Pith. sign in

REVIEW 2 major objections 5 minor 43 references

Modulated Poisson-Dirichlet diffusions arising from inclusion processes with a slow phase

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Inclusion processes with a slow phase have a two-component Poisson–Dirichlet diffusion limit.

desk verdict New two-component scaling limit, mostly solid proof, but the well-posedness of the limit diffusion is not fully proven and the gap is load-bearing. read the letter →

arxiv 2507.13799 v1 pith:BV7UGWMJ submitted 2025-07-18 math.PR cond-mat.stat-mech

Simon Gabriel This is my paper · ORCID
classification math.PRcond-mat.stat-mech MSC 60K3560J2582C2282C26
keywords modulatedPoisson–DirichletdiffusioninclusionprocessslowphasecondensationthermodynamiclimitinstantaneousFellermartingaleproblem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a mean-field inclusion process with an additional slow phase has a well-defined dynamic scaling limit in the thermodynamic limit $N/L \to \rho > 0$. The limit is a two-component Feller diffusion: a Poisson–Dirichlet diffusion $X$ whose cluster sizes describe the condensed fast phase, and a deterministic control process $Y$ describing the occupations of the slow phase, which modulates both the total mass $\gamma(Y)$ and the drift $\theta(Y)$ of $X$. The paper proves that the fast phase condenses instantaneously, so the fraction of sites occupied by fast-phase particles vanishes for almost every positive time, and this supplies the missing probabilistic ingredient needed to compare generators. If the theorem is right, condensation with a slow phase is not a static equilibrium phenomenon but a dynamical one, with nontrivial mass exchange between the phases occurring on the same time scale as cluster evolution.

What carries the argument

The load-bearing object is the modulated Poisson–Dirichlet diffusion $(X,Y)$ on the compact state space $S = \{(x,y): \sum_i x_i \le \gamma(y)\}$, generated by $L = A_{\gamma(y),\theta(y)} + G$, where $A_{\gamma,\theta}$ is the Ethier–Kurtz generator with coalescence parameter $\gamma$ and drift $\theta$, and $G = \sum_{k=0}^{A-1} b_k(y)\partial_{y_k}$ drives the deterministic control $Y$; for the inclusion process $G$ is the explicit generator in (27) and the modulation functions are $\gamma$ and $\theta$ from (28). The proof is carried by two estimates that fit together: Lemma 4.1 approximates the particle-system generator applied to $f\circ\pi$ by $Lf(\pi(\eta))$ up to an error proportional to the fraction of sites occupied by fast-phase particles, and Proposition 4.2 shows that this fraction integrates to zero, using a coupling of the tagged occupation $\eta_1(t)$ to a comparison random walk whose excursions above level $A$ have almost surely vanishing duration.

What would settle it

Simulate the inclusion process with $A=1$ and supercritical density $\rho>1/2$, starting almost all particles in the fast phase, and measure $(1/T)\int_0^T \#_{>A}\eta(t)/L\,dt$ as $N/L\to \rho$; if the integrated fraction of fast-occupied sites does not vanish, Proposition 4.2 and Theorem 2.2 fall. Alternatively, exhibit two distinct solutions to the martingale problem for $L$ under Assumption 1 with $\gamma$ only $C^1$, which would falsify the claimed well-posedness.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.2: for inclusion processes whose rates satisfy Assumption 2, if the embedded initial configurations $\pi(\eta^{(L,N)}(0))$ converge to $(x,y)\in S$ as $N/L\to \rho > 0$, then the embedded processes converge in law in $D([0,\infty), \nabla \times S_Y)$ to the modulated Poisson–Dirichlet diffusion $(X,Y)$ started from $(x,y)$, with generator $L = A_{\gamma(y),\theta(y)} + G$. Here $G$ is the generator in (27), $\gamma(y) = (1 - \rho^{-1}\sum_{k=0}^A k y_k)_+$, and $\theta(y) = \sum_{k=0}^A (r_k - q_k)y_k$ with $y_A = 1 - \sum_{k=0}^{A-1} y_k$. Thus cluster sizes in the fast phase evolve by the Poisson–Dirichlet diffusion while the slow phase follows an explicit system of ODEs. The paper also proves well-posedness of this modulated diffusion as a Feller process on a compact state space and, as its key probabilistic input, Proposition 4.2, which shows that the integrated fraction of sites occupied by fast-phase particles vanishes in the thermodynamic limit.

