The asymmetric simple exclusion process (ASEP) with open boundaries — particles hopping right at rate 1 and left at rate q in (0,1) on a segment of length N, with reservoirs of fixed rates attached to both ends — is the paradigmatic non-reversible, boundary-driven interacting particle system. For the totally asymmetric case (q=0) the mixing-time phase diagram is completely known: linear mixing with cutoff in the high and low density phases, order N^{3/2} without cutoff in the maximal current phase, and order N^2 without cutoff on the coexistence line. Elboim and Schmid's Conjecture 1.3 (2024) predicts that the same three-regime picture holds for the ASEP with q>0, where the exactly solvable last-passage-percolation machinery behind the TASEP proofs is unavailable. The conjecture remains open: for fixed q>0 only order-N (pre-cutoff) bounds in the high and low density phases (Gantert–Nestoridi–Schmid, 2023) and the mixing-time order at the triple point itself (Ferrari–Schmid, order N^{3/2}) are known, while cutoff in the high and low density phases, the maximal current phase, and the coexistence line are unresolved.

Background

The asymmetric simple exclusion process (ASEP) is a continuous-time Markov chain modeling one-dimensional transport with hard-core exclusion: each lattice site carries at most one particle, and a particle at site x attempts to jump to x+1 at rate 1 and to x-1 at rate q in [0,1), the jump being suppressed when the target site is occupied. On the finite segment {1,…,N} with open boundaries, the ends of the segment are coupled to particle reservoirs: at the left end a particle enters at rate α when site 1 is empty and leaves leftward at rate γ when it is occupied, while at the right end a particle leaves at rate β and enters at rate δ. The totally asymmetric case q=0, typically with γ=δ=0, is the TASEP. For α,β>0 the open ASEP is an irreducible chain on {0,1}^N with a unique stationary measure μ_N; unlike on the full line or the closed segment this measure has no product form, and is described by the matrix product ansatz of Derrida, Evans, Hakim and Pasquier, which represents μ_N through traces of products of infinite non-commuting matrices satisfying a quadratic algebra.

The large-scale behavior of μ_N is governed by Liggett's boundary-induced phase diagram, best phrased through two effective reservoir densities ρ_L and ρ_R in (0,1) determined by the left and right boundary rates together with q (under Liggett's condition γ=(1−α)q and δ=(1−β)q one has simply ρ_L=α and ρ_R=1−β, which is also the TASEP convention). The process is in the low density phase if ρ_L<1/2 and ρ_L+ρ_R<1 (bulk density about ρ_L), in the high density phase if ρ_R>1/2 and ρ_L+ρ_R>1 (bulk density about ρ_R), in the maximal current phase if ρ_L≥1/2≥ρ_R (bulk density about 1/2, the density carrying the maximal stationary current), on the coexistence line if ρ_L+ρ_R=1 with ρ_L<1/2 (the stationary measure is then a superposition of low- and high-density profiles separated by a shock at a random location), and at the triple point if ρ_L=ρ_R=1/2.

The quantity of interest is the speed of convergence to μ_N. For a probability measure ν on {0,1}^N write ‖ν−μ_N‖_TV for the total-variation distance, and define the ε-mixing time t_mix^N(ε) as the first time t by which the law of the chain started from the worst initial configuration is within ε of μ_N in total variation. A family of chains exhibits cutoff when convergence is abrupt: t_mix^N(ε)/t_mix^N(ε′)→1 as N→∞ for all fixed ε,ε′ in (0,1), meaning the distance to equilibrium stays near its maximum until a sharp threshold and then collapses. 'No cutoff' means the ε-dependence persists at first order, as for the TASEP on the coexistence line where t_mix^N(ε) is of order N^2 log(1/ε).

For the open TASEP this picture is now complete. Elboim and Schmid proved that in the bulk of the high and low density phases mixing occurs in linear time with cutoff, with an explicit limiting constant, and that on the coexistence line the mixing time is of order N^2 with no cutoff. The maximal current phase was settled earlier by Schmid: there the mixing time is of order N^{3/2} with no cutoff, a sub-diffusive scale reflecting the Kardar–Parisi–Zhang (KPZ) universality class, the class of one-dimensional stochastic growth models whose fluctuations grow like t^{1/3}; Schmid and Sly determined the analogous picture on the circle. These proofs rest on a representation of the open TASEP as last passage percolation (LPP) on a slab — a directed model of maximal-weight paths, in which mixing is controlled by the coalescence of geodesics (optimal paths) and is analyzable through exact determinantal formulas connected to random matrix theory. By contrast, on the closed particle-conserving segment the ASEP mixes in linear time with cutoff (Labbé and Lacoin), so the three-regime structure is genuinely a boundary effect.

