Random partitions in statistical mechanics

We consider a family of distributions on spatial random partitions that provide a coupling between different models of interest: the ideal Bose gas; the zero-range process; particle clustering; and spatial permutations. These distributions are invariant for a"chain of Chinese restaurants"stochastic process. We obtain results for the distribution of the size of the largest component.


Introduction
We study systems of random integer partitions that are independent except for a global constraint on their total mass. Such setting appears, directly or indirectly, in diverse systems of statistical mechanics: the ideal quantum Bose gas, the zero-range process, particle clustering, certain coagulation-fragmentations processes, and some models of spatial permutations. A common feature is the possibility of a Bose-Einstein condensation; namely, under some conditions, a phase transition takes place that is accompanied with the formation of large objects. In the language of probability, a single random variable realizes the large deviation required to satisfy the constraint on its sum; this behavior is a well-known hallmark for sums of heavy-tailed random variables, and in fact many of our results can be read as abstract results for (conditioned) sums of independent random variables.
We introduce the setting in Section 2. The random objects are "spatial partitions", that is, collections of integer partitions indexed by locations. The distribution has a product structure but for the global constraint. Two marginals play an important rôle. The first marginal deals with the site occupation numbers; the resulting distribution is that of the ideal Bose gas or of the zero-range process. The second marginal deals with the total integer partitions; the resulting distribution is that of the particle clustering and of spatial permutations. The present study originated in an attempt at unifying the latter two systems, and the links with the former systems were rather unexpected. It is useful to establish connections since many results and properties of one system can be transferred to the others.
The special cases are described in Section 3. The ideal Bose gas can be found in Section 3.1, the zero-range process in Section 3.2, particle clustering in Section 3.3, coagulationfragmentations processes in Section 3.4, and spatial permutations in Section 3.5.
The measures considered in this article are invariant measures for interesting Markov processes. One process represents customers in a "chain of Chinese restaurants" which combines the usual Chinese restaurant process with the zero-range process. It is described in Section 4.1. This actually holds only when the parameters satisfy certain consistency properties. Another process is a coagulation-fragmentation process which is a variant of the Becker-Döring model, see Section 4.2.
The possible occurrence of Bose-Einstein condensation and its consequences are addressed in Section 5. The relevant asymptotic is the thermodynamic limit of statistical mechanics, and the critical density is given by an explicit formula. We study more specific settings in the last two sections, namely, the case of the trap potential in Section 6 and the case of the square potential in Section 7.

2.1.
Setting. An integer partition 1 λ of the integer n ∈ N is a finite decreasing sequence λ 1 λ 2 · · · λ k 1, of arbitrary length k ∈ N, whose elements add up to n: k j=1 λ j = n; one often writes "λ n". We refer to the length k of the sequence as the number of components of the partition λ. Every partition is uniquely determined by the numbers r j (λ) = #{i = 1, . . . , k | λ i = j}, j ∈ N, that count how many times a given integer m appears in the partition λ; they are often called occupation numbers of the partition. We also define the partition of n = 0 as the empty sequence (length 0, all occupation numbers equal to 0). We are interested in random integer partitions that have additional structure. A spatial partition of the integer n is a collection λ = (λ x ) x∈Z d of integer partitions at each site x ∈ Z d . That is, each λ x is a k-tuple (λ x1 , λ x2 , . . . , λ xk ), where k varies and where the integers λ xj satisfy λ x1 λ x2 . . . λ xk 1 (2.1) (and we allow for the "empty" sequence with k = 0). The spatial partition λ is uniquely determined by the numbers r xj that count how many times a given integer j appears in the partition at site x, r xj = #{i = 1, 2, · · · : λ xi = j}.
(2.2) We can form two different types of marginals, summing over sites x or integers j; this gives rise to two different types of occupation numbers. We let r = (r j ) j 1 denote the sums over all the sites, i.e., r j = x∈Z d r xj . The site occupation numbers η = (η x ) x∈Z d are given by Thus for every x, λ x is a partition of the integer η x . Sometimes this aspect is stressed and one calls λ a vector partition of the vector η, written λ η (see e.g. [53]). These definitions are illustrated in Figure 1. The intuition is as follows: we think of x ∈ Z d as sites in space (hence the name spatial partitions). "Space" and "site" should be taken loosely -x can be a particle position, a moment vector x = k in a Fourier transformed picture, or a label for an energy level, see the examples and Table 1 in Section 3. Let Λ n denote the set of spatial partitions such that (2. 4) In the language of statistical mechanics, Λ n is a canonical configuration space with total particle number n. Notice that Λ n is a countable set. For later purpose we also define N n ⊂ N Z 0 as the set of η's with x∈Z d η x = n, and R n ⊂ N N 0 as the set of r's with j∈N r j = n. Let π 1 : Λ n → N n , λ → r(λ) and π 2 : Λ n → R n , λ → η(λ) be the natural projections.
Apart from the space dimension d, the relevant parameters for our probability distribution on Λ n are the following: (i) A potential function V : R d → (−∞, ∞]. (ii) A parameter ρ ∈ [0, ∞) which represents the density of the system. We set L d = n/ρ. (iii) A sequence of nonnegative parameters θ = (θ j ) j 1 . The probability distribution on Λ n is defined as . (2.5) The number Z L,n is the normalization; it actually depends on V and θ, but we alleviate the notation by neglecting to make it explicit. We assume that Z L,n < ∞ The relevant asymptotic is the thermodynamic limit where both n and L tend to infinity, with L such that n = ρL d . We will propose different interpretations of the measure P L,n later. One such interpretation is that P L,n is a canonical Gibbs measure for particles moving in the trap potential V (x/L), forming groups at each site. Particles from different groups do not interact; the parameter θ j is a Boltzmann weight for intra-group interactions. 2 See Section 3 for details.

