistent and long-standing persistent AF by cryoablation.

METHOD

We prospectively included all consecutive patients who underwent cryo-PVI for the treatment of persistent drug-refractory AF from the introduction of second generation cryoballoon ablation to the Amiens University Hospital in September 2012 to November 2019. Patients provided their signed informed consent to participate in the registry and the study was approved by the local ethics committee. Data files of the registry were declared to and authorized by the French data protection committee (Commission Nationale Informatique et Liberté, CNIL) (PI2020_843_0120).

The primary objectives of the study were to assess the success rates and complication rates after one cryo-PVI procedure. The secondary objective was to establish the long-term predictive factors of ATa.

Patients were included if they presented symptomatic persistent AF, documented by 12-lead ECG, refractory to at least one class I or III antiarrhythmic drug. AF was classified according to the 2020 ESC Guidelines for the management of AF. [START_REF] Hindricks | ESC Guidelines for the diagnosis and management of atrial fibrillation developed in collaboration with the European Association for Cardio-Thoracic Surgery (EACTS)[END_REF] The diagnosis-to-ablation time was defined by the time between the first episode of AF and ablation. Exclusion criteria were (i) prior left atrial (LA) ablation, (ii) LA tachycardia at the time of cryoablation, (iii) having a prosthetic heart valve, (iv) being a minor (< 18 years old), and (v) no follow-up after the cryo-PVI procedure.

LA area was measured during echocardiography in a four-chamber view. LA volume was obtained from multi-detector computed tomography and segmentation by AW VolumeShare 5 software (General Electric Company, Fairfield, Connecticut, USA). The first centimeter of each pulmonary vein and the LA appendage were included in the determination of LA volume. Hypertension was defined as repeated office or out-of-office systolic blood pressure values ≥ 140 mmHg and/or diastolic blood pressure values ≥ 90 mmHg [START_REF] Williams | ESC/ESH Guidelines for the management of arterial hypertension: The Task Force for the management of arterial hypertension of the European Society of Cardiology and the European Society of Hypertension: The Task Force for the management of arterial hypertension of the European Society of Cardiology and the European Society of Hypertension[END_REF] prior to treatment for hypertension.

Cryoballoon ablations were preceded by at least three weeks of treatment with an oral anticoagulant and then contrast-enhanced multi-slice CT of the left atrium. The cryo-PVI procedure has been described in detail elsewhere. 19 Briefly, access to the left atrium was gained after fluoroscopy-guided transseptal puncture (SL1, St Jude Medical).. Only one fourminute application was performed with a second-or third generation cryoballoon (the 23-or 28-mm Advance balloon, Cryocath, Medtronic, Dublin, Republic of Ireland). An intraluminal catheter (Achieve, Medtronic) was used for all procedures. A second cryoapplication was performed (to achieve a complete block) only if there was persistent atriovenous conduction.

A quadripolar catheter was used to continuously pace the phrenic nerve adjacent to the superior vena cava during cryoablation over the right pulmonary veins. Cryoablation was discontinued immediately if there was loss of diaphragmatic contraction to avoid right phrenic nerve palsy. A temporary subcutaneous venous figure-of-eight suture was used for all patients to achieve hemostasis.

Oral anticoagulation was maintained for patients treated with a vitamin K antagonist.

Oral anticoagulation was withdrawn 48 h before the procedure for patients treated with a direct oral anticoagulant (until October 2017) or maintained (from November 2017 onwards).

For all patients, heparin was administered as an initial bolus after the transseptal puncture and then administered continuously to maintain the target activated clotting time (> 300 s).

Data collected during the procedure included the duration of the time to effect (TTE), minimal temperature reached during application in each PV, cumulative duration of cryoapplication in each PV, number of cryo-applications for each PV, and the duration of the time under fluoroscopy and fluoroscopy exposure.

All adverse events occurring during the procedure or within the following month were recorded and described. Vascular complications included vascular pseudo aneurysms, arteriovenous fistulae, hematomas, and puncture-side hemorrhages. After ablation, antiarrhythmic drugs were maintained for at least the blanking period of three months.

