Mean-Field Inclusion Processes

• Mean-field inclusion processes are conservative Markov jump models on complete graphs that incorporate a slow phase to dynamically modulate clustering and condensation.
• A threshold parameter A distinguishes fast and slow particle dynamics, leading to instantaneous condensation where a macroscopic mass condenses on very few sites.
• The limiting process is characterized by a coupled system where a modulated Poisson-Dirichlet diffusion interacts with deterministic slow-phase statistics, ensuring continuous mass exchange.

Mean-field inclusion processes are conservative Markov jump processes on complete-graph geometries in which attractive interaction drives clustering and, at high density, condensation. In the formulation with a slow phase studied in "Modulated Poisson-Dirichlet diffusions arising from inclusion processes with a slow phase" (Gabriel, 18 Jul 2025), the particles are split into a fast phase and a slow phase by a threshold $A\in \mathbf N_0$: sites with occupation above $A$ retain the usual inclusion-process clustering dynamics, while sites with at most $A$ particles interact only at a vanishing rate of order $L^{-1}$. The thermodynamic limit at fixed density produces a two-component infinite-dimensional diffusion in which a condensed phase and a microscopic fluid phase co-evolve, exchange mass, and are coupled through a deterministic control process. The limiting object is a modulated extension of the classical Poisson-Dirichlet diffusion, with instantaneous condensation as the key probabilistic mechanism enabling convergence (Gabriel, 18 Jul 2025).

1. Microscopic definition and slow-phase modification

At the microscopic level, the process is defined on the configuration space

$$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$

where $L$ is the number of sites, $N$ is the number of particles, and $\eta^{i,j}=\eta+e_i-e_j$ denotes the configuration obtained by moving one particle from site $i$ to site $j$ (Gabriel, 18 Jul 2025). In the classical mean-field inclusion process, the generator is

$A$0

The slow-phase model generalizes this by introducing a threshold $A$1. Sites with at most $A$2 particles are assigned to the slow phase, and sites with more than $A$3 particles are assigned to the fast phase. A leading example of the modified generator is

$A$4

The general formulation uses rate functions $A$5 satisfying, for $A$6,

$A$7

with $A$8 in the main theorem, and for $A$9,

$A$0

Accordingly, slow-phase transitions are scaled down by $A$1, whereas fast-phase rates remain of order one in the excess occupation $A$2 (Gabriel, 18 Jul 2025).

This separation of rates is the structural novelty of the model. The slow phase does not remove the inclusion-process condensation mechanism; instead, it modulates it dynamically by introducing a microscopic reservoir whose occupation statistics remain relevant in the limit.

2. Condensation and instantaneous projection onto the fast phase

The condensation mechanism is the same high-density phenomenon known from the inclusion process: at density $A$3, mass above the threshold $A$4 rapidly aggregates into clusters of diverging size (Gabriel, 18 Jul 2025). In the slow-phase setting, however, the limiting picture is no longer solely a condensate evolving autonomously. The system also retains a nontrivial fluid or slow component.

A central result is that condensation is instantaneous in the thermodynamic limit. For any positive time, the fraction of sites in the fast phase is already negligible, even though the condensate carries a macroscopic fraction of the mass. This is formalized by

$A$5

where

$A$6

The statement means that the number of sites carrying a true condensate is a vanishing fraction of the volume immediately after time $A$7 (Gabriel, 18 Jul 2025). The condensed phase therefore occupies negligible volume but non-negligible mass. This feature is the decisive probabilistic input in the scaling limit: the system effectively projects at positive times onto configurations in which condensed mass is concentrated on very few sites, while the remainder of the system is described by slow-phase occupation statistics.

A plausible implication is that the model separates geometric support from mass support particularly sharply: volume concentration and mass concentration become asymptotically distinct observables. That interpretation is consistent with the paper’s emphasis on vanishing fast-site fraction together with macroscopic condensed mass.

3. The limiting modulated Poisson-Dirichlet diffusion

The scaling limit is a two-component infinite-dimensional diffusion $A$8, termed the modulated Poisson-Dirichlet diffusion (Gabriel, 18 Jul 2025). Its state space is

$A$9

where

$L^{-1}$0

The component $L^{-1}$1 describes the ordered sizes of macroscopic clusters above threshold $L^{-1}$2, while

$L^{-1}$3

records the occupation statistics of the slow phase. The two components are coupled through

$L^{-1}$4

Here, $L^{-1}$5 is the limiting fraction of mass available to the condensed fast phase, and $L^{-1}$6 modulates the drift parameter of the diffusion (Gabriel, 18 Jul 2025).

The generator of the limiting process is

$L^{-1}$7

with

$L^{-1}$8

and

$L^{-1}$9

For the inclusion model,

$$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$0

with $$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$1 and $$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$2.

Thus $$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$3 evolves deterministically by a closed ODE system, whereas $$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$4 is a Poisson-Dirichlet-type diffusion whose coefficients are modulated by the current slow-phase state (Gabriel, 18 Jul 2025). The resulting structure is neither a purely autonomous infinite-dimensional diffusion nor a purely deterministic hydrodynamic limit; it is a coupled stochastic-deterministic system in which the stochastic sector is continuously controlled by the deterministic one.

4. Relation to the classical Poisson-Dirichlet diffusion

The limiting diffusion extends the classical Ethier-Kurtz Poisson-Dirichlet diffusion in a precise sense (Gabriel, 18 Jul 2025). When $$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$5, there is no slow phase, $$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$6, and the limit reduces to the usual infinite-dimensional Poisson-Dirichlet diffusion generated by $$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$7. In that regime, the new framework collapses to the standard mean-field inclusion-process scaling limit.