2.2.
Marginals and conditional probabilities. An advantage of the probability measure (2.5) is that it allows us to switch between Gibbs random partitions and sums of independent random variables (to be precise, integer-valued, infinitely divisible variables). The latter play an important rôle as stationary measures of the zero-range process. The connection is that these measures arise as marginals of P L,n . To the best of our knowledge, this connection has not been noticed before. It allows us to deduce limit laws for random partitions from limit laws for sums of independent variables.
Let P L,n • π −1 1 and P L,n • π −1 2 be the push-forwards of P L,n under the projections onto N n and R n . Set where the first sum is over the partitions λ of the integer m. We can think of h m as the analogue of the normalization Z L,n for a single site (no product over x, no background potential V (x/L)). Later we will discuss the properties of the map (θ j ) → (h m ).
Proposition 2.1. 2 Our interpretation is consistent with the examples given in Section 3 but different from that of Vershik [53] and Pitman [49], Chapter 1.5, who consider ZL,n as a microcanonical rather than a canonical partition function.
(a) The measure P L,n • π −1 1 has the product form The measure P L,n • π −1 2 is of the Gibbs partition form Proposition 2.1 has natural probabilistic proof and interpretation, which we defer to Section 2.4. For now, suffice it to say that the first marginal (2.7) arises as the stationary measure of the zero-range process.
In order to recover the full measure from the marginals, we give below the conditional measures P L,n (λ|η(λ) = η) and P L,n (λ|r(λ) = r). Let ν m be the measure on integer partitions of m given by . (2.9) The normalization h m is defined in (2.6). The measure ν m is the analogue of the measure (2.5) for a single site x. It is an example of a Gibbs random partition which we describe in more details in Section 2.3. The next proposition contains expressions of the conditional probabilities. We leave the elementary proof to the reader. Notice that P L,n (λ|η) does not depend on V , and P L,n (λ|r) does not depend on θ. Part (a) says that, conditioned on the site occupation numbers, the partitions λ x become independent Gibbs partitions. Part (b) says that given r j , the r xj 's are multinomial: each of the r j components of size j chooses the site x with probability p xj .
2.3. Gibbs random partitions. The measure ν m defined in (2.9) can be viewed as describing Gibbs random partitions, which have been studied in detail before, see [6,33,16,31,48,55,45,47,25,7] and Chapters 1.5 and 2.5 in [49]. The main questions deal with the number of elements and their typical size, for given weights (θ j ). The probability that the typical size is equal to is given by See Section 5 for more discussion about the random variable for the typical size of elements, where the random elements is picked with probability that is proportional to its size. We now study the relation between between the θ j 's and the h m 's. This is obviously useful in view of the relation above. But this relation is also conceptually interesting since the θ j s are related to the Lévy measure of a process on N 0 given by the h n s.
Recall that a measure µ is infinitely divisible if for all n ∈ N, there is a measureμ n such that µ =μ n * · · · * μ n is the n-fold convolution of µ n . Similarly we say that a sequence (h m ) m 0 of non-negative numbers h m 0 is infinitely divisible if for all n ∈ N there is a sequence of non-negative numbers (h (n) m ) such that (h m ) is the n-fold convolution of (h (n) m ) m 0 , i.e., h = h (n) * · · · * h (n) . There is a rich theory for infinitely divisible measures [39], closely related to the topic of Lévy processes. Proposition 2.3. Let (h m ) m 0 be a sequence of nonnegative numbers such that h 0 = 1 and m h m z m has nonzero radius of convergence. Then there is a unique sequence (θ j ) j 1 of real numbers such that Eq. (2.6) holds for all m ∈ N. Moreover, we have θ j 0 for all j 1 if and only if (h m ) m 0 is infinitely divisible.
Proof. We note the power series identity Here, t satisfies z e t < R. We recognize the Lévy-Khinchin representation for an infinitely divisible measure in the special case of a measure on N 0 ; it follows from classical results (see e.g. Chapter 18 in [39]) that (p m ) is infinitely divisible if and only if θ j 0 for all j 1. If this is the case, the weights α j = θ j j z j define a measure on N, the Lévy measure of (p m ).
Another question of interest is the relation between the asymptotic behaviors of the sequences (θ m ) and (h m ) as m → ∞. This has been investigated in the references cited at the beginning of this subsection. Relevant results can also be found in the probability literature on the relation between the tails of an infinitely divisible measure and its Lévy measure, e.g. in [30,35]. Here we quote a result of Embrechts and Hawkes [30] on the tail equivalence of an infinitely divisible measure and its Lévy measure; equivalently, on the relation between the tails of (h m ) and (θ j /j). Recall that the convolution between two sequences is defined by (a * b) n = n j=0 a j b n−j . Theorem 2.4. (Embrechts and Hawkes [30]) Let (p n ) n 0 define an infinitely divisible law on N 0 with Lévy measure (α j ) j 1 . Suppose that α j > 0 for all j 1. Let α j = α j /( k 1 α k ). The following are equivalent as n → ∞: (i) (p * p) n = 2(1 + o(1))p n and p n+1 /p n → 1.
We note that the Lévy measure of an integer-valued random variable has always finite mass, hence j 1 α j < ∞.
A probability measure on N satisfying (p * p) n ∼ 2p n is called discrete subexponential. This property suggests that, if two independent variables are conditioned so their sum takes some big value, one variable will take a small value and the other variable will take the appropriate big value. Discrete subexponential variables are necessarily heavy-tailed, that is, n p n z n has radius of convergence 1. We can apply this theorem if j θ j /j < ∞ and θ j+1 /θ j → 1. Let p m = h m exp(− j 1 θ j /j). The condition (ii) becomes The first equality is valid if n is even, there is an unimportant correction in the case of n odd. An immediate consequence of Theorem 2.4 and of the dominated convergence theorem is the following.
Theorem 2.5. Assume that θ n+1 /θ n → 1 and that θ n−j /θ n c j for 1 j n 2 and all n, with c j satisfying θ j c j /j < ∞. Then, as n → ∞, Notice that c j is necessarily greater or equal to 1, so the theorem requires j θ j /j < ∞. It was actually proposed in [16] but the connection with infinitely divisible laws and with Theorem 2.4 had not been noticed. It applies in particular to stretched exponential weights, θ j /j = exp(−j γ ) with 0 < γ < 1.

2.4.
Gibbs partitions and Poisson random variables. We conclude this section by explaining in more details the probabilistic picture behind Proposition 2.1. To this aim we generalize a well-known relationship between Gibbs partitions and Poisson variables, see for example Eq. (1.53) in [49] or the conditioning relation (3.1) in [6].
Let us assume that there exists z > 0 such that for all L > 0, (2.14) Let (R xj (ω)) x∈Z d ,j∈N be a family of independent random variables defined on some common probability space (Ω, F, P z L ). Under the a priori or grand-canonical measure P z L the variables R xj (ω) are independent Poisson variables, The occupation numbers are random variables We also define the total number of particles by A moment of thought shows that the law P L,n is recovered by conditioning on the event x,j jR xj (ω) = n, P L,n (λ) = P z L ∀x ∈ Z d ∀j ∈ N : R xj (ω) = r xj (λ) N (ω) = n , (2.18) and the normalization satisfies Note that in Eq. (2.18) the right-hand side is independent of z; this is related to the invariance of the measure P L,n under rescalings θ j → θ j z j . Under the a priori measure P z L , the variables R j (ω) are independent Poisson variables 20) and the H x (ω) are independent variables with cumulant generating function Put differently, H x (ω) is an integer-valued, infinitely divisible random variable with Lévy measure ν(j) = (θ j /j)z j exp(−jV (x/L)), compare with the proof of Proposition 2.3. Furthermore, with h m as defined in Eq. (2.6). Indeed, .