Load-bearing premise

The load-bearing premise is that the modulating functions are smooth enough for the limiting generator to define a genuine Feller process: $\gamma(y) = (1 - \rho^{-1}\sum_{k=0}^A k y_k)_+$ is only $C^1$ on its support, while Lemma 3.2 applies the generator to compositions that need $C^2$ regularity, and Remark 3.4 acknowledges this gap with only a sketched stopping-time repair.

Editorial extensions

If this is right

  • The full co-evolution of condensed clusters and the slow phase is captured by a Feller diffusion, so cluster-size dynamics and slow-phase mass exchange can be analyzed without resolving the microscopic particle system.
  • For $A=0$ the control process disappears, $\gamma\equiv 1$, and Theorem 2.2 reduces to the known convergence of the inclusion process to the classical Poisson–Dirichlet diffusion $A_{1,\Theta}$.
  • The deterministic limit $\gamma(Y(t))$ gives an explicit law of motion for the mass in the condensed phase; for the leading example with $A=1$ it evolves by $\gamma(t) = 1 - 2\rho/(e^{\Theta t(1-2\rho)}-2\rho)$, revealing a critical density $\rho_c = 1/2$.
  • Whenever $\rho > A$, the slow phase converges to its stationary profile $y$ and the mass in the condensed phase converges to $1 - \rho_c/\rho$, so the equilibrium phase transition of [CGG22] persists in the non-equilibrium dynamics.
  • Positive-time concentration on the boundary $\sum_i X_i(t) = \gamma(Y(t))$ is a corollary, so at every positive time the condensed phase exactly fills the mass allowed by the slow phase.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same two-step architecture—generator approximation up to an error controlled by the fraction of fast-occupied sites, plus an instantaneous-condensation estimate—should transfer to other condensing systems such as zero-range processes with two rate scales, provided an analogue of Lemma 5.2 can be built.
  • Removing the hard threshold $A$ would plausibly produce a doubly infinite-dimensional limit in which the slow occupations become a continuum profile and the Poisson–Dirichlet component is modulated by an infinite-dimensional control; the hard cut-off appears to be a technical convenience rather than a necessity.
  • The difficulty in proving generator convergence directly, discussed in Remark 4.4, suggests that the projection from $D([0,\infty),\nabla \times S_Y)$ to $S$ is discontinuous at time zero, a feature that may also appear in other instantaneous-condensation limits and could be studied as a projection phenomenon.
  • A quantitative version of Proposition 4.2 tracking $\zeta_L\int_0^T \#_{>A}\eta(t)\,dt$ would extend the theorem to rate perturbations slower than $1/L$ and give a testable threshold on how slow the slow phase may be.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper defines a two-component infinite-dimensional diffusion, the modulated Poisson–Dirichlet diffusion, in which the classical Poisson–Dirichlet generator has parameters γ(Y(t)) and θ(Y(t)) driven by a deterministic ODE control Y. Proposition 2.1 claims this process is well-posed as a Feller process on a compact state space, and Theorem 2.2 claims that the inclusion process with a slow phase, embedded via ordered fast-phase masses and slow-phase occupation fractions, converges in law to this diffusion. The proof proceeds by finite-dimensional Wright–Fisher approximation, a generator-difference estimate, and a new direct proof of instantaneous condensation of the fast-phase mass. Explicit formulas (27)–(28) give the limiting generator and modulation parameters in terms of the microscopic rates, with no fitted parameters.

Significance. If the gaps in the paper are repaired, this is a substantial contribution: it provides a genuinely new infinite-dimensional limit process, gives an explicit and parameter-free derivation of the limiting generator, and supplies a direct probabilistic proof of instantaneous condensation rather than deriving it from the limit. The manuscript is also unusually honest in flagging its own weak points: Remark 3.4 concedes a domain-regularity omission, and Remark 4.4 concedes that direct generator convergence was not achieved. These self-identified issues are exactly where the load-bearing work remains.

