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A condensation transition is said to occur when a non-zero fraction of all the par- ticles typically accumulate on a single lattice site (or vanishing volume fraction). This phenomenon has been the subject of recent research interest, as an interesting class of phase transitions that can occur even in one dimensional systems. As suggested in Section 2.2.2, condensation transitions in particle systems with unbounded local state space can be interpreted as ‘jamming’ transitions in exclusion models with long-range interactions. To be more specific we discuss details of the transition in the context of the homogeneous zero-range process, however this class of transitions is not restricted to this model (as is observed in Chapter 6). Condensation due to spatial inhomogeneities has been studied for zero-range processes in [9, 45], in relation to jamming in exclusion models [87], and in models with a single defect site [3].

Condensation can occur in a homogeneous zero-range process if the jump ratesg(n) asymptotically decay with the number of particles n. In terms of the exclusion process representation in Section 2.2.2 this would correspond to long range dependence of the

jump rates. A prototypical model with rates

g(n) = 1 + b

nγ forn= 1,2, . . . (2.35)

has been introduced in [42], where condensation occurs for parameter valuesγ ∈(0,1), b > 0 or γ = 1, and b > 2. If the particle density ρ exceeds a critical value ρc, the

system phase separates into a homogeneous background (fluid phase) with density ρc

and a condensate (condensed phase), where the excess particles accumulate on a single randomly located lattice site. This transition has been established on a rigorous level in a series of papers [4, 5, 69, 80] for the thermodynamic limit, as well as on a finite system as the total number of particles diverges [52]. Dynamic aspects of the transition such as equilibration and coarsening [59, 69] and the stationary dynamics of the condensate [61] are well understood heuristically. For the latter first rigorous results have been achieved recently [8]. In this work it was shown that on a finite system as the particle number diverges the time dependent location of the condensate converges to an effective Markov process, and they were able to calculate the transition rates in the case of reversible dynamics.

2.3.1 Equivalence of ensembles

By choice of the jump rates (2.35) the grand canonical single site partition functionz(µ) (see (2.14)) turns out to converge on the boundary of its domain, and its first derivative is finite. This implies that the average density under the grand canonical measures is in- creasing on (−∞, µc] andρc=R(µc)<∞(see (2.16)). So the grand canonical measures

exist only for densities up to (and including) ρc, and product stationary distributions

do not exist with higher average density. The grand canonical measure with average density ρc is referred to as the ‘critical measure’, and the critical single site marginals

decay sub-exponentially,

lim

n→∞

1

nlogνµc[n] = 0.

SinceR(µ) is strictly increasing on (−∞, µc] it is invertible, and we denote the inverse by

µ(ρ) (with slight abuse of notation). In this way we can parametrise the grand canonical measures by densities ρ∈ [0, ρc]. We extend this function to higher densities by fixing

the chemical potential constant atµc,

¯ µ(ρ) =    µ(ρ) forρ≤ρc µc forρ > ρc. (2.36)

The first rigorous results on the equivalence of ensembles were given in [69]. This can be stated in terms of the relative entropy (see Appendix A) as in the following theorem.

Theorem 2.7. Let µ¯ be defined as in (2.36). Then the specific relative entropy between

πL_bρLc andν_µ(ρ)L_¯ asymptotically vanishes,

lim L→∞ 1 LH(π L bρLc|νµ(ρ)L¯ ) = 0 .

For x∈R+ we denote the largest integer less thanx by bxc (the integer floor).

Proof. See proof of Theorem 1 in [69].

Convergence in relative entropy implies weak convergence of the canonical measures to the grand canonical measure on finitely many lattice sites. We generalise this concept of equivalence of ensembles in Chapter 3 in order to deal with more refined scaling limits and generalized models with size-dependent parameters. This result implies that below ρcthe canonical measures converge locally to the grand canonical measures. Above the

critical density the canonical measures converge locally to a product of the critical grand canonical marginals, with density ρc, and the excess mass accumulates on a vanishing

volume fraction.

2.3.2 Condensation

Further to the equivalence of ensembles and phase separation, it has been proved that the condensed phase occupies only a single lattice site [69] which is located uniformly at random on the lattice and typically contains all of the excess mass. The result can be stated in terms of the normalised maximum site occupation which satisfies a weak law of large numbers, ML:= 1 Lxmax∈ΛL ηx πL_bρLc −−−→(ρ−ρc) asL→ ∞, where πL bρLc

−−−→denotes convergence in probability, i.e.

π_bL_ρLc[|ML−(ρ−ρc)|> ]→0 asL→ ∞ .

The equivalence of ensembles in the case of supercritical densitiesρ > ρc, for the bulk

of the system after removing the maximum component was strengthened by Armend´ariz and Loulakis [4] (with partial results already in [80]). Here it was shown that, after removing the condensate, the canonical measure converges in total variation to the critical grand canonical measures on the rest of the system. This result was first shown for a fixed number of sites as the total number of particles diverges [52]. To state the result we define the cut operator that removes the maximum occupied site, C :XL →

XL−1, (C(η))x=    ηx ifx < i(η), ηx+1 ifx≥i(η) , (2.37)

wherei(η) is the (smallest) index of the lattice site containing the maximum. Then the strong form of equivalence of ensembles above the critical density can be stated as,

For eachρ > ρc, π L bρLc◦C −1_−νL−1 µc T.V.→0 asL→ ∞,

wherek·k_T.V.is the total variation norm, defined in (A.4). This gives rise to fluctuations of the maximum, which have already been studied in [80], heuristically in [59], and using saddle point calculations in [47]. It has been shown that in the condensed regime ρ > ρc, if the critical grand canonical measure has finite variance, the fluctuations of the

maximum are Gaussian, otherwise they are given by a completely asymmetric stable law. For ρ ≤ρc the fluctuations are given by standard extreme value statistics for maxima

of independent random variables [47], and the transition between different distributions at the critical point has been studied recently in [5].