For $$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$8, the diffusion is no longer autonomous because the additional process $$\Omega_{L,N}=\Big\{\eta\in \mathbf N_0^L:\sum_{i=1}^L \eta_i=N\Big\},$$9 modulates both the effective mass parameter $L$0 and the drift parameter $L$1. The paper further establishes the rescaling relation

$L$2

where $L$3 is a classical Poisson-Dirichlet diffusion driven by $L$4 with a time-dependent parameter $L$5 (Gabriel, 18 Jul 2025).

This relation identifies the new limit as a controlled or modulated version of the Ethier-Kurtz diffusion rather than a fundamentally different diffusion class. The significance of that observation is conceptual as well as technical. Conceptually, the familiar Poisson-Dirichlet mechanism for ranked condensate sizes survives intact, but only after rescaling by the currently available condensed mass. Technically, it explains why methods adapted to classical Poisson-Dirichlet diffusions remain relevant, even though the full system now includes a deterministic control channel.

5. Mass exchange and coupled phase evolution

A defining feature of the model is nontrivial mass exchange between the condensed phase and the slow phase on the same macroscopic time scale as the Poisson-Dirichlet diffusion of cluster sizes (Gabriel, 18 Jul 2025). The slow phase traps particles for longer times, but it does not freeze the fast phase. Instead, it feeds back into the condensed mass through the quantity $L$6.

The paper emphasizes that the condensed mass evolves deterministically according to the slow-phase ODEs. In the example $L$7,

$L$8

and therefore the condensed fraction $L$9 changes dynamically (Gabriel, 18 Jul 2025). Condensation is therefore not presented merely as an equilibrium phenomenon; the condensate and the fluid phase co-evolve and exchange mass over time.

This dynamical coupling rules out a common simplification according to which the condensate can be treated as a closed subsystem after formation. In the present setting, the ranked cluster process $N$0 depends continuously on the slow-phase statistics, while those statistics determine how much mass is available to be condensed at all. A plausible implication is that metastable or near-stationary descriptions based only on condensate coordinates would be incomplete whenever the slow phase evolves on the same macroscopic scale.

6. Generator approximation, martingale formulation, and well-posedness

A major technical point is that generator convergence does not hold in the usual strong sense (Gabriel, 18 Jul 2025). The finite-$N$1 generator can be approximated by the limiting generator only up to an error controlled by the number of fast sites: $N$2 This error does not vanish deterministically at the generator level. It vanishes only probabilistically, because instantaneous condensation implies that $N$3 in time average.

The paper explicitly notes that classical estimates of generator differences yield non-vanishing deterministic error bounds in this setting if compared naively (Gabriel, 18 Jul 2025). The missing ingredient is therefore not a sharper deterministic estimate but a probabilistic mechanism eliminating the residual error. For that reason, the convergence proof proceeds through martingale problems rather than direct Trotter-Kurtz convergence.

The limiting dynamics are shown to be a well-posed Feller process on the compact, metrizable state space $N$4. Under Lipschitz assumptions on $N$5, $N$6, and the ODE drift $N$7, the closure of $N$8 generates a Feller process with continuous paths,

$N$9

Existence is proved via finite-dimensional Wright-Fisher approximations together with a rescaling argument, while uniqueness is obtained from a triangular moment hierarchy for the $\eta^{i,j}=\eta+e_i-e_j$0-coordinates (Gabriel, 18 Jul 2025).

The process is supported on the boundary

$\eta^{i,j}=\eta+e_i-e_j$1

meaning that for every positive time the condensed mass saturates the available fraction $\eta^{i,j}=\eta+e_i-e_j$2. This boundary concentration is described as the modulated analogue of the classical fact that the Ethier-Kurtz Poisson-Dirichlet diffusion lives on $\eta^{i,j}=\eta+e_i-e_j$3 for $\eta^{i,j}=\eta+e_i-e_j$4 (Gabriel, 18 Jul 2025). In effect, instantaneous condensation persists at the level of the limiting diffusion as an invariant structural constraint: all mass available for the fast phase is immediately realized as condensed mass.

7. Position within inclusion-process theory

Within the theory of inclusion processes, the slow-phase model isolates a regime in which clustering remains dominant but is no longer the sole asymptotically relevant mechanism (Gabriel, 18 Jul 2025). The thermodynamic limit retains the hallmark condensation phenomenon of the mean-field inclusion process, yet augments it with a deterministic microscopic sector whose state modulates the stochastic condensate diffusion.

The model therefore refines the standard dichotomy between microscopic fluid and macroscopic condensate. Instead of treating the fluid sector as asymptotically negligible, it identifies a class of systems in which the fluid phase survives in the limit through finitely many occupation statistics $\eta^{i,j}=\eta+e_i-e_j$5, and these statistics govern both the amount of condensed mass and the effective drift of the ranked-cluster diffusion (Gabriel, 18 Jul 2025).

This suggests a broader interpretation of mean-field condensation models with separated time scales. When a vanishing but not negligible interaction mechanism is built into the microscopic dynamics, the limiting object need not be purely stochastic or purely deterministic. In the present case, the outcome is a coupled macroscopic limit in which the solid condensed and microscopic fluid phase evolve together, with condensation instantaneous, mass exchange nontrivial, and the limiting dynamics well posed as a Feller process on a compact state space (Gabriel, 18 Jul 2025).