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The lower tail of $q$-pushTASEP

Ivan Corwin, Milind Hegde

Abstract

We study $q$-pushTASEP, a discrete time interacting particle system whose distribution is related to the $q$-Whittaker measure. We prove a uniform in $N$ lower tail bound on the fluctuation scale for the location $x_N(N)$ of the right-most particle at time $N$ when started from step initial condition. Our argument relies on a map from the $q$-Whittaker measure to a model of periodic last passage percolation (LPP) with geometric weights in an infinite strip that was recently established in [arXiv:2106.11922]. By a path routing argument we bound the passage time in the periodic environment in terms of an infinite sum of independent passage times for standard LPP on $N\times N$ squares with geometric weights whose parameters decay geometrically. To prove our tail bound result we combine this reduction with a concentration inequality, and a crucial new technical result -- lower tail bounds on $N\times N$ last passage times uniformly over all $N \in \mathbb N$ and all the geometric parameters in $(0,1)$. This technical result uses Widom's trick [arXiv:math/0108008] and an adaptation of an idea of Ledoux introduced for the GUE [Led05a] to reduce the uniform lower tail bound to uniform asymptotics for very high moments, up to order $N$, of the Meixner ensemble. This we accomplish by first obtaining sharp uniform estimates for factorial moments of the Meixner ensemble from an explicit combinatorial formula of Ledoux [Led05b], and translating them to polynomial bounds via a further careful analysis and delicate cancellation.

The lower tail of $q$-pushTASEP

Abstract

We study -pushTASEP, a discrete time interacting particle system whose distribution is related to the -Whittaker measure. We prove a uniform in lower tail bound on the fluctuation scale for the location of the right-most particle at time when started from step initial condition. Our argument relies on a map from the -Whittaker measure to a model of periodic last passage percolation (LPP) with geometric weights in an infinite strip that was recently established in [arXiv:2106.11922]. By a path routing argument we bound the passage time in the periodic environment in terms of an infinite sum of independent passage times for standard LPP on squares with geometric weights whose parameters decay geometrically. To prove our tail bound result we combine this reduction with a concentration inequality, and a crucial new technical result -- lower tail bounds on last passage times uniformly over all and all the geometric parameters in . This technical result uses Widom's trick [arXiv:math/0108008] and an adaptation of an idea of Ledoux introduced for the GUE [Led05a] to reduce the uniform lower tail bound to uniform asymptotics for very high moments, up to order , of the Meixner ensemble. This we accomplish by first obtaining sharp uniform estimates for factorial moments of the Meixner ensemble from an explicit combinatorial formula of Ledoux [Led05b], and translating them to polynomial bounds via a further careful analysis and delicate cancellation.
Paper Structure (35 sections, 28 theorems, 198 equations, 3 figures)

This paper contains 35 sections, 28 theorems, 198 equations, 3 figures.

Table of Contents

  1. Introduction, main results, and proof ideas
  2. Principal objects and models of study
  3. Some notation and distributions
  4. The model of $q$-pushTASEP
  5. Law of large numbers and asymptotic Tracy-Widom fluctuations of $q$-pushTASEP
  6. Main results
  7. Lower tails of KPZ observables
  8. The relative difficulty of upper and lower tail bounds
  9. Work using determinantal representations of Laplace transforms
  10. Coupling and geometric methods in polymer models
  11. Last passage percolation
  12. Relation between $q$-pushTASEP and LPP
  13. Proof ideas
  14. Uniform LPP control
  15. Meixner ensemble analysis
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $q, u\in(0,1)$ and let $a_i=b_j = u$ for all $i,j$. There exist positive absolute constants $c'$, C, and $N_0$ (independent of $q$ and $u$) such that, with $\theta_0 = C(1-u)^{-1/3}(1\vee (\log q^{-1})^{-2/3})$ and $c = (1-u)^{1/2}c'$, and for $N\geq N_0$ and $\theta>\theta_0$,

Figures (3)

  • Figure 1: A depiction of one step in the evolution of $q$-pushTASEP. The dotted circle is the position of the $k$th particle at time $T-1$, and the solid black circle is it after it moves to its position at time $T$. The left red circle is the $(k+1)$th particle at time $T$ and the right one the same at time $T+1$. The movement of the $k$th particle in the previous step effects the $P_{k,T}$ contribution to the total jump size of $2$ of the $(k+1)$th particle at time $T$.
  • Figure 2: The environment in which the infinite last passage percolation occurs. The dashed arrow on top indicates the direction in which the squares wrap around, and the solid green line on the left is a downward path which wraps around the strip. In the left panel we have