Relationship with existing models
Our setting is closely related to several models of interest, namely the ideal Bose gas, the zero-range process, particle clustering, coagulation-fragmentation, Becker-Döring, spatial permutations, and population genetics. The relations are explained in this section. Each situation comes with its own language; the keywords and their correspondence is summarised in Table 1.  Zero-range  particle  site  -Chinese restaurant customer restaurant  table  ideal Bose gas,  particle Fourier mode, cycle spatial permutation energy level particle clustering, particle site in space cluster (droplet) nucleation population genetics individual colony same-allele group within colony Table 1. Language of the different models and their correspondence.
3.1. Ideal Bose gas. Although it was not fully appreciated at the time, Bose-Einstein condensation is the first description of a phase transition in statistical mechanics. The ideal Bose gas is a quantum system whose description involves a complex Hilbert space and the Schrödinger equation; but its equilibrium state is a probability distribution of occupation numbers of Fourier modes. It fits our setting, by choosing V (x) = β x 2 and θ ≡ 1; β is the inverse temperature. With the change of variables k = 2π L x, writing λ kj instead of λ xj , P L,n becomes a probability measure on spatial partitions (λ kj ). Summing over j and writing and η k instead of η x = η kL , we get the familiar probability measure on occupation numbers Summing over k we get that the marginal P L,n • π −1 2 is the probability distribution of cycle lengths associated with the ideal Bose gas [52,10]. Thus P L,n provides a coupling of the distribution of momenta and cycle lengths for the ideal Bose gas. This generalizes to independent bosons in a trap, when the weight exp(−β k k 2 η k ) is replaced with exp(−β r∈I E r η r ) where I is a countable index set replacing Z d and E r (r ∈ I) are the eigenvalues of the Schrödinger operator in the trap. Results in this case were recently obtained in [21] (see also [18]).
The interacting Bose gas is much more complicated and it does not fit the present setting. But a partial mean-field approach for the dilute regime suggests that interactions can be approximated by cycle weights θ j [14].

3.2.
Zero-range process. This describes a system of classical particles with stochastic dynamics. There are n particles moving on the sites {1, . . . , L} and we let η ∈ H n denote the occupations of the sites. The dynamics is as follows. A particle exits the site x at rate g(η x ), where g is a given function N → (0, ∞), and it chooses a new site uniformly at random among the neighbors. As it turns out, the spatial dimension does not appear in the stationary measure, so the model is usually studied in d = 1. The invariant measure is where the function h k is related to the rates g by .
It fits the setting studied in this article by choosing the potential V such that e −V (s) = χ [0,1] (s); see Eq. (2.7) . It was noticed by Evans [36] that for certain rates g, the system possesses a critical density where a sort of Bose-Einstein condensation takes place. See also [41,5,4,22] for further studies. Variants of the model allow for motion on graphs other than Z d and hopping mechanisms different from the simple random walk, see [36,54,40] and the references therein. When e −V differs from the characteristic function, the marginal (2.7) is the stationary measure of an inhomogeneous zero-range process, compare with Eq. (2.4) of [40].
3.3. Particle clustering. The measure P L,n describes approximately the droplet size distributions for a system of interacting particles in the canonical Gibbs ensemble, see [51,44] and the references therein. Let v be a pair potential with a finite range R > 0. Thus particles at mutual distance larger than R do not interact. With each configuration (x 1 , . . . , x n ) ∈ [0, L] dn we associate a graph by drawing a line between x i , x j whenever |x i − x j | R, and we call N k (x) the number of connected components having k particles. We have N k=1 kN k (x) = n. We put on [0, L] dn the canonical Gibbs measure at inverse temperature β > 0. If we neglect the constraint that particles belonging to different connected components have mutual distance larger than R, the probability of seeing a given droplet size distribution (N k ) becomes approximately where Z j (β) is a partition function over droplet-internal degrees of freedom [44]. Eq. (3.4) corresponds to a measure of the form (2.8) with x exp(−V (x/L)) replaced by L d , and θ j /j = Z j (β).

3.4.
Coagulation-fragmentation processes and Becker-Döring equations. The Becker-Döring system of coupled ordinary differential equations [9] is a popular model for the dynamics of nucleation, with interesting long-time behavior [8]. A natural stochastic variant of the model is the following. Let (a j ) j 1 and (b j ) j 1 be positive numbers such that for all j, j+1 . We define a continuous-time Markov chain with state space {λ : λ n} and two types of transition: • Coagulation: a monomer decides to join a j-cluster. The transition (r 1 , r j , r j+1 ) → (r 1 − 1, r j−1 − 1, r j+1 + 1) happens at rate a j r 1 r j /L d if j = 1, and a 1 r 1 (r 1 − 1)/L 2 for j = 1. • Fragmentation: a monomer decides to depart from a j-cluster, resulting in the transition (r 1 , r j−1 , r j ) → (r 1 + 1, r j−1 + 1, r j − 1). This happens at rate b j r j .
One can check through detailed balance that the marginal (2.8) with the square potential exp(−V (s)) = 1 [0,1] (s) is a stationary measure of this process. The dynamics is a stochastic version of the Becker-Döring equations much in the same way as the Marcus-Lushnikov coalescent is a stochastic version of the Smoluchowski coagulation equations [2]. The model can be easily generalized to allow for joining and departure of groups of size larger than one, and falls into the class of well-studied coagulation-fragmentation processes, see e.g. [29,12]. In Section 4.2 we propose a "spatial" version of the process for which our measure P L,n is stationary.
3.5. Spatial permutations. Models of spatial permutations are motivated by Feynman's approach to the Bose gas and by Sütő's work on cycles [52]. A more general framework was proposed in [13], which was studied further in [15,17]. Spatial permutations involve a distribution jointly over points in R d and over permutations of these points, with penalties that discourage long jumps. More precisely, the probability space is Λ n × S n , where Λ is the box [0, L] d (with periodic boundary conditions), n is the number of points, and S n is the group of permutations of n elements. Let ξ : The probability density for having points x 1 , . . . , x n and permutation σ is then (3.6) The marginal obtained after summing over permutations is a permanental point process, which we do not discuss here despite interest of its own. We rather focus on properties of permutation cycles. We make the extra assumption that e −ξ has nonnegative Fourier transform, which we write e −V , with V a real function on R d . Then the marginal over the cycle lengths, after integrating over positions and summing over compatible permutations, is precisely the marginal (2.8). This follows from the Poisson summation formula, see [13] for details. Notice that the special case ξ(x) = β x 2 corresponds to the homogeneous ideal Bose gas described in Section 3.1.
3.6. Population genetics. Consider n individuals that carry different alleles of a given gene, and live in different locations or colonies x. Inside each colony, we group individuals that carry the same allele. This gives rise to a spatial partition. In the Ewens case θ j ≡ θ, one might imagine that our measure is stationary for a population model that includes migration (with some colonies possibly more attractive than others) and mutation. Remember that the Ewens sampling formula appears naturally in the infinite alleles model with mutation rate θ; see [28] and references therein for more background.

Stochastic processes
Here we propose two continuous-time Markov processes with state space Λ n that have P L,n as a reversible (hence stationary) measure. The first process combines the zerorange process with the Chinese restaurant process; it has the nice structural property that the vector of site occupation numbers η(λ(t)) is Markovian (regardless of the starting point λ(0)) and evolves as the zero-range process. However we are able to define the Chinese restaurant part only when the weights θ j are j-independent. For non-constant θ j 's, we replace the Chinese restaurant step by instant reshuffling. The second process is a coagulation-fragmentation process that is very natural from the point of view of the Becker-Döring model of nucleation explained in Section 3.4.

