Instantaneous Condensation in Particle Systems

• Instantaneous condensation is defined as the phenomenon where a positive fraction of fast-phase mass concentrates on a vanishing fraction of sites in the thermodynamic limit.
• The inclusion process features a two-phase structure with slow and fast dynamics, resulting in a macroscopic Poisson–Dirichlet diffusion coupled to a deterministic slow profile.
• Rigorous analysis shows that the spatial support of fast-phase sites vanishes over time, ensuring convergence from microscopic dynamics to the limiting two-component process.

Instantaneous condensation is the regime in which a positive fraction of the mass in the fast phase of a condensing particle system concentrates on a vanishing fraction of sites immediately in the thermodynamic limit. In the mean-field inclusion process with a slow phase, this phenomenon is formulated as a vanishing space-time density of sites carrying any fast-phase occupation, even though the fast-phase mass itself can remain positive; it is the key probabilistic input that makes the limiting macroscopic dynamics a modulated Poisson–Dirichlet diffusion rather than a merely formal generator ansatz (Gabriel, 18 Jul 2025).

1. Microscopic setting and two-phase structure

The relevant model is a mean-field inclusion process on a complete graph of size $L$ with $N$ indistinguishable particles and generator

$$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$

on

$$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$

A fixed threshold $A\in\mathbb{N}$ separates a slow phase from a fast phase. For occupations $k\le A$, the rates are of order $1/L$, while for $n>A$ they are asymptotic to $n-A$, so particles above threshold behave as in a classical inclusion process up to a small perturbation (Gabriel, 18 Jul 2025).

This induces a decomposition of each configuration into microscopic slow occupations and macroscopic excess masses. Writing

$\eta_i^+ := (\eta_i-A)_+,$

ordering $N$0 decreasingly as $N$1, and defining

$N$2

the embedding

$N$3

takes values in $N$4, where $N$5 records ranked fast-phase cluster sizes and $N$6 records the slow profile (Gabriel, 18 Jul 2025).

The effective fast-phase mass fraction associated with a slow profile $N$7 is

$N$8

with $N$9. At the microscopic level the corresponding quantity is

$$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$0

This quantity measures how much mass is available for macroscopic clustering after the slow occupations have been accounted for (Gabriel, 18 Jul 2025).

2. Definition of instantaneous condensation

In this setting, condensation refers to the attractive inclusion interaction causing a macroscopic fraction of the total mass to be carried by a vanishing fraction of sites. Earlier mean-field inclusion models without a slow phase already exhibited clusters of order $$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$1, with normalized cluster sizes converging to a Poisson–Dirichlet distribution in the thermodynamic limit (Gabriel, 18 Jul 2025).

With a slow phase present, instantaneous condensation is the sharper statement that if the initial fast-phase mass fraction converges,

$$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$2

then for every $$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$3,

$$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$4

where

$$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$5

Thus the set of sites carrying any fast-phase occupation has vanishing density in space-time (Gabriel, 18 Jul 2025).

The statement is not that the fast phase disappears. On the contrary, the fast-phase mass can remain positive while its support collapses to measure zero. The paper makes this distinction explicit: condensation is “instantaneous in space,” meaning that once the fast phase is present, its spatial support shrinks to a vanishing fraction of sites while the macroscopic fast mass is retained (Gabriel, 18 Jul 2025).

3. Mechanism behind the phenomenon

Two ingredients drive the result. First, the fast phase is dynamically stable on the time scale relevant for the diffusion limit. If $$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$6, then with high probability the fast-phase mass fraction stays bounded away from zero on finite time intervals; if $$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$7, it remains small with high probability. This a priori control prevents the fast phase from being washed out before macroscopic clustering can occur (Gabriel, 18 Jul 2025).

Second, individual sites do not remain in the fast phase for macroscopic times. For a fixed site $$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$8, the occupation process $$\mathfrak{L}_{L,N} f(\eta) = \sum_{i,j=1}^L u_1(\eta_i)\,u_2(\eta_j)\,[f(\eta^{i,j}) - f(\eta)],$$9 can be viewed as a continuous-time inhomogeneous random walk. The analysis couples it to a homogeneous biased random walk $$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$0 whose rates above $$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$1 are of order $$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$2, so excursions above the threshold are very short. Conditioned on $$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$3 staying away from zero,

$$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$4

uniformly over bounded initial occupations at the site. By symmetry, this implies the vanishing space-time density of fast sites (Gabriel, 18 Jul 2025).