Anticoagulants were continued for at least the first three months.

Patients were followed up in our institution at 3, 6, and 12 months with a physical examination, questions about arrhythmia-related symptoms, a 12-lead ECG, and 24 h-Holter monitoring. After the first year, patients were followed up by their cardiologists at least once a year with a 12-lead ECG and 24 h-Holter monitoring. Any symptoms following ablation were evaluated by 12-lead ECG and 24 h-Holter monitoring. Previously implanted pacemakers or dual-chamber implantable cardioverted defibrillators were assessed at each visit.

The primary endpoint was the recurrence of atrial tachyarrhythmia (ATa), defined as a documented episode of AF, LA flutter, or atrial tachycardia (whether symptomatic or not) lasting for 30 s or more on any cardiac rhythm recording made during a planned or symptomdriven consultation after the three-month blanking period. Recurrences of right atrial flutter (confirmed by an electrophysiological study) were not included.

The secondary endpoint was the frequency of complications of any type during the procedure or within the following month. Continuous variables are expressed as the mean ± standard deviation (SD) or median [interquartile range (IQR)] for normally and non-normally distributed data, respectively, and were compared using Student's t-test or the Mann-Whitney-Wilcoxon test, as appropriate. Categorical variables are expressed as the frequency (percentage) and were compared using a chi-squared test or Fisher's exact test.

The cumulative probability of ATa-free survival was estimated using the Kaplan-Meier method. Survival curves were compared using the Log-rank test. Cox proportional hazard models were used to identify factors associated with ATa recurrence, with an estimation of the hazard ratios (HRs) and 95% confidence intervals. Variables with a p-value < 0.10 in univariate analysis were included in a multivariate Cox model. All tests were twotailed and the limit for statistical significance was set at P < 0.05. Statistical analysis was performed using SPSS for Windows software, version 22.0 (SPSS Inc., Chicago, IL, USA).

The only missing data concerned the LA area (27%), procedure time (6%), balloon in/out (5%), fluoroscopy time (9%), fluoroscopy exposure (11%) and cryotherapy time (8%).

RESULTS

In total, 417 patients met the inclusion criteria (Table 1). Two patients were in LA flutter during the procedure, seven were lost to follow-up immediately after the procedure, and nine had a prosthetic heart valve; all 18 were excluded. Cryo-PVI was performed on 399 patients with persistent AF (mean age: 61 ± 9 years; males = 316, 79%). Among them, 52 (13%) had long-standing persistent AF (Table 1). The mean time between the first AF episode and ablation (diagnosis to ablation time) was 17 ± 19 months.

Patients with long-standing persistent AF had a significantly larger left atrium than patients with persistent AF: area: 30 ± 4 vs. 26 ± 5 cm² (p < 0.001), volume: 197 ± 37 vs. 161 ± 41 ml (p < 0.001), and indexed volume: 89 ± 16 vs. 77 ± 20 ml/m² (p < 0.001).

Eighty-six persistent AF patients (25%) were in sinus rhythm at the beginning of the procedure (Supplementary Table 1). A 28-mm cryoballoon was used for 322 patients (93%).

A left PV block was achieved for all patients. A right superior PV block was achieved for 394 patients (99%) and a right inferior PV block for 396 (99%).

We observed 10 cases (2.5%) of unresolved phrenic nerve palsy at discharge, one of tamponade (0.25%), two of stroke (0.5%), two of pericardial effusion (0.5%), and five of vascular complications (1.25%). There was no procedure-related death. There was no diminution/attenuation of diaphragm contractions in any of the 10 phrenic nerve palsy cases.