4.1.
Chain of Chinese restaurants. Recall that, in the Chinese restaurant process, customers enter the restaurant one by one. The (n + 1)th customer sits next to an existing customer with probability 1 n+θ , and starts a new table with probability θ n+θ . The table occupation is a random partition and the distribution after n customer is given by the Ewens measure. That is, take θ j ≡ θ in Eq. (2.9). We refer to Chapter 3 in [49] for background and details.
We adapt the dynamics to spatial partitions as follows. To each site x ∈ Z d is associated a restaurant. The total number of customers is n and it is conserved. Let η x denote the number of customers at site x, and λ x η x denote the table occupation. We consider a continuous-time Markov dynamics where • A customer exits restaurant x at rate g(η x ); he is chosen uniformly among the η x customers at x. • The new restaurant y is chosen with probability t(x, y). It may depend on L. We assume that y t(x, y) = 1, and that • In the restaurant y, the new customer sits next to another customer with probability 1 n+θ , and at an empty table with probability θ n+θ . With h n defined in (2.6), we take the rate to be g(n) . It is clear that this defines a Markov process on spatial partitions. Let q(λ, λ ) denote the rate of the transition λ → λ . In order to write it explicitly, observe that it is zero unless there exist x, y ∈ Z d and j, k ∈ N (with k = 0) such that and all other occupation numbers r zi unchanged. In this case, the rate is Here, the probability that the customer leaving restaurant x leaves a j-table is and the probability that the customer joins a k- The stationary measure of this process is precisely the measure defined in Eq. (2.5). Indeed, it satisfies the detailed balance condition.
Proof. We only need to consider the case λ = λ . When the customer changes restaurants, we have to check that Using (4.1), one can check that for all λ, λ and all η, we have the identities The detailed balance property follows from Proposition 2.2.
There is a unique Markov semi-group (P t (λ, λ )) t 0 solving the backwards Kolmogorov equations for the rates (4.3). It defines a Markov process (λ(t)) t 0 with state space Λ n that has P L,n as a reversible (hence stationary) measure.
(P t ) defines a strongly continuous contraction semi-group in ∞ (Λ n ) with infinitesimal generator Q: P t = exp(tQ).
Proof. This follows from Lemmas 4.1 and 4.2, and the non-explosion criterion given in [23], Corollary 3.8.
Hence the site occupation numbers evolve according to a (possibly inhomogeneous) zero-range process; the total rate for a customer to leave a restaurant is g(η shows thatP t := exp(tQ) uniquely defines a Markov semi-group on N n , which is an inhomogeneous zero-range process. Eqs. (4.12) and (4.13) become for all λ ∈ Λ n , η ∈ N n . The latter identity can be conviently recast in operator language: This condition is sufficient to ensure that η(λ(t)) t 0 is Markovian with transition rates q(λ, λ ), see Theorem 4 in [19] (the theorem is formulated for finite state spaces but holds in our set-up as well).
Thus we have constructed a process combining the zero-range and Chinese restaurant process for which the measures P L,n are reversible. Finally, let us address the non-Ewens case where the parameters (θ j ) depend on j. There exist Markov processes such that the measure (2.5) is stationary. For instance, take a zero-range process on the (η x ), together with an "instant reshuffling" of the partitions at the two locations where a customer has been lost or gained. The transition rate of the process is One can check that Propositions 4.3 and 4.4 remain true. It would be nice to discover other, more natural Markov processes.

4.2.
Coagulation-fragmentation processes. For our second process, let (a j ) j 1 and (b j ) j 2 be positive sequences such that for all j, In addition to (4.1), we assume that for some z > 0 satisfying Eq. (2.14), we have As the process presented here is a generalization of the stochastic version of the Becker-Döring model presented in Sec. 3.4, we use the vocabulary of nucleation/particle clustering and we speak of particles and clusters (or droplets) instead of customers or restaurants.
There are many possible generalizations: for example, we could allow for the jumping of whole clusters rather than only monomers, or allow two groups of size larger than one to coalesce, or take coagulation-fragmentation rates proportional to powers of the occupation numbers r γ xj , . . . , see [29]. We stick to the process presented here for notational simplicity, and also because it is closest in spirit to the original Becker-Döring model of nucleation [9] set in the framework of kinetic gas theory: monomers are mobile particles of a dilute gas, clusters of size j 2 are small chunks of condensed precipitate and they are deemed immobile (or at least very slow compared to gas particles).
Proposition 4.5. The transition rates specified above define uniquely a Markov semigroup with state space Λ n . The measure P L,n is reversible, hence stationary, for the associated Markov process.
Proof. We leave the proof of the detailed balance conditions to the reader and look at the non-explosion criterion from Lemma 4.2. Let z > 0 be such that Eq. (2.14) holds. The contribution to λ =λ P z L (λ)q(λ, λ ) from coagulation of j-clusters (j = 1) with monomers is estimated as We have used that under P z L the R xj (ω)'s are independent Poisson variables, and we have used the assumption (4.16). The contributions of coagulation of monomers, fragmentation, and random walk can be estimated in a similar way. We conclude as in Lemma 4.2 and Proposition 4.3.
It is natural to ask about the evolution of the marginals η(λ(t)) and r(λ(t)). Let us first look at the site occupation numbers. The rate for a particle to leave a site depends on the number of monomers at this site, i.e., it is not a function of the total number of particles at the site alone. Therefore η(λ(t)) is not Markovian. However, it is natural to approximate the process by replacing the jump rate r x1 t(x, y) by the conditional expectation which is the transition rate for a zero-range process (for the last equality, see Proposition 2.1 in [31]). Now look at the cluster size counts r j = x r xj . Remember that the conditional law of r xj is binomial with r j trials and success probability p xj ∝ exp(−jV (x/L)). Again, r(λ(t)) t 0 will not be Markovian, but it is natural to approximate by a process with effective coagulation rate x∈Z d a j E L,n r x1 (λ)r xj (λ) r 1 (λ) = r 1 , r j (λ) = r j = a j r 1 r j x∈Z d p x1 p xj , (4.20) for j = 1. Similar expressions hold for coagulation of two monomers and for the effective fragmentation rate. In the case of square traps, exp and the effective rates are exactly those of the Becker-Döring process from Section 3.4.

