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Matrix ansatz and large deviations of the density in exclusion processes · 2006-prob-203

Limit theorems in probability theory · ICM 2006: Matrix ansatz and large deviations of the density in exclusion processes

MSC 60 · Probability theory and stochastic processes

Inferred from the community field; finer classification has not been reviewed.

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Problem statement

Fix an integer d≥2d\ge 2. For each integer L≥1L\ge 1, let

ΩL,d={η⊆{1,…,L}:∣i−j∣≥d for all distinct i,j∈η}. \Omega_{L,d}=\bigl\{\eta\subseteq\{1,\ldots,L\}: |i-j|\ge d\text{ for all distinct }i,j\in\eta\bigr\}.

An element i∈ηi\in\eta represents the left endpoint of a hard rod occupying the sites i,i+1,…,i+d−1i,i+1,\ldots,i+d-1 in the physical lattice {1,…,L+d−1}\{1,\ldots,L+d-1\}. On ΩL,d\Omega_{L,d}, consider the continuous-time open totally asymmetric exclusion process with the following transitions:

  • if η∩{1,…,d}=∅\eta\cap\{1,\ldots,d\}=\varnothing, insert a rod at left endpoint 11, replacing η\eta by η∪{1}\eta\cup\{1\}, at rate 11;
  • for each i∈ηi\in\eta with i<Li<L, replace ii by i+1i+1 at rate 11 whenever the resulting configuration belongs to ΩL,d\Omega_{L,d};
  • if L∈ηL\in\eta, remove that rod, replacing η\eta by η∖{L}\eta\setminus\{L\}, at rate 11.

Let πL,d\pi_{L,d} be the unique stationary distribution of this finite Markov chain, and sample η\eta according to πL,d\pi_{L,d}. This choice of boundary rates places the system in its maximal-current phase. Define the bulk density of rod left endpoints and the equilibrium density-variance coefficient by

ρd=1d (d+1)andDd(ρ)=ρ(1−ρ)−(d−1)ρ2(2−dρ). \rho_d=\frac{1}{\sqrt d\,(\sqrt d+1)} \qquad\text{and}\qquad D_d(\rho)=\rho(1-\rho)-(d-1)\rho^2(2-d\rho).

For 0≤x≤10\le x\le 1, let

NL(x)=∣η∩{1,…,⌊Lx⌋}∣andHL(x)=NL(x)−ρd⌊Lx⌋LDd(ρd). N_L(x)=\bigl|\eta\cap\{1,\ldots,\lfloor Lx\rfloor\}\bigr| \qquad\text{and}\qquad H_L(x)=\frac{N_L(x)-\rho_d\lfloor Lx\rfloor}{\sqrt{L D_d(\rho_d)}}.

Let B=(B(x))0≤x≤1B=(B(x))_{0\le x\le1} be a Brownian motion with B(0)=0B(0)=0 and

E[(B(x)−B(x′))2]=∣x−x′∣2, \mathbb E\bigl[(B(x)-B(x'))^2\bigr]=\frac{|x-x'|}{2},

and let Y=(Y(x))0≤x≤1Y=(Y(x))_{0\le x\le1} be an independent normalized Brownian excursion associated with the same diffusion normalization; that is, YY has the law of a Brownian path with increment variance ∣x−x′∣/2|x-x'|/2, conditioned to remain positive on (0,1)(0,1) and to satisfy Y(0)=Y(1)=0Y(0)=Y(1)=0.

Prove that, as L→∞L\to\infty,

HL ⟹ B+Y. H_L\ \Longrightarrow\ B+Y.

Establish this convergence in the Skorokhod J1J_1 topology on D([0,1])D([0,1]), or, at minimum, establish convergence of all finite-dimensional distributions.