major comments (2)
  1. [§3.1, Lemma 3.2 and Remark 3.4] The existence proof applies L^(M) to f composed with Π, but for the model's own γ(y) = (1 − ρ^{-1} Σ_{k=0}^A k y_k)_+, the function f∘Π is not C^2 across the kink {γ = 0}. Remark 3.4 explicitly acknowledges this and sketches a stopping-time construction at T = inf{t : γ(Y(t)) = 0}, but the remark does not prove that the stopped Wright–Fisher martingale problem is well-posed, that the generator identity remains valid on the stopped process, or that the post-T extension is a genuine solution to the original martingale problem. Because Theorem 2.2 defines its target process through Proposition 2.1, this gap is load-bearing and must be closed with a complete argument, not a sketch.
  2. [§3.1, Lemma 3.5] The uniqueness proof in Lemma 3.5 establishes uniqueness of the one-dimensional time marginals µ_t of X(t), but Proposition 2.1 asserts well-posedness of the martingale problem, i.e. uniqueness of the law of the full process (X,Y) in C([0,∞),S). The citation [EK86, Theorem 4.4.2(a)] concerns well-posed martingale problems and does not imply process-level uniqueness from uniqueness of one-dimensional marginals. Since Theorem 2.2 identifies every subsequential limit as a solution of this martingale problem, absence of genuine process-level uniqueness leaves the limit object ambiguous. The proof needs an argument for uniqueness of finite-dimensional distributions or an equivalent direct proof of martingale-problem well-posedness.
minor comments (5)
  1. [§2.2, after Eq. (29)] For the leading example (7), Assumption 2 gives q_k = r_k = Θ for k = 1,...,A and q_0 = 0, r_0 = Θ, so the theorem's θ(y) = Σ_{k=0}^A (r_k − q_k)y_k equals Θ y_0, not the constant Θ. The sentence "θ(·) = Θ > 0 is fixed" is therefore inconsistent with the theorem's own formula; please either correct the example or clarify the different rate convention being used.
  2. [§3.1, Remark 3.4] There is a typo: "We ommited this complication" should be "We omitted this complication". More importantly, the remark's stopping-time argument is only one paragraph and should either be expanded into a proof or moved to the main text as a lemma.
  3. [§5.2, Eq. (66)] The display ending with "T M/N L" is ambiguous; from the surrounding text it appears the intended term is T · (N/L)/M, which vanishes in the limit N/L → ρ before M → ∞. Please rewrite this bound with explicit fractions.
  4. [Appendix A] The appendix states that uniqueness of the martingale problem for the multi-loci Wright–Fisher diffusion "is expected to hold" but no proof or precise reference is given. Since the main text does not use this uniqueness, the statement is not load-bearing, but it should be labeled as conjectural or supported by a citation.
  5. [Assumption 1(c) and Eq. (26)] The notation supp γ is used as if it were the open set {γ > 0}: Assumption 1(a) writes S_Y^* = S_Y \ supp γ. Standard notation would make supp γ the closure of this set, on which the positive part function is not C^1. Please define supp γ explicitly to avoid confusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the limiting diffusion is constructed from the model rates by explicit generator expansion, with the only flagged issue being an acknowledged smoothness gap in the well-posedness proof, a rigor concern rather than a circular reduction.

full rationale

Theorem 2.2's limit is not fitted or presupposed: Section 6 (Lemma 4.1) obtains Lf(pi(eta)) by direct Taylor expansion of the inclusion generator L^{L,N}, and all limit coefficients—G in (27), gamma and theta in (28), and the state-space constraint (26)—emerge algebraically from the model rates q_k, r_k and the embedding pi; no parameter is calibrated to the claimed output. The error term O_f(#_{>A}eta / L) is controlled by the independently proved instantaneous-condensation estimate (Proposition 4.2, Section 5), which is established by explicit random-walk coupling rather than by assuming the limit. Well-posedness of the modulated PD diffusion is argued through finite-dimensional Wright-Fisher approximations (Lemma 3.2) using external existence results [Eth76], [EK86], and uniqueness via an explicit moment hierarchy (Lemma 3.5); the classical PD facts imported from [EK81] are standard external support. The only substantive caveat is the paper's own Remark 3.4: f∘Pi may fail to be C^2 at the kink of gamma on {gamma=0}, because gamma is only C^1 on supp gamma, and the proposed stopping-time extension is sketched rather than proved. This is an acknowledged correctness/rigor gap in the well-posedness argument for Proposition 2.1, and hence a load-bearing weakness for Theorem 2.2, but it is not a circularity: no claim is defined in terms of itself, no fitted quantity is relabeled as a prediction, and the self-citations [CGG22], [CGG24] are used for context, the A=0 special case, and the stationary phase-transition discussion rather than as premises in the convergence proof. Score 1 reflects the noted gap and minor self-citations, not a circular reduction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper's central claim rests on the rate-scaling assumption (Assumption 2), the regularity assumptions (Assumption 1), and standard infinite-dimensional diffusion theory. No parameters are fitted; all constants in the proof are explicit functions of the model data.