Condensation
We study the asymptotic behavior in the thermodynamic limit n → ∞, L → ∞ at fixed density ρ = n/L d . The model may undergo a phase transition when the density varies. Above a certain transition density ρ t , a condensation occurs that is characterised by the presence of large components.
There are two natural definitions of condensation, one in terms of site occupation numbers (number of customers of a restaurant), used in the zero-range process, and the other in terms of random partitions (number of occupants of a table). We shall actually observe that both definitions are equivalent. The results of this section can be found in Theorems 5.1 and 5.2. The expressions are familiar for the zero-range process and for spatial permutations. The motivation of this section is to present a novel, unified perspective.
Condensation is measured by the following "order parameters": (The existence of the limits over n is proved in the proof of Theorem 5.2 below. The existence of the limits over K is guaranteed by monotonicity.) We say that condensation occurs if µ ∞ > 0 or ν ∞ > 0, and refer to µ ∞ or ν ∞ as the condensate fraction. In principle, it might happen that ν ∞ < µ ∞ . But Theorem 5.2 below shows that µ ∞ = ν ∞ ; the two definitions of condensation are equivalent. Under some assumptions, we prove the existence of a transition density ρ t such that the order parameters are zero for ρ ρ t and positive for ρ > ρ t . In order to define the transition density, let Let z t ∈ (0, ∞] be the radius of convergence of the power series above. The transition density is ρ t = ρ(z t ).
(5.4) It is possible that ρ t is infinite, meaning that no transition takes place. In the case z t = 1 we get The transition density may be finite for two reasons (or a combination of both). First, a trap such as V (x) = x δ yields e −jV ∼ j −d/δ , which is summable if δ < d. This is the case of the ideal Bose gas of Section 3.1, where θ j ≡ 1 and δ = 2. Eq. (5.5) is then the famous formula derived by Einstein in 1924. Second, the parameters θ j may be summable. This is the case of the invariant measure of the zero-range process of Section 3.2 in the regime studied by Evans [36]. There is no confining potential here, e −jV ≡ 1. A very different situation is particle clustering, see Section 3.3, where the parameters θ j are given by certain cluster integrals [43]. The general form of (5.3) with θ j ≡ 1 and V ≡ 0 appeared in the context of spatial random permutations with cycle weights [14,15] discussed in Section 3.5. A variant of the alternative expression (5.3) for zero-range processes with disorder can be found e.g. in [40] (Eq. (4.2)).
If we are given parameters h n instead of the θ j , we can consider the alternative expression n 0 h n z n e −nV (x) dx, (5.6) where the integrand is by definition equal to 0 when V (x) = ∞. Indeed, the right side of Eq. (5.6) is equal to and using (2.11) with z → z e −V (x) we get ρ(z) in (5.3). The expression (5.6) is the density-activity series for the (inhomogeneous) zero-range processes.
We make the following assumptions throughout this section.
It follows from this assumption that the series j 1 θ j j z j has the same radius of convergence z t as the series (5.3) for ρ(z). The next theorem claims that ρ t is indeed the transition density.
Theorem 5.1 (Condensation). Under Assumption 1, we have ν ∞ = µ ∞ = max 0, 1 − ρt ρ . The proof of this theorem can be found immediately after Theorem 5.2 below. It is possible to provide more detailed results about "typical sizes" of components. Choose a customer at random, and consider the size of the element of the partition that he belongs to. Informally, we have Prob(typical size of element is j) = E L,n jR j n . (5.7) Indeed, there are jR j customers in tables with j people. Similarly, let N k denote the number of restaurants with k people: The probability that a random customer finds himself in a restaurant with k people is Prob(typical site occupation is k) = E L,n N k n . (5.9) The next theorem gives the asymptotic behavior of typical sizes. In order to state it, we introduce z 0 (ρ) as the unique solution of the equation ρ(z) = ρ when ρ < ρ t ; we set z 0 (ρ) = z t when ρ ρ t .

Theorem 5.2 (Typical sizes).
Under Assumption 1, we have for all j 1 and k 0, Here, =⇒ denotes the convergence in probability as n, L → ∞ with ρL d = n.
The proof of this theorem can be found at the end of the section. We first use it to prove Theorem 5.1.
Proof of Theorem 5.1. We have and the definitions of ρ t and z 0 (ρ) imply that ν ∞ = max(0, 1 − ρt ρ ). The proof for µ ∞ is similar and is based on the alternative formula (5.6) for ρ(z).
The rest of this section is devoted to the proof of Theorem 5.2. The method relies on the conditioning relation (2.18); we first prove limit laws for the grand-canonical measures P z L and then show that the conditioning merely picks the right activity z = z 0 (ρ). Technicalities arise because the grand-canonical probability to have exactly n particles goes to 0, and because the power series at finite L need not converge at z = z t . Lemma 5.3 below and the limit law for the R j 's are closely related to large deviations results [11,52,10,41,43]; we provide a complete proof because our setting is more general.
Proof. Using Eq. (2.19) but neglecting the probability term, we get For any z < z t , we have by dominated convergence lim sup for any choice of nonnegative integers (r j ) such that jr j = n. Let us introduce the following entropy function on sequences a = (a j ) such that a j 0 and j ja j 1: We have 1 n log Z L,n −I 1 n r 1 , 1 n r 2 , . . . + j 1 log r j ! (r j /e) r j .
(5.15) Let a j be equal to the right side of Theorem 5.2 (a). We choose a sequence r (n) such that jr → a j as n → ∞. By continuity of V around 0, we have const, (5.16) where the constants do not depend on L, j. The logarithm in the first sum in Eq. (5.15) is less than a constant times j, so the summand is less than a constant times ja j . We can then use the dominated convergence theorem and the first sum in Eq. (5.15) vanishes in the limit n → ∞. The last sum also vanishes: Using 1 and a calculation shows that −I(a) is equal to the right side of the expression in Lemma 5.3.
Lemma 5.4. For any z < z t , we have 20) and the limit (a) exists by Assumption 1. For the limit (b), we have We can expand the last exponential so as to get a convergent series of terms of the form e −nV ( 1 L x) with n ∈ N. The Riemann sums converge in the limit L → ∞ by Assumption 1, and the result follows from dominated convergence.
Lemma 5.5. Let z < z t and a > 0. There exist L 0 and b > 0 such that for all L L 0 , we have Proof. We bound the cumulant generating functions and then deduce bounds with the help of Markov's inequality. Since random variables at different locations are independent, we have

(5.22)
Using log(1 + s) s, we find The latter expectation converges by Lemma 5.4 (b) so that everything is bounded by c(z)t 2 for t 1 with c(z) depending on z only. Then .

(5.24)
We can choose a b > 0 indeed. A similar bound can be found for P z L ( 1 L d (N k −E z L N k ) < −a) and this completes the proof of (a). The same method can be used to prove the claim (b) for R j .
Proof of Theorem 5.2. We only prove the claim (b), as the proof for (a) is the same. Let m k be the limit in (b). We have We choose z close to z 0 (ρ) in such a way that (5.27) (If ρ < ρ t , we can choose z = z 0 (ρ); if ρ ρ t , we need to choose z close enough to z t .) From Lemma 5.5 there exist b > 0 and L 0 such that for all L > L 0 , we have Choosing a < b, we get the claim.