The minimum temperature for these patients was 52 ± 6°C (vs. 48 ± 8°C, p = 0.12) for RIPV and 51 ± 7°C (vs. 50 ± 10°C, p = 0.83) for RSPV. Eight patients had persistent phrenic nerve palsy at the first follow up visit and only one at the one year follow up. There was no significant difference in the duration of the procedure between the two patients who experienced stroke and those who did not. There were no long-term complications directly relatable to the cryoablation procedure identified during follow up, in particular, atriooesophageal fistulae. Eleven patients died during follow-up (3 cancer, 4 terminal heart failure, 1 car accident, 1 stroke, 2 no information). Three patients experienced a late thromboembolic event.

All patients completed the six-month follow-up visit and 359 (90%) the 12-month follow-up visit. The median follow-up was 24 months (IQR 43-13). Among all patients, ATa recurred for 211 (53%) within a mean of 12 ± 14 months (Figure 2). Seventy-one patients (34%) relapsed after 12 months. The 1-, 2-, and 3-year probabilities of freedom from ATa recurrence were 63% [60; 65], 48% [45; 51], and 38% [35; 42], respectively. Fifty-six patients (30%) without recurrence were still on an antiarrhythmic drug at last follow-up (31 on Amiodarone, 18 on Flecaïnide, 7 on Sotalol). Among the 347 patients with persistent AF, the 1-, 2-and 3-year probabilities of freedom from ATa recurrence were 66% [64; 69], 51%

[48; 54], and 42% [38; 45], respectively. Among the 52 patients with long-standing persistent AF, the 1-, 2-and 3-year probabilities of freedom from ATa recurrence were 40% [33; 47], 27% [20; 33], and 18% [10; 25], respectively.

The results of the univariate analysis concerning the risk of ATa recurrence are shown in Table 2. After multivariate analysis that included BMI, long-standing persistent AF, hypertension, coronary artery disease, CHA2DS2-VASc score, and LA area, the predictors of recurrence of ATa were long-standing persistent AF (HR = 1.83, CI [1.22; 2.75], p = 0.003) and LA area (HR = 1.05, CI [1.02; 1.09], p = 0.002). After multivariate analysis that included the same variables but in which LA area was replaced with LA volume, the predictors of recurrence of ATa were long-standing persistent AF (HR = 1.68, CI [1.16; 2.44], p = 0.006) and LA volume (HR =1.006, CI [1.002; 1.009], p = 0.001).

Among the 211 patients who had a recurrence, 139 (66%) underwent at least one repeat procedure (Figure 2). Patients who had a recurrence of paroxysmal AF underwent radiofrequency (RF) repeat PVI. Patients who had a recurrence of persistent AF or left atrial flutter underwent linear RF ablation. During the first repeat procedure, 80 patients (58%) had at least one reconnected PV. The mean number of reconnected PVs was 0.94. Along these 139 patients, 104 (75%) had not experienced a recurrence at the last follow-up. Finally, 292 patients (188 after a single procedure and 104 after repeat procedures, 73%) were free of recurrence at the last follow-up.

DISCUSSION

In this large cohort, we establish that cryo-PVI alone for persistent AF is a safe and effective strategy. Long-term predictors of recurrence were long-standing persistent AF and LA dimension.