assumptions (4)
  • standard math Classical Poisson-Dirichlet diffusion exists and has the uniqueness and invariance properties stated in [EK81].
    Used as the building block in Lemma 3.6 and Section 3.2; it supplies the invariant measures PD(beta*) and the convergence to the boundary of the simplex for positive times.
  • standard math Multi-loci Wright-Fisher diffusions with Lipschitz drift admit martingale problem solutions in C([0,infty), K_d) (Appendix A, [Eth76]).
    Lemma 3.2's existence proof approximates the modulated PD diffusion by these finite-dimensional diffusions.
  • domain assumption Rate scaling Assumption 2: u_iota(n) is close to n-A for n>A and u_iota(k)L is close to q_k, r_k for k<=A, with zeta_L = O(1/L).
    Defines the class of particle systems to which Theorem 2.2 applies; all constants in the limit formulas are built from q_k, r_k, rho, A.
  • domain assumption Regularity Assumption 1: Lipschitz gamma and theta, absorbing gamma=0 set, and a Lipschitz non-negative extension of beta from supp gamma.
    Needed for well-posedness of the limit process; the author verifies it for the inclusion-process model, but the positive-part gamma is only C^1 on supp gamma, causing the gap acknowledged in Remark 3.4.
invented entities (1)
  • Modulated Poisson-Dirichlet diffusion (X,Y) independent evidence
    purpose: The conjectured and then proven scaling limit describing the co-evolution of condensed and fluid phases
    Not pulled from a hat: its generator (16) is derived from the microscopic rates via (27)-(28), and it reduces to the classical PD diffusion when A = 0. Its stationary measures and boundary behavior are characterized, providing falsifiable handles.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Modulated Poisson-Dirichlet diffusions arising from inclusion processes with a slow phase." pith.science (2026). https://pith.science/paper/BV7UGWMJ

@misc{pith2026250713799,
  author       = {Pith},
  title        = {Pith review of: Modulated Poisson-Dirichlet diffusions arising from inclusion processes with a slow phase},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BV7UGWMJ}},
  note         = {Machine review of arXiv:2507.13799}
}
read the original abstract

We study mean-field inclusion processes with an additional slow phase, in which particle interactions occur at a vanishing rate proportional to the inverse system size. In the thermodynamic limit, such systems exhibit condensation at high particle density, forming clusters of diverging size. Our main result provides convergence in law of inclusion processes to a novel two-component infinite-dimensional stochastic diffusion, describing the co-evolution of the solid condensed and microscopic fluid phase. In particular, we establish non-trivial mass exchange between the two phases. The resulting scaling limit extends the Poisson-Dirichlet diffusion (Ethier and Kurtz, 1981), introducing an additional control process that modulates its parameters. Our result builds on classical estimates of generator differences, which in this setting yield non-vanishing deterministic error bounds. We provide the missing probabilistic ingredient by showing instantaneous condensation, with particle clusters concentrating on a vanishing volume fraction immediately. We further establish the well-posedness of the limiting dynamics generally as Feller processes on a compact state space.

Figures

Figures reproduced from arXiv: 2507.13799 by the authors.

Figure 1
Figure 1. Sample particle configuration η with a slow phase. For instance, let us consider rates of the form LL,N f(η) = X L i,j=1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. below depicts this non–trivial mass evolution of the fast phase. 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 t γ(t) [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. On the left we see a possible initial particle configuration, [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Evolution of η1(t) (in black) and Z(t) (in red). For rates satisfying (77), there exists a coupling of the two processes such that Tn ⩽ T ′ n and Rn ⩾ R′ n , i.e. the duration of η1 spent away from A is bounded by the duration of Z away from A, see (79). 28 [PITH_FULL…
Figure 5
Figure 5. Figure 5: The left figure shows the coupling of the embedded discrete–time random walks [PITH_FULL_IMAGE:figures/full_fig_p034_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Ayala, G

    M. Ayala, G. Carinci, and F. Redig. Condensation of SIP Particles and Sticky Brownian Motion . Journal of Statistical Physics , 183(3):1--42, may 2021

    work page 2021

  2. [2]

    Bianchi, S

    A. Bianchi, S. Dommers, and C. Giardin \` a . Metastability in the reversible inclusion process . Electronic Journal of Probability , 22:1--34, jan 2017

    work page 2017

  3. [3]

    Bonorino, R

    L. Bonorino, R. De Paula, P. Gon c alves, and A. Neumann. Hydrodynamics of porous medium model with slow reservoirs. Journal of Statistical Physics , 179(3):748--788, 2020

    work page 2020

  4. [4]