Trap potentials
In this section we investigate the shape and location of the condensate for a class of potential functions that have a unique minimum at the origin; square traps are treated in Section 7. Theorem 6.1 states that the condensate is located at the origin (the trap's minimum), and Theorems 6.3-6.6 provide the fluctuations of the occupation of the origin. The fluctuations are governed by infinitely divisible laws that need not be normal or α-stable, as expected for sums of independent random variables that are not identically distributed [39]. Theorems 6.1, 6.3 and 6.4 generalize known results [15,18,21], Theorems 6.5 and 6.6 are new. A graphical overview is given in Figure 2. Section 6.3 discusses the distribution of the partition elements at the origin: the condensate can concentrate on a single large element or be distributed according to some non-trivial law, e.g. a Poisson-Dirichlet distribution.
6.1. Macroscopic occupation of the origin. In this subsection we consider the marginal measure on η and we establish that the excess mass is concentrated at x = 0. We also prove a central limit theorem.
(iii) For every a > 0, there exists a constant C a such that we also assume that the left side converges to e −aV as L → ∞.
For the weights, we assume that θ j > 0 for all j, that lim j→∞ 1 j log θ j = 0, and that j 1 by e −jV ( 1 L x) . It follows from our assumptions that R d exp(−jV (x))dx is bounded from above and below by a constant times j −d/δ , whence z t = 1 and ρ t < ∞. Additional regularity conditions on the θ j s, formulated as conditions on the tails of the h n s, will be imposed in the theorems. They are loosely related to regularity conditions for heavy-tailed random variables [35] and can be checked, in part, with the help of Theorem 2.5.
The assumption δ < d will be important in our proofs. We suspect that a condition on δ is necessary, as for large δ the potential may be too shallow in order to confine the condensate to a single site; in fact the limit δ → ∞ corresponds, formally, to the square traps from Section 7, where all sites are alike and the condensate chooses uniformly among them. It is not clear how large δ should be in order for this to occur.
Recall that H 0 is the random variable for the total occupation of the origin.
Theorem 6.1. Suppose that Assumption 2 holds true, and that there exist constants C, c 0 and a 1 2 (1 − δ d ) such that for all n 1, Assume also that ρ > ρ t . Then as n, L → ∞ with fixed ρ = n/L d , we have The theorem applies to algebraically decaying weights θ j = j −γ , γ > 0, by Theorem 2.5. It also applies to stretched exponential weights θ j /j = e −j γ and algebraically growing weights θ j = j γ with γ > 0 small enough. In the latter case, we indeed have for suitable constants c, C. Eq. (6.2) was proven by Erlihson and Granovsky, see Eq.
(4.66) in [33]. It also follows from results proven independently in [31]. (The proof in [31] was given for specific weights that satisfy θ j = (1 + o(1))j γ , but it can be extended with the help of known results on polylogarithms [38,Chapter VI.8].
For the proof of Theorem 6.1 we follow [18,15], see also [21]. First we express the canonical expectations with respect to the grand-canonical measure at z = z t = 1 with the help of the conditioning relation (2.18). A difficulty here is that j θ j /j can be infinite, in which case P zt L (H 0 = ∞) = 1. Therefore we give a special treatment to x = 0. Let be the number of particles not at 0. 3 Let ρ L t = 1 L d E zc L (M ); this is approximately equal to the critical density: which converges to ρ t by dominated convergence.
Lemma 6.2. Under the same assumptions as in Theorem 6.1, there exist C, c > 0 such that for all L > 0 and all B > 0, If in addition j jθ j /j d/δ < ∞, the same estimate holds with exponent min(d/2, d − δ) instead of (d − δ)/2.
As we shall see later, the additional condition means that M has normal fluctuations of order L d/2 ; if the confining potential is quite shallow, δ > d/2, our large deviations estimate kicks in at fluctuations of the order of L δ only.

Proof. Recall that the cumulant generating function of a Poisson variable
Using | e s − 1 − s| 1 2 s 2 e |s| , the interior sum is bounded by Then there is a constant C > 0 such that Using Markov's inequality,

9)
3 A careful inspection of the definitions shows that the measure µΛ used by Buffet and Pulé [18] and Betz and Ueltschi [15] is, in our notation, the law of M under P z t L . The variable M is also used by Chatterjee and Diaconis [21]. and a similar bound for the probability that 1 to bound the square bracket in (6.6) by a constant times j. The right side in (6.8) is then replaced by const(t 2 L /L d ) jθ j /j d/δ and it is bounded when t L = O(L d/2 ).
Proof of Theorem 6.1. We have θ j j (6.10) so that .

(6.12)
We use the convention that h m = 0 when m < 0. We show that E L,n exp( t L d (n − H 0 )) → e tρt for any t. Convergence in distribution follows from the pointwise convergence of the Laplace transform and we get the claim of the theorem. It is enough to show that for any ε > 0, we have We have (6.14) The last inequality follows from Lemma 6.2. The last term vanishes indeed as L → ∞.