Among the 347 patients with persistent AF, the 2-and 3-year probabilities of freedom from ATa recurrence were 51% and 42%. Sawhney et al. [START_REF] Sawhney | Cryoablation for persistent and longstanding 15 persistent atrial fibrillation: results from a multicentre European registry[END_REF] recently published a long-term follow-up study of 487 patients with persistent AF, with a success rate of 64% at 24 months after a single cryoballoon PVI procedure. The difference from our reported success rate of 51% at two years, while all patients were off-AADs (vs. 30% on AADs in our study), may be explained by more evolved AF in our study, with a mean atrial area of 26 cm². Four other studies with a long follow-up of 22 [START_REF] Tscholl | Two years outcome in patients with persistent atrial fibrillation after pulmonary vein isolation using the second-generation 28-mm cryoballoon[END_REF] , 20 [START_REF] Guhl | Efficacy of Cryoballoon Pulmonary Vein Isolation in Patients With Persistent Atrial Fibrillation[END_REF] , and 18 months [START_REF] Tondo | Pulmonary vein isolation cryoablation for patients with persistent and long-standing persistent atrial fibrillation: Clinical outcomes from the real-world multicenter observational project[END_REF][START_REF] Reissmann | Predictors of freedom from atrial arrhythmia recurrence after cryoballoon ablation for persistent atrial fibrillation: A multicenter study[END_REF] reported recurrence rates similar to ours of 56, 50, 51.5, and 53%, respectively. Other studies published on this subject 2-4,7- 9,12,13,16 reported results after short-term follow-up of approximately one year, with logically better success rates than ours (from 54.8 to 72%, Supplementary Table 2). An alternative strategy for persistent atrial fibrillation ablation is to add substrate modification to PVI. In the metanalysis of Clarnette et al. [START_REF] Clarnette | Outcomes of persistent and long-standing persistent atrial fibrillation ablation: a systematic review and meta-analysis[END_REF] , the single-procedure success rate was 46% for both PVI plus complex fractionated atrial electrograms ablation and PVI ablation plus linear lesions, within the same range as our success rate Studies of Tondo et al. [START_REF] Tondo | Pulmonary vein isolation cryoablation for patients with persistent and long-standing persistent atrial fibrillation: Clinical outcomes from the real-world multicenter observational project[END_REF] and Sawhney et al. [START_REF] Sawhney | Cryoablation for persistent and longstanding 15 persistent atrial fibrillation: results from a multicentre European registry[END_REF] provide data about one procedure of cryoballoon PVI for long-standing persistent AF. Tondo et al. [START_REF] Tondo | Pulmonary vein isolation cryoablation for patients with persistent and long-standing persistent atrial fibrillation: Clinical outcomes from the real-world multicenter observational project[END_REF] reported a probability of one-year freedom from AF recurrence of 63.1% for 52 patients with long-standing persistent AF. Sawhney [START_REF] Sawhney | Cryoablation for persistent and longstanding 15 persistent atrial fibrillation: results from a multicentre European registry[END_REF] reported an arrhythmia-free survival rate of 57% at 24 months follow-up. We report worse results, with only 27% of patients free of ATa after two years. This difference could be explained by the longer follow-up in our study (vs. Tondo). Moreover, the mean LA volume was 74.8 ml in the study of Tondo et al., whereas it was 165 ml in ours, which confirms that we included more evolved patients. In any case, the difference cannot be explained by the rate of AAD maintenance, which was similar between the study of Tondo and ours. Our results lead to two conclusions. First, cryoballoon PVI as an index procedure for long-standing persistent AF should, in most cases, be the first step of a stepwise approach, which should include (an)other procedure(s). Second, our results provide a supplementary argument for the strategy of early intervention to improve the outcome after AF ablation.

Kirchhof et al. [START_REF] Kirchhof | Early Rhythm-Control Therapy in Patients with Atrial Fibrillation[END_REF] recently showed that early rhythm-control therapy was associated with a lower risk of adverse cardiovascular outcomes.

In the meta-analysis of Andrade [START_REF] Andrade | Efficacy and safety of cryoballoon ablation for atrial fibrillation: a systematic review of published studies[END_REF] , as well as in the Fire and Ice trial [START_REF] Kuck | Cryoballoon or Radiofrequency Ablation for Paroxysmal Atrial Fibrillation[END_REF] , complication rates of cryoballoon ablation were found to be very low. The phrenic-nerve palsy rate was 4.73% and 2.7%, respectively. Our results are in the same range, with a rate of 2.5%. In our study, 0.5% of patients had strokes, 0.25% tamponade, and 1.25% vascular complications.

Our results are very close to those of the aforementioned studies, which showed 0.57% and 0.5% thromboembolic events, 0.47 and 0.3% tamponade, and 1.79% and 1.9% vascular complications, respectively.