For the ASEP with q>0 none of these tools is available: there is no exact LPP representation, the stationary measure is known only implicitly, the chain is non-reversible, and the exact spectral gaps computed by de Gier and Essler control only the asymptotic decay rate, not the pre-equilibrium mixing behavior. Gantert, Nestoridi and Schmid obtained the first bounds: order-N mixing in the high and low density phases (pre-cutoff only, constants may depend on ε) and an order-N^3 upper bound at the triple point; they conjectured cutoff with an explicit constant in the high density phase, order N^{3/2} without cutoff in the maximal current phase, and posed the order on the coexistence line as an open question. Ferrari and Schmid determined the mixing time at the triple point itself — of order N^{3/2} for fixed q<1, and more generally of order N^{3/2+κ} when 1−q is of order N^{−κ} for κ in [0,1/2) — confirming the predicted order at that single point of the phase diagram. Elboim and Schmid's Conjecture 1.3, the subject of this problem, asks for the full three-regime picture for the open ASEP with fixed parameters.

Problem Statement

Prove or refute Conjecture 1.3 of Elboim–Schmid. Consider the open ASEP on the segment {1,…,N}: particles jump right at rate 1 and left at rate q in (0,1) with hard-core exclusion, the left boundary admits entry at rate α and exit at rate γ, and the right boundary admits exit at rate β and entry at rate δ, with all parameters fixed (independent of N) and α,β>0, γ,δ≥0. Let t_mix^N(ε) denote the ε-mixing time in total variation with respect to the unique stationary measure, and classify the parameters by the effective reservoir densities ρ_L, ρ_R, defined by ρ_L=1/(1+C) with C=C(α,γ,q)=(1/(2α))(1−q−α+γ+√((1−q−α+γ)^2+4αγ)) and ρ_R=A/(1+A) with A=A(β,δ,q)=(1/(2β))(1−q−β+δ+√((1−q−β+δ)^2+4βδ)), so that ρ_L depends only on (α,γ,q) and ρ_R only on (β,δ,q) (Ferrari–Schmid, arXiv:2501.03951, (1.2)–(1.4)) (low density phase: ρ_L<1/2 and ρ_L+ρ_R<1; high density phase: ρ_R>1/2 and ρ_L+ρ_R>1; maximal current phase: ρ_L≥1/2≥ρ_R; coexistence line: ρ_L+ρ_R=1 with ρ_L<1/2; for the TASEP, ρ_L=α and ρ_R=1−β). Establish the following three claims. (1) In the high and low density phases, the mixing time is of order N, and cutoff occurs (the first-order mixing time is asymptotically independent of ε, as proven for the TASEP). (2) In the maximal current phase, the mixing time is of order N^{3/2}, and no cutoff occurs. (3) On the coexistence line, the mixing time is of order N^2, and no cutoff occurs. The single overarching objective is the sharp convergence-to-equilibrium phase diagram of the two-reservoir open ASEP, extending the completed TASEP picture; the three claims are the three regimes of that one diagram. A complete solution settles all three regimes, each with matching upper and lower bounds for the order together with a determination of the presence or absence of cutoff; a resolution of a single regime alone is a partial result, and a refutation of any one of the three claims refutes the conjecture. A full resolution also settles Gantert–Nestoridi–Schmid's Conjecture 1.9 and their Question 1.10.

References

  1. Dor Elboim and Dominik Schmid, Mixing times and cutoff for the TASEP in the high and low density phase, Probability and Mathematical Physics 5 (2024) 413–459. DOI: 10.2140/pmp.2024.5.413; arXiv:2208.08306, https://arxiv.org/abs/2208.08306
  2. Nina Gantert, Evita Nestoridi, and Dominik Schmid, Mixing times for the simple exclusion process with open boundaries, Annals of Applied Probability 33 (2023) 1172–1212. DOI: 10.1214/22-aap1839; arXiv:2003.03781, https://arxiv.org/abs/2003.03781
  3. Dominik Schmid, Mixing times for the TASEP in the maximal current phase, Annals of Probability 51 (2023) 1342–1379. DOI: 10.1214/22-aop1620; arXiv:2104.12745, https://arxiv.org/abs/2104.12745
  4. D. Schmid and A. Sly, Mixing times for the TASEP on the circle, Probability Theory and Related Fields 194 (2026) 1161–1233. DOI: 10.1007/s00440-026-01466-0; arXiv:2203.11896, https://arxiv.org/abs/2203.11896
  5. Cyril Labbé and Hubert Lacoin, Cutoff phenomenon for the asymmetric simple exclusion process and the biased card shuffling, Annals of Probability 47 (2019) 1541–1586. DOI: 10.1214/18-aop1290; arXiv:1610.07383, https://arxiv.org/abs/1610.07383
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  8. Dominik Schmid and Zongrui Yang, The open ASEP with light particles, Probability Theory and Related Fields 195 (2026) 1455–1503. DOI: 10.1007/s00440-026-01486-w; arXiv:2409.00849, https://arxiv.org/abs/2409.00849
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