6.2.
Fluctuations of the condensate. We now study the fluctuations of the condensate. The goal is to find the correct scaling α and the limiting random variable X such that For simplicity, we fix the potential V (x) = c x δ in this subsection. It turns out that the phase diagram of the fluctuations is very rich. It is pictured in Fig. 2. We only provide partial results, see the regions in dark colors. The lightly colored region is left to future studies; it would certainly be interesting to know what happens there. The first result deals with normal fluctuations. Let The first result holds in the case where the variance above is finite.  h n+m h n = 1.
Then, as L, n → ∞ with n = ρL d , The conditions on the h n s can be easily verified for many given (θ j ) with the help of Theorem 2.5 and Eq. (6.2). In the case of stretched exponential weights θ j = e −j α , where h n ≈ e −n α , we have h n+m /h n ≈ e −αm/n 1−α so we need b < 1 − α. This is possible only if α < 1 2 min( 1 2 , 1 − δ d ). For larger α, or for weights θ j = j γ with large γ, we expect a deterministic shift in the average occupation number of the origin, small compared to n but large compared to √ n, and then normal fluctuations around the shifted mean; compare with Theorem 7.1 (c).
Proof. This is similar to Theorem 6.1. Since H 0 = ρL d − M , it is enough to show that As in Eq. (6.12), and again using the convention h m = 0 when m < 0, we have We have (6.20) The square bracket is less than j 2 u 2 /σ 2 t L d . From our assumptions and dominated convergence, we get There remains to verify that the second term in (6.19) vanishes in the limit L → ∞. Let α = min( d 2 , d − δ) and ν < α − ad. As in the proof of Theorem 6.1, it follows from Lemma 6.2 that We also have (6.23) We choose ν > d (1 − b), which is possible by the assumptions of the theorem. Then (1)) and the second term in (6.19) vanishes in the limit L → ∞.
The central limit theorem applies to the ideal Bose gas in dimensions d 5. It was noted by Buffet and Pulé that the fluctuations were non-normal in d = 3, of order L 2 [18]. We propose here three theorems about non-normal fluctuations larger than L d/2 . They always apply to situations where the variance σ 2 t defined in (6.16) is infinite. In order to guess the scale of the fluctuations, we observe that This can be shown by controlling the sum over j as in the proof of Lemma 6.2. The first theorem deals with the Ewens case where the weights θ j are constant. The critical density is finite and the variance infinite when 1 < d δ − γ 2. Let (G x ) x∈Z d be i.i.d. Gamma variables with parameters (θ, 1) (that is, the probability density function is Γ(θ) −1 t θ−1 e −t for t 0). Theorem 6.4. Suppose that V (x) = c x δ with d 2 < δ < d, and that θ j ≡ θ is constant. Then if ρ > ρ t , we have When θ = 1, G x is exponential with mean 1 and we recover the result from [18,21]. The limiting random variable of Theorem 6.4 requires some explanations (see [21] for a closely related situation). Each G x has mean θ. Consider Since 2δ > d and x∈Z d \{0} x −2δ < ∞, a standard theorem applies that concerns random series [27, Chapter 2.5]: For every increasing sequence (Λ n ) n∈N of domains with ∪ n Λ n = Z d , the limit Y (ω) = lim n→∞ Y Λn (ω) exists almost surely. The dominated convergence theorem further shows that the limiting random variable has characteristic function It follows in particular that the law of Y is independent of the precise choice of the sequence of domains. The infinite sum in Theorem 6.4 is by definition a variable equal to Y in law.
Next, let Z be the random variable whose Laplace transform for t 0 is Theorem 6.6. Suppose that V (x) = x δ , θ j = j γ with γ < 0, and that 1 < δ d < γ + 2. Then if ρ > ρ t , we have Remark. If δ d, an analogous result for the fluctuations of M in the grand-canonical ensemble at z = z t still holds, but we can no longer use Lemma 6.2 in order to pass from the grand-canonical to the canonical ensemble. We suspect that Theorem 6.6 can be shown with an adequate replacement of Lemma 6.2 (e.g., algebraic deviations bounds based on finite moments), but leave the proof or disproof as an open problem.
We prove these theorems by showing convergence of Laplace or Fourier transforms. Let Combining Eq. (6.30) with asymptotic expansions of I −1 ( e −µ ), I 0 ( e −µ ) around µ = 0, we obtain two lemmas on single-site fluctuations.
Lemma 6.7. Suppose that V (x) = x δ and that θ j = θ for all j. Then under P zt L , for every fixed x ∈ Z d \{0}, we have where G x is a Gamma random variable with parameter (θ, 1).
Now we come to the proofs of the theorems. The proof of Theorem 6.4 is based on Lemma 6.7 and is analogous to the proofs in [18,21] and to the proof of Theorem 6.5, and it is therefore omitted. In contrast with algebraically growing weights no additional condition on γ is needed in order to go from the grand-canonical to the canonical ensemble. This is because h n = θ(θ + 1) · · · (θ + n − 1)/n! is a slowly varying function of n.
Proof of Theorem 6.5. We treat first the fluctuations of M in the grand-canonical ensemble at z = z t . To this aim we look at the Fourier tranform of the law of L −δ(γ+2) (M −E zt L M ) under P zt L ; it is given by a sum over x analogous to Eq. (6.29). We split the sum in two parts. Revisiting the proof of Lemma 6.8, we find that for suitable c and ε, all µ ε δ , and all |t| We apply the inequality to µ = x δ /L δ with x εL, and t = s/L δ(γ+2) , with L large enough so that |s|L −δ(γ+2) We have δ(γ + 2) > d, so the right side can be summed over x ∈ Z d \ {0}. From Lemma 6.8 and dominated convergence, we get Write M ε for the sum of the H x over x εL. We have, for every κ > 0, Proof of Theorem 6.6. For large x , the integrand in Eq. (6.27) behaves like a constant times x −δ(2−|γ|) , which is integrable at ∞. For small x the dominant contribution to the integral comes from t x −(1−|γ|) , which is integrable around 0. Therefore the integral in Eq. (6.27) is well-defined and finite.
Next we treat the fluctuations of M under P zt L . Fix ε > 0. Just as in the proof of Theorem 6.5, one can show that sites x with x εL are irrelevant. Abbreviate t L = t/L α 0 and write f (x; t) for the integrand in Eq. (6.27). Eq. (6.39) stays true [38,Chapter VI.8]. Therefore as ε → 0, uniformly in L, We conclude with the help of Eq. (6.12) as in the proof of Theorems 6.1 and Theorems 6.3, using that by Theorem 2.5, h n = (1 + o(1))θ n /n = (1 + o(1))n −|γ|−1 is a regularly varying function of n.
6.3. Size of partition elements. In this section we have so far discussed the occupation number at x = 0. We now turn to the partition structure of the condensate. Again we consider the potential V (x) = x δ and the weights θ j = j γ . In view of Proposition 2.2 (a), and since by Theorem 6.1 we have H 0 /L d → ρ − ρ t when ρ > ρ t , it is natural to assume that the limiting structure of the partition λ 0 under P L,n is the same as that of a single-site Gibbs partition ν m when m ∼ (ρ − ρ t )L d → ∞. We do not not wish to carry out the proof here (see however the proof of Proposition 7.2 in Section 7) and we base our discussion on the assumption that this is true. Recall the definition (5.1) of ν ∞ , and let be the fraction of particles in partition components of macroscopic size, i.e., of the order of n. It is clear that ν ∞ ν macro . There are three phases in the (γ, ρ)-plane, as illustrated in Fig. 3. For ρ < ρ t , we have ν ∞ = ν macro = 0 by Theorem 5.1.
For ρ > ρ t , the site 0 is macroscopically occupied and the partition structure results from the Gibbs measure on partitions, ν m . There are three possibilities, depending on the parameter γ: γ < 0 The partition displays a unique large element that contains all the indices save a (random) finite number [16]. This gives a phase with macroscopic elements. γ = 0 (Uniform random permutations or the Ewens measure.) The number of partition elements is logarithmic, they have typically a size proportional to m and their joint law is asymptotically a Poisson-Dirichlet law, as for the lengths of cycles in uniform random permutations; see e.g. [6]. Here also, ν macro = ν ∞ . γ > 0 It was proved in [31,25] that typical elements have size of order m 1/(γ+1) , with Gamma distribution. Moreover the number of elements divided by m γ/(γ+1) converges in law to some constant [33,31], and the limit behavior of the partition is described by a deterministic limit shape [34,20]: there is a function w(x) such that for all x > 0, we have the convergence in law under ν m : 1 m γ/(γ+1) #{j : λ j xm 1/(γ+1) } → w(x). (6.51) Remark. The Poisson-Dirichlet process appearing for γ = 0 has been generalized and belongs to a two-parameter family introduced in [50], called two-parameter Poisson-Dirichlet or Pitman-Yor processes. They can be constructed using subordinators or stick-breaking schemes [50] and can appear as limits of more general Gibbs partitions where the weight is allowed to depend explicitly on the number of cycles [49].  The transition line bordering the finite phase is given by the transition density ρ = ρ t defined in (5.5).