After multivariate analysis, we found long-standing persistent AF and LA area/LA volume to be predictors of recurrence. It is logical that we found the LA area/volume to be predictors of recurrence, as published studies already identified LA diameter [START_REF] Sawhney | Cryoablation for persistent and longstanding 15 persistent atrial fibrillation: results from a multicentre European registry[END_REF][START_REF] Reissmann | Predictors of freedom from atrial arrhythmia recurrence after cryoballoon ablation for persistent atrial fibrillation: A multicenter study[END_REF] and LA area [START_REF] Akkaya | Second-generation cryoballoon ablation for treatment of persistent atrial fibrillation: Three-year outcome and predictors of recurrence after a single procedure[END_REF] as predictors. Only two studies [START_REF] Koektuerk | Cryoballoon Ablation for Pulmonary Vein Isolation in Patients With Persistent Atrial Fibrillation: One-Year Outcome Using Second Generation Cryoballoon[END_REF][START_REF] Akkaya | Second-generation cryoballoon ablation for treatment of persistent atrial fibrillation: Three-year outcome and predictors of recurrence after a single procedure[END_REF] included LA area (measured in echocardiography) in their Cox Model and none included LA volume, which is less subject to inter-observer variability. [START_REF] Ratajczak | Anatomical Evaluation of the Pulmonary Veins and the Left Atrium Using Computed Tomography Before Catheter Ablation: Reproducibility of Measurements[END_REF][START_REF] Mahabadi | Quantitative assessment of left atrial volume by electrocardiographic-gated contrast-enhanced multidetector computed tomography[END_REF] Long-standing persistent AF has already been described as a strong predictive factor of recurrence. Thus, this item is included in the MB LATER [START_REF] Mujovic | Prediction of very late arrhythmia recurrence after radiofrequency catheter ablation of atrial fibrillation: The MB-LATER clinical score[END_REF] and CAAP AF [START_REF] Winkle | Predicting atrial fibrillation ablation outcome: The CAAP-AF score[END_REF] scores.

Similarly, diagnosis-to-ablation time [START_REF] Ciconte | Pulmonary vein isolation as index procedure for persistent atrial fibrillation: One-year clinical outcome after ablation using the second-generation cryoballoon[END_REF][START_REF] Boveda | Outcomes after cryoballoon or radiofrequency ablation for persistent atrial fibrillation: a multicentric propensity-score matched study[END_REF][START_REF] Akkaya | Second-generation cryoballoon ablation for treatment of persistent atrial fibrillation: Three-year outcome and predictors of recurrence after a single procedure[END_REF][START_REF] Sawhney | Cryoablation for persistent and longstanding 15 persistent atrial fibrillation: results from a multicentre European registry[END_REF] has been shown to be predictor of recurrence after cryoballoon PVI for persistent AF, in numerous studies. Relapse during the blanking period [START_REF] Koektuerk | Cryoballoon Ablation for Pulmonary Vein Isolation in Patients With Persistent Atrial Fibrillation: One-Year Outcome Using Second Generation Cryoballoon[END_REF][START_REF] Ciconte | Pulmonary vein isolation as index procedure for persistent atrial fibrillation: One-year clinical outcome after ablation using the second-generation cryoballoon[END_REF][START_REF] Straube | Cryoballoon ablation for persistent atrial fibrillation -Large single-center experience[END_REF][START_REF] Yalin | Safety and efficacy of persistent atrial fibrillation ablation using the second-generation cryoballoon[END_REF] is widely cited as an independent predictor, but we chose not to test it in our study because these data are available only after the decision to perform the procedure. It has no clinical relevance in the decision of whether to perform the procedure nor how to perform the first procedure.

This was a single-center study, and its conclusion can therefore not be extrapolated to other centers. However, we present the third largest cohort of cryoballoon ablation for persistent AF. Our protocol of atrial arrhythmia monitoring after cryo-PVI does not include continuous monitoring but corresponds to that of other studies and to real-life practice. Given the fact that 30% of the patients without recurrence were still on antiarrhythmic drugs at the last follow-up, we cannot exclude that the success rates described here are due to the combined effects of the cryo-PVI and continued medication. The measurement of the LA area in echocardiography is subject to inter-observer variability and conclusions based on these data need to be made with caution. However, volume measurements by tomography are readily available but rarely mentioned in published studies. They represent an original parameter that is less subject to imprecision.