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Question 1

Fix an integer d≥2d\ge2. For each L≥1L\ge1, consider the continuous-time Markov chain on

ΩL,d={η⊆{1,…,L}:∣i−j∣≥d for all distinct i,j∈η}, \Omega_{L,d}=\bigl\{\eta\subseteq\{1,\ldots,L\}: |i-j|\ge d\text{ for all distinct }i,j\in\eta\bigr\},

where i∈ηi\in\eta is the left endpoint of a rod occupying i,i+1,…,i+d−1i,i+1,\ldots,i+d-1. A rod is inserted at left endpoint 11 at rate 11 when η∩{1,…,d}=∅\eta\cap\{1,\ldots,d\}=\varnothing; a rod with left endpoint i<Li<L moves to i+1i+1 at rate 11 when the resulting configuration belongs to ΩL,d\Omega_{L,d}; and a rod with left endpoint LL is removed at rate 11. Let πL,d\pi_{L,d} be the stationary distribution, and let

JL,d=πL,d{η:L∈η} J_{L,d}=\pi_{L,d}\{\eta:L\in\eta\}

be the stationary particle current through the right boundary. Prove that

lim⁡L→∞JL,d=1(d+1)2. \lim_{L\to\infty}J_{L,d}=\frac{1}{(\sqrt d+1)^2}.

Source reference: Greg Lakatos and Tom Chou, “Totally Asymmetric Exclusion Processes with Particles of Arbitrary Size,” Journal of Physics A: Mathematical and General 36 (2003), 2027–2041; maximal-current formula (11) and the concluding conjecture that the Tonks/refined mean-field current formulas are exact. DOI: 10.1088/0305-4470/36/8/302; arXiv: cond-mat/0211555.

Progress & source assessment

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The exact assertion remains unresolved. No citable proof or counterexample was found for the open hard-rod process with any fixed rod length d≥2d\ge2, even at the level of all finite-dimensional distributions.

The originating paper, “The Asymmetric Exclusion Process and Brownian Excursions,” Bernard Derrida, C. Enaud, and Joel L. Lebowitz, Journal of Statistical Physics 115 (2004), 365–382, DOI 10.1023/B:JOSS.0000019833.35328.b4, arXiv cond-mat/0306078, proves the Brownian-motion-plus-Brownian-excursion description for ordinary open TASEP (one-site particles). Section 3 then formulates precisely the extended-particle model and the scaling used here, but tests it only by Monte Carlo simulations for d=1,3,5d=1,3,5, comparing the second and fourth moments of particle counts with the conjectured law. Thus it supplies numerical evidence, not a proof, for d≥2d\ge2.

The strongest rigorous later result is “Limit Fluctuations for Density of Asymmetric Simple Exclusion Processes with Open Boundaries,” Włodzimierz Bryc and Yizao Wang, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 55 (2019), 2169–2194, DOI 10.1214/18-AIHP945, arXiv 1707.07350. Its Theorem 1.2 proves finite-dimensional convergence to an independent sum of Brownian motion and Brownian excursion throughout the maximal-current phase of open ASEP. Its state space is {0,1}n\{0,1\}^n, so it concerns particles occupying one site and does not cover the hard-rod state space ΩL,d\Omega_{L,d} for d≥2d\ge2.

For extended particles, “Totally Asymmetric Exclusion Processes with Particles of Arbitrary Size,” Greg Lakatos and Tom Chou, Journal of Physics A: Mathematical and General 36 (2003), 2027–2041, DOI 10.1088/0305-4470/36/8/302, arXiv cond-mat/0211555 derives the stated maximal-current density using Tonks-gas/refined mean-field arguments and supports it by simulation, but explicitly presents the relevant formulas as conjecturally exact and proves no density-fluctuation limit. “Exclusion Process on an Open Lattice with Fluctuating Boundaries,” S. L. Narasimhan and A. Baumgaertner, Physica A: Statistical Mechanics and its Applications 533 (2019), 121930, DOI 10.1016/j.physa.2019.121930, arXiv 1807.10667 develops a rod-to-monomer representation with a particle-number-dependent effective lattice and compares mean-field predictions with Monte Carlo data; it likewise gives no stationary invariance principle.

Accordingly, established progress consists of the complete d=1d=1 analogue, the equilibrium/ring calculation of the variance coefficient, mean-field and mapping results for extended particles, and numerical evidence for selected moments. What remains open is precisely the proof, for every fixed d≥2d\ge2, that the stationary cumulative rod density has the stated B+YB+Y scaling limit (including even finite-dimensional convergence).

References
  1. [1]ICM Conjectures · 2006-prob-203 link
  2. [2]ICM 2006: Matrix ansatz and large deviations of the density in exclusion processes link

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