Square potentials
This last section is devoted to square traps, e −V (x) = 1 (− 1 variables. And it is relevant in mathematical physics because it describes the stationary measure of the zero-range process, and certain equilibrium measures in droplet formations. The results of Theorems 5.1 and 5.2 are valid here. In the case z t = 1, the transition density (5.5) takes the simple form We refer to our previous discussion of the zero-range process for background and literature behind this formula. In contrast to the confining potentials of Section 6, condensation does not take place at x = 0 but on a random location picked uniformly at random. On that site, the partition has one large element. Instead of H 0 , we consider here the following random variables: The main theorem of this section is due to Nagaev [46], and to Armendáriz, Grosskinsky, and Loulakis [4], see Theorem 7.1 below. It concerns the random variable M L and we complement it with a result for T L . Before this we discuss the grand-canonical measure P zt L and the connection with Cramér series. At the end of the section, we comment on the relation with surface tension and the critical point of fluids.
For concreteness we focus on algebraic and stretched exponential weights, θ j = j −α with α > 1 or θ j = j exp(−j γ ) with 0 < γ < 1. Under P zt L the random variables H x are i.i.d. and their distribution satisfies 3) The last relation follows from Theorem 2.5. The means and variance of H x are The variance is consistent with Eq. (6.24). It turns out that the limit law for stretched exponential weights with fast decay ( 1 2 γ < 1) involves a deterministic shift ∆ L . In order to define it, consider first the cumulant generating function It is finite for t 0. Given τ < 0, let t τ be the unique solution of the equation ϕ (t) = ρ t + τ . Then as τ 0, the Legendre transformφ(τ ) = sup t∈R (tτ − ϕ(t)) has an asymptotic expansion of the form The coefficients λ k are rational functions of the cumulants and they form the Cramér series of H x [42]. In a more general context, the series provides corrections to the central limit theorem when looking at moderate deviations that are larger than √ n but smaller than n; see Chapter 7 in [42]. For heavy-tailed variables, the Cramér series diverges and we truncate it at t := (1 − γ) −1 − 2 + 1. Let It follows from the results of Nagaev [46] that for ρ > ρ t and 1 2 γ < 1, The interpretation of Eq. (7.8) is that one of the L d sites captures almost all of the excess particles (ρ − ρ t )L d , the remaining L d − 1 sites have a small excess ∆ of particles, and the shift ∆ minimizes the free energy penalty f L (∆). In order to compute ∆, it is convenient to change variables and write ∆ = a(ρ − ρ t )L d ; setting the derivative equal to zero yields the equation It has a solution a ∼ γσ 2 L d(γ−1) (ρ − ρ t ) γ−2 . The minimizer ∆ L satisfies The last identity follows from Theorem 2.5). Eq. (7.11) also holds in the case of the algebraic weights θ j = j α with α > 1 [26]. Before formulating the theorem on M L we recall the definition of α-stable random variables. They are needed in the case of slowly decaying algebraic weights (1 < α < 2). They are conveniently defined via their Laplace transform: For t 0, where M (α) = ∞ 0 ( e −y −1+y)y −1−α dy > 0. The random variable Y has zero expectation E(Y ) = 0; its right tail is P(Y x) = (1 + o(1))α −1 x −α as x → ∞, and its left tail P(Y −x) is exponentially small [39]. Recall that ρ t = j 1 θ j and σ 2 t = j jθ j . The results from conditioned sums of random variables can be formulated in our context as follows.
(b) If θ j = j −α with α > 2 or θ j = exp(−j γ ) with 0 < γ < 1 2 , then where Y L converges in law to a standard normal variable. (c) If θ j = j exp(−j γ ) with 1 2 γ < 1, then where Y L converges in law to a standard normal variable, and ∆ L is the deterministic shift of order L dγ L d/2 defined in (7.10).
The next proposition is a variant of the non-spatial Theorem 3.2 in [16].
Proposition 7.2. Suppose that ρ > ρ t and that the weights are from one of the cases in Theorem 7.1. Let (N j ) j 1 be independent Poisson random variables with parameters θ j /j. Then j 1 jN j is almost surely finite, and under P L,n we have the convergence in law The proof is given at the end of this section. Proposition 7.2 implies that for all sequences a L → ∞, P L,n M L − T L a L → 0. Therefore T L and M L have the same fluctuations, and we have proved the following: Thus we have proved a limit law for the largest component of a Gibbs partition of n with probability weights 1 Z L,n j 1 1 r j ! L d θ j j r j (7.14) (see Proposition 2.1), and Eqs. (7.8) and (7.11) provide, up to the factor exp(−L d j θ j /j), the precise asymptotics of the partition function Z L,n .
From a mathematical point of view, Theorem 7.3 is new 4 and it highlights similarities between limit laws for the Gibbs partition (7.14) and sums of i.i.d. variables. These similarities might seem surprising at first, but we have given a natural explanation by providing a coupling (the measure P L,n ) to a model with i.i.d. variables (the H x s).
From a physical point of view, the coupling P L,n establishes an explicit relation between stationary measures of the zero-range process on the one hand and the ideal Bose gas and classical droplet models the other hand. The former is often used for driven systems that are out of equilibrium, the latter belong to equilibrium statistical mechanics. It is amusing to note, moreover, that the intricacies of the large deviations of stretched exponential variables with γ Indeed, it is common to model the free energy of a droplet of volume j as a bulk term proportional to j plus a surface term j γ with γ = (d − 1)/d [51]. Assume that the bulk term is zero and set µ sat = 0 (the chemical potential at which the gas saturates). Then 4 Bogachev and Zeindler prove the existence of a giant cycle of size of order (1 − ρt/ρ)n for a related model of random permutations, but without results on its fluctuations [17, Corollary 5.14]. p(µ) = j (θ j /j) e µj is the pressure of the gas, and Eq. (7.6) gives an asymptotic expansion of the free energy as ρ ρ sat . The variance σ 2 is proportional to the isothermal compressibility κ T of the saturated gas [51]: A formal manipulation in thermodynamics notation shows that which converges to ρ 2 sat σ 2 as ρ ρ sat . Here, V is the volume, N = ρV the particle number, and F = V f (N/V ) the extensive free energy. Eq. (7.7) is up to an additive constant the free energy of supersaturated gas in coexistence with a large droplet; the shift ∆ L ∝ L d−1 ∝ n (d−1)/d is natural because surface tension tends to make the condensate droplet smaller. This gives a physical interpretation to the large deviations for stretched exponential variables with γ = (d − 1)/d 1/2. The anomalous fluctuations arise when the compressibility κ T ∝ σ 2 becomes infinite, i.e., at the critical point of the fluid. In the Fisher droplet model, this is achieved by assigning to each droplet a "surface" free energy more general than j (d−1)/d , with temperaturedependent parameters; see the review [51] and the references therein.
Proof of Proposition 7.2. Observe that for k 0, P j 1 jN j = k = h k e − j 1 θ j /j (7.17) (with h 0 = 1). By Eq. (2.11) the right side adds up to 1 when k is summed over k 0, so that j 1 jN j is almost surely finite. Let S n be the size of the largest partition element, at the site of higher occupation. Clearly S L T L M L ; we show below that S L = T L with high probability. Fix k ∈ N 0 and let δ > 0 with δ < 1 − ρ t /ρ. Choose n large enough so that δn − k > k. Using Proposition 2.2, we get The first term vanishes because 1 n M L → 1 − ρt ρ > δ; the second term vanishes because (θ j /j) is discrete subexponential (Theorems 2.4 and 2.5). Thus we have shown that for all k 0, lim n→∞ ρL d =n P L,n S L = M L − k = P j 1 jN j = k , (7.20) that is, M L −S L converges in law to j 1 jN j . It follows in particular that 1 n S n → 1− ρt ρ . To conclude, we note that since S L T L , we have P L,n T L = S L ) P L,n T L > S L | S L δn + P L,n S L < δn . (7.21) The last term vanishes because 1 n S L → 1 − ρt ρ and δ < 1 − ρ t /ρ. The first term in the right side is bounded by the conditional probability that there is a second site, in addition to the most occupied one, whose occupation is at least δn. With the help of Eqs. (7.8) and (7.11) one can show that the probability of the latter event goes to zero. Hence P L,n (T L = S L ) → 0 and we conclude with the help of Eq. (7.20).
In forthcoming work [32] we will introduce analytic continuations, known as Lindelöf integrals, of the grand canonical ensembles studied in this paper and, by methods of Hardy space analysis and contour deformation, show how the results of this section may be extended to a broader class of decaying weights in which the weights θ j arise as the values, at positive integers j, of a function p(ξ) analytic for ξ ∈ C/ {(−∞, 0]}.