In conclusion, two-year ATa-free survival rates were 51 and 27% for persistent and long-standing persistent AF patients, respectively. Complications were rare. The independent long-term predictors of recurrence were long-standing persistent AF and LA area/volume.

Treatment by cryo-PVI for persistent AF appears to be reasonable, but insufficient for longstanding persistent AF.

Figure 1 .

 1 Figure 1. ATa-free survival for persistent vs. long-standing persistent AF.

Figure 2 .

 2 Figure 2. Results after one cryo-PVI and repeat procedures.

Table 1 .

 1 Patient characteristics at baseline

	Variable	All patients	Persistent AF	Long-Standing Persistent AF	p
			(n = 347)	(n = 52)
	Age (years ± sd)	61 ± 9	60 ± 9	61 ± 9	0.82
	Men	316 (79%) 272 (78%)	44 (85%)	0.36
	Height (m)	1.75 ± 0.1	1.75 ± 0.1	1.76 ± 0.1	0.46
	Weight (kg)	91 ±18	90 ± 17	98 ± 18	0.002
	Body mass index (kg/m 2 )	30 ± 5	29 ± 5	32 ± 5	0.002
	Body surface area (m 2 )	2.1 ± 0.2	2.1 ± 0.2	2.2 ± 0.2	0.003
	Diagnosis-to-ablation time (months)	17 ± 19	16 ± 20	28 ± 16	< 0.001
	Number of cardioversions	0.8 ± 0.9	0.7 ± 0.9	1.2 ± 0.9	0.001
	Clearance (ml/min)	81 ± 21	81 ± 21	83 ± 26	0.51
	Clearance < 60ml/min	61 (15%)	51 (15%)	10 (19%)	0.41
	Hypertension	193 (48%) 164 (47%)	29 (56%)	0.30
	Diabetes mellitus	52 (13%)	44 (13%)	8 (15%)	0.66
	Stroke	28 (7%)	25 (7%)	3 (6%)	1
	Heart failure	78 (19%)	67 (19%)	11 (21%)	0.71
	Coronary artery disease	46 (11%)	38 (11%)	8 (15%)	0.35
	Absence of SHD	259 (65%) 228 (66%)	31 (60%)	0.44
	Antiarrhythmic drug therapy	232 (58%) 208 (60%)	24 (46%)	0.07
	CHA2DS2-VASc score	1.7 ± 1.4	1.7 ± 1.4	1.7 ± 1.5
	0	96 (24%)	86 (25%)	10 (19%)
	1	115 (29%)	97 (28%)	18 (35%)
	2 3	77 (19%) 64 (16%)	68 (20%) 53 (15%)	9 (17%) 11 (21%)	0.07
	4	33 (8%)	31 (9%)	2 (4%)
	≥ 5	14 (4%)	12 (3%)	2 (4%)
	Previous cavotricuspid isthmus ablation	29 (7%)	26 (7%)	3 (6%)	1
	LA and LV parameters
	LA area (cm 2 )	26 ± 5	26 ± 5	30 ± 4	< 0.001
	LA volume (ml)	165 ± 42	161 ± 41	197 ± 37	< 0.001
	LA volume Index (ml/m 2 )	78 ± 20	77 ± 20	89 ± 16	< 0.001
	LVEF (%)	54 ± 12	54 ± 12	53 ± 12	1
	LVEF ≤ 40%	74 (18%)	62 (18%)	12 (23%)	0.68
	PV anatomy
	LSPV diameter (mm)	19 ± 2	19 ± 2	20 ± 2	0.08
	LIPV diameter (mm)	17 ± 2	17 ± 2	18 ± 2	0.02
	RSPV diameter (mm)	20 ± 2	20 ± 2	20 ± 2	0.06
	RIPV diameter (mm)	19 ± 2	19 ± 2	20 ± 3	0.004
	LCT	63 (16%)	54 (16%)	9 (17%)	0.69
	Accessory vein	51 (13%)	48 (14%)	3 (6%)	0.12

Table 2 .