    Baldasso, A

    R. Baldasso, A. Menezes, O.and Neumann, and R.R. Souza. Exclusion process with slow boundary. Journal of Statistical Physics , 167(5):1112--1142, 2017

    work page 2017

  5. [5]

    Costantini, P

    C. Costantini, P. De Blasi, S.N. Ethier, M. Ruggiero, and D. Span \` o . Wright Fisher construction of the two-parameter Poisson Dirichlet diffusion . The Annals of Applied Probability , 27(3):1923--1950, jun 2017

    work page 1923

  6. [6]

    Chleboun and S

    P. Chleboun and S. Grosskinsky. Condensation in stochastic particle systems with stationary product measures. Journal of Statistical Physics , 154:432--465, 2014

    work page 2014

  7. [7]

    Chleboun, S

    P. Chleboun, S. Gabriel, and S. Grosskinsky. Poisson-Dirichlet asymptotics in condensing particle systems . Electronic Journal of Probability , 27:1 -- 35, 2022

    work page 2022

  8. [8]

    Chleboun, S

    P. Chleboun, S. Gabriel, and S. Grosskinsky. Size-biased diffusion limits and the inclusion process. Electronic Journal of Probability , 29(none), January 2024

    work page 2024

Show all 43 references
  1. [9]

    den Hollander and S

    F. den Hollander and S. Nandan. Spatially inhomogeneous populations with seed-banks: I. duality, existence and clustering. Journal of Theoretical Probability , 35(3):1795--1841, 2022

    2022

  2. [10]

    Ethier and R.C

    S.N. Ethier and R.C. Griffiths. The Infinitely-Many-Sites Model as a Measure-Valued Diffusion . The Annals of Probability , 15(2):515--545, apr 1987

    1987

  3. [11]

    Evans and T

    M.R. Evans and T. Hanney. Nonequilibrium statistical mechanics of the zero-range process and related models. Journal of Physics A: Mathematical and General , 38(19):R195, 2005

    2005

  4. [12]

    Ethier and T.G

    S.N. Ethier and T.G. Kurtz. The infinitely-many-neutral-alleles diffusion model . Advances in Applied Probability , 13(3):429--452, 1981

    1981

  5. [13]

    Ethier and T.G

    S.N. Ethier and T.G. Kurtz. Markov Processes: Characterization and Convergence . Wiley, March 1986

    1986

  6. [14]

    Ethier and T.G

    S.N. Ethier and T.G. Kurtz. The Infinitely-Many-Alleles Model with Selection as a Measure-Valued Diffusion . In Lecture Notes in Biomathematics , pages 72--86. Springer Berlin Heidelberg, 1987

    1987

  7. [15]

    S.N. Ethier. A class of degenerate diffusion processes occurring in population genetics. Communications on Pure and Applied Mathematics , 29(5):483–493, September 1976

    1976

  8. [16]

    S.N. Ethier. A limit theorem for two-locus diffusion models in population genetics. Journal of Applied Probability , 16(2):402–408, June 1979

    1979

  9. [17]

    S.N. Ethier. Eigenstructure of the infinitely-many-neutral-alleles diffusion model. Journal of Applied Probability , 29(3):487–498, September 1992

    1992

  10. [18]

    Evans and B

    M.R. Evans and B. Waclaw. Condensation in stochastic mass transport models: beyond the zero-range process. Journal of Physics A: Mathematical and Theoretical , 47(9):095001, 2014

    2014

  11. [19]

    W. Feller. An Introduction to Probability Theory and Its Applications, Vol. 1 . An Introduction to Probability Theory and Its Applications. Wiley, 1968

    1968

  12. [20]

    S. Feng. The P oisson- D irichlet Distribution and Related Topics . Springer Berlin Heidelberg, 2010

    2010

  13. [21]

    Franceschini, P

    C. Franceschini, P. Gon c alves, and F. Sau. Symmetric inclusion process with slow boundary: Hydrodynamics and hydrostatics. Bernoulli , 28(2):1340--1381, 2022

    2022

  14. [22]

    Forman, S

    N. Forman, S. Pal, D. Rizzolo, and M. Winkel. Diffusions on a space of interval partitions: Poisson–Dirichlet stationary distributions . The Annals of Probability , 49(2), March 2021

    2021

  15. [23]

    Forman, S

    N. Forman, S. Pal, D. Rizzolo, and M. Winkel. Ranked masses in two-parameter Fleming–Viot diffusions . Transactions of the American Mathematical Society , October 2022