The resulting picture is a separation of scales. The slow phase occupies a positive fraction of sites and evolves deterministically in the limit, whereas the fast phase carries positive mass but is confined to finitely many macroscopic clusters. This suggests that the thermodynamic limit is naturally described by a macroscopic cluster partition coupled to a deterministic slow background.

4. Role in the convergence proof

Instantaneous condensation is not an auxiliary regularity statement; it is the missing probabilistic ingredient in the convergence from the microscopic inclusion process to the limiting two-component diffusion. The obstacle is that strong generator convergence fails. For $$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$5 in the core of the limiting generator,

$$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$6

uniformly in $$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$7 (Gabriel, 18 Jul 2025).

The non-vanishing deterministic term $$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$8 comes from the fact that the embedding $$\Omega_{L,N} := \{\eta \in \mathbb{N}_0^L : \sum_i \eta_i = N\}.$$9 tracks $A\in\mathbb{N}$0 only for $A\in\mathbb{N}$1 and the total fast mass, but not $A\in\mathbb{N}$2. Since

$A\in\mathbb{N}$3

the mismatch cannot be eliminated uniformly on the state space (Gabriel, 18 Jul 2025).

Instantaneous condensation resolves exactly this issue. Along trajectories,

$A\in\mathbb{N}$4

and the first term tends to zero in probability by instantaneous condensation. The generator difference therefore vanishes only after time integration along paths, not uniformly. This is the mechanism by which the martingale problem for the limiting operator $A\in\mathbb{N}$5 is recovered (Gabriel, 18 Jul 2025).

5. Limiting dynamics and modulated Poisson–Dirichlet structure

The limiting process is a two-component diffusion $A\in\mathbb{N}$6 on

$A\in\mathbb{N}$7

with generator

$A\in\mathbb{N}$8

Here $A\in\mathbb{N}$9 is a deterministic control process on $k\le A$0, and $k\le A$1 is the ranked fast-phase cluster process. The Poisson–Dirichlet-type part is

$k\le A$2

while $k\le A$3 governs the slow-phase ODE (Gabriel, 18 Jul 2025).

A central consequence is that for all positive times the limit process lies on the boundary

$k\le A$4

Equivalently, for any initial condition,

$k\le A$5

almost surely. The paper identifies this as the analytic embodiment of instantaneous condensation at the level of the limit process (Gabriel, 18 Jul 2025).

Conditionally on $k\le A$6, the fast-phase cluster process can be represented as a classical Poisson–Dirichlet diffusion rescaled by $k\le A$7: $k\le A$8 This shows that the condensed phase retains Poisson–Dirichlet structure while exchanging mass nontrivially with the slow phase through the evolving control variable $k\le A$9 (Gabriel, 18 Jul 2025).

6. Interpretation, analogies, and broader context

Instantaneous condensation clarifies what “condensation” means in this class of models. It does not mean that one distinguished site deterministically captures all excess mass, nor that the number of fast sites is literally finite at every finite $1/L$0. The precise asymptotic statement is weaker and more structural: the fast phase occupies vanishing volume fraction in space-time, while its total mass can stay strictly positive (Gabriel, 18 Jul 2025).

This phenomenon also sharpens the connection between condensing particle systems and Poisson–Dirichlet asymptotics. In condensing stochastic particle systems with stationary product measures, the condensed phase converges to Poisson–Dirichlet statistics under broad heavy-tail conditions on the stationary weights (Chleboun et al., 2021). The modulated inclusion-process limit furnishes a dynamic realization of that connection in which the Poisson–Dirichlet component is coupled to a slow phase and justified by an explicit instantaneous-concentration mechanism rather than by equilibrium asymptotics alone (Gabriel, 18 Jul 2025).

A further analogy is with entrance-boundary phenomena in infinite-dimensional Poisson–Dirichlet diffusions. For Petrov’s diffusion, the subset of the Kingman simplex with $1/L$1 acts like an entrance boundary: for every initial state, the process is in the full-mass simplex for all $1/L$2 almost surely (Ethier, 2014). The limiting inclusion-process result is not stated in the same language, but the concentration property

$1/L$3

suggests an analogous boundary mechanism: configurations with delocalized fast mass are instantaneously projected onto the condensed boundary of the state space (Gabriel, 18 Jul 2025).

In that sense, instantaneous condensation is both a microscopic and a macroscopic concept. Microscopically, it is the vanishing space-time density of fast sites. Macroscopically, it is the statement that the condensed phase is immediately confined to a boundary manifold carrying a Poisson–Dirichlet-type diffusion.