 2 Predictive factors of ATa recurrence after univariate analysis

		No ATa	ATa	HR univariate
	Variable	recurrence	recurrence	analysis	p
		(n = 188)	(n = 211)	CI [95% CI]
	Age (years ± sd)	60 ± 9	61 ± 9	1.01 [0.99; 1.02]	0.18
	Men	146 (78%)	170 (81%)	0.96 [0.68; 1.35]	0.81
	Height (m)	1.76 ± 0.1	1.75 ± 0.1	0.91 [0.19; 4.18]	0.90
	Weight (kg)	89 ± 18	93 ± 17	1.006 [0.99; 1.01]	0.14
	Body Mass Index (kg/m 2 )	29 ± 5	30 ± 5	1.03 [1.00; 1.05]	0.051
	Body Surface Area (m 2 )	2.1 ± 0.2	2.1 ± 0.2	1.47 [0.86; 2.53]	0.16
	Diagnosis-to-ablation-time (months)	15 ± 18	19 ± 21	1.005 [0.99; 1.01]	0.15
	Long standing persistent AF	13 (7%)	39 (18%)	2.06 [1.45; 2.93]	< 0.001
	Clearance (ml/min)	81 ± 21	82 ± 22	0.99 [0.99; 1.004]	0.52
	Clearance ≤ 60 ml/min	30 (16%)	31 (15%)	1.02 [0.70; 1.49]	0.92
	Hypertension	82 (44%)	111 (53%)	1.37 [1.05; 1.80]	0.022
	Diabetes mellitus	23 (12%)	29 (14%)	1.26 [0.85; 1.87]	0.25
	Heart Failure	46 (24%)	32 (15%)	0.69 [0.47; 1.007]	0.054
	Coronary artery disease	14 (7%)	32 (15%)	1.42 [0.98; 2.08]	0.07
	Absence of SHD	125 (66%)	134 (63%)	0.92 [0.70; 1.22]	0.57
	CHA2DS2-VASc score	1.6 ± 1.4	1.7 ± 1.4	1.09 [0.99; 1.20]	0.08
	Previous cavotricuspid isthmus ablation	14 (7%)	15 (7%)	0.94 [0.55; 1.59]	0.81
	LA area (cm 2 )	25 ± 5	27 ± 5	1

.07 [1.03; 1.10] < 0.001

	LA volume (ml)	158 ± 43	172 ± 40	1

.007 [1.004; 1.01] < 0.001

  : atrial fibrillation, ATa: atrial tachyarrhythmia, LA: left atrium, LCT: left common trunk, LV: left ventricle, LVEF: left ventricle ejection fraction, PVI: pulmonary vein isolation, SHD: structural heart disease

	LA volume Index (ml/m 2 )	76 ± 20	81 ± 20	1.01 [1.006;1.02]	< 0.001
	LVEF (%)	53 ± 12	54 ± 12	1.001 [0.99; 1.01]	0.82
	LVEF < 50%	53 (28%)	46 (22%)	0.86 [0.62; 1.20]	0.37
	LVEF ≤ 40%	39 (21%)	35 (17%)	0.92 [0.64; 1.32]	0.66
	LCT	28 (15%)	35 (17%)	1.02 [0.71; 1.46]	0.92
	Accessory vein	20 (11%)	31 (15%)	1.39 [0.95; 2.03]	0.09
	Third-generation balloon	24 (13%)	26 (12%)	0.92 [0.61; 1.39]	0.69
	Fourth-generation balloon	51 (27%)	27 (13%)	0.98 [0.65; 1.48]	0.93
	Cavotricuspid isthmus ablation at PVI	31 (16%)	37 (17%)	0.91 [0.64; 1.30]	0.61

AF

ACKNOWLEDGEMENTS

The authors thank Helene Duquenne for her help with the data acquisition.

FUNDING SOURCES

No funding sources.