Structure of the Condensed Phase in the Inclusion Process

Abstract

We establish a complete picture of condensation in the inclusion process in the thermodynamic limit with vanishing diffusion, covering all scaling regimes of the diffusion parameter and including large deviation results for the maximum occupation number. We make use of size-biased sampling to study the structure of the condensed phase, which can extend over more than one lattice site and exhibit an interesting hierarchical structure characterized by the Poisson-Dirichlet distribution. While this approach is established in other areas including population genetics or random permutations, we show that it also provides a powerful tool to analyse homogeneous condensation in stochastic particle systems with stationary product distributions. We discuss the main mechanisms beyond inclusion processes that lead to the interesting structure of the condensed phase, and the connection to other generic particle systems. Our results are exact, and we present Monte-Carlo simulation data and recursive numerics for partition functions to illustrate the main points. Keywords Condensation • Inclusion process • Poisson-Dirichlet distribution • Size-biased sampling 1 Introduction Condensation phenomena in stochastic particle systems (SPS) continue to be a topic of major research interest. They can be caused by spatial inhomogeneities (see e.g. [1,2] and references therein) or attractive particle interaction in spatially homogeneous systems, which is the focus of this paper. If the total density of particles exceeds a critical value, the system Communicated by Abishek Dhar.

Poisson-Dirichlet asymptotics in condensing particle systems

“…Our goal here is to understand details of the condensed phase when it extends over more than one site and exhibits a non-trivial structure. Such structures have previously been observed as a result of spatial correlations [WE12, WSJMO09, TTCB10] and as a result of L-dependent stationary weights, with a soft cut-off for site occupation numbers under zero-range dynamics [SEM08] or in the inclusion process [JCG19].…”

“…Our main result provides a generalization to models with more general weights that do not lead to exact expressions for Z L,N , and with non-trivial bulk distribution where 0 < ν ρ (η x ) < ρ. In our proof, we not only make use of the Poisson-Dirichlet distribution's characterisation via size-biased sampling, which was essential for the arguments in [JCG19], but also use the characterisation as the unique reversible distribution under split-merge dynamics as explained in Section 2.1. This allows us to avoid explicit expressions or approximations of the partition function Z L,N which are not always at hand.…”

“…and analogously for GEM [0,α] (θ). Since its introduction in [Kin75] the PD distribution emerged first in population biology [Kin75,EK81], before appearing in statistical mechanics [KMRT + 07, GUW11, BU11, GLU12, IT20] and interacting particle systems [JCG19].…”

Condensation of SIP Particles and Sticky Brownian Motion

“…Remark 6. 7 The choice of the space of all compactly supported smooth functions C := C ∞ k (R) as dense set for our Hilbert space turns out to be particularly convenient since it is a core of the Dirichlet form associated to the Brownian motion. As a consequence, we can make use of the same set also for proving that (41) is satisfied.…”

“…In the context of the SIP on a finite lattice, the authors of [6] showed the emergence of condensates as the parameter k → 0, and rigorously characterize their dynamics. We also mention the recent work [7] where the structure of the condensed phase in SIP is analyzed in stationarity, in the thermodynamic limit. More recently in [8], condensation was proven for a large class of inclusion processes for which there is no explicit form of the invariant measures.…”