    2022

  16. [24]

    Forman, D

    N. Forman, D. Rizzolo, Q. Shi, and M. Winkel. Diffusions on a space of interval partitions: the two-parameter model. Electronic Journal of Probability , 28(none), January 2023

    2023

  17. [25]

    Feng and W

    S. Feng and W. Sun. Some diffusion processes associated with two parameter Poisson–Dirichlet distribution and Dirichlet process . Probability Theory and Related Fields , 148(3–4):501–525, July 2009

    2009

  18. [26]

    Greven, F

    A. Greven, F. den Hollander, and M. Oomen. Spatial populations with seed-bank: well-posedness, duality and equilibrium. Electronic Journal of Probability , 27:1--88, 2022

    2022

  19. [27]

    Grosskinsky and W

    S. Grosskinsky and W. Jatuviriyapornchai. Derivation of mean-field equations for stochastic particle systems. Stochastic Processes and their Applications , 129(4):1455–1475, April 2019

    2019

  20. [28]

    Grosskinsky and A

    S. Grosskinsky and A. Koutsimpela. Tagged particles and size-biased dynamics in mean-field interacting particle systems . Electronic Communications in Probability , 30(none):1 -- 19, 2025

    2025

  21. [29]

    Giardin \`a , J

    C. Giardin \`a , J. Kurchan, and F. Redig. Duality and exact correlations for a model of heat conduction. Journal of mathematical physics , 48(3):033301, 2007

    2007

  22. [30]

    Godr \` e che

    C. Godr \` e che. Condensation for random variables conditioned by the value of their sum. Journal of Statistical Mechanics: Theory and Experiment , 2019(6):063207, 2019

    2019

  23. [31]

    Grosskinsky, F

    S. Grosskinsky, F. Redig, and K. Vafayi. Condensation in the Inclusion Process and Related Models . Journal of Statistical Physics , 142(5):952--974, feb 2011

    2011

  24. [32]

    Grosskinsky, F

    S. Grosskinsky, F. Redig, and K. Vafayi. Dynamics of condensation in the symmetric inclusion process . Electronic Journal of Probability , 18:1--23, jan 2013

    2013

  25. [33]

    Hoeksema, C

    J. Hoeksema, C. Y. Lam, and A. Schlichting. Variational convergence of exchange-driven stochastic particle systems in the thermodynamic limit. arXiv preprint arXiv:2401.06696 , 2024

    2024 arXiv

  26. [34]

    Jatuviriyapornchai, P

    W. Jatuviriyapornchai, P. Chleboun, and S. Grosskinsky. Structure of the Condensed Phase in the Inclusion Process . Journal of Statistical Physics , 178(3):682--710, dec 2019

    2019

  27. [35]

    Jatuviriyapornchai and S

    W. Jatuviriyapornchai and S. Grosskinsky. Condensation in zero-range processes with a fast rate. Journal of Statistical Physics , 191(11):1--18, 2024

    2024

  28. [36]

    Kallenberg

    O. Kallenberg. Foundations of Modern Probability . Springer International Publishing, 2021

    2021

  29. [37]

    S. Kim. Second time scale of the metastability of reversible inclusion processes. Probability Theory and Related Fields , 180(3-4):1135--1187, 2021

    2021

  30. [38]

    J.F.C. Kingman. Random Discrete Distributions . Journal of the Royal Statistical Society: Series B (Methodological) , 37(1):1--15, 1975

    1975

  31. [39]

    Kim and I

    S. Kim and I. Seo. Condensation and metastable behavior of non-reversible inclusion processes. Communications in Mathematical Physics , 382:1343--1401, 2021

    2021

  32. [40]

    P.A.P. Moran. Random processes in genetics. Mathematical Proceedings of the Cambridge Philosophical Society , 54(1):60--71, 1958

    1958

  33. [41]

    L.A. Petrov. Two-parameter family of infinite-dimensional diffusions on the Kingman simplex . Functional Analysis and Its Applications , 43(4):279--296, dec 2009

    2009

  34. [42]

    Schmuland

    B. Schmuland. A result on the infinitely many neutral alleles diffusion model . Journal of Applied Probability , 28(2):253--267, jun 1991

    1991

  35. [43]

    Schlichting

    A. Schlichting. The exchange-driven growth model: Basic properties and longtime behavior. Journal of Nonlinear Science , 30(3):793–830, November 2019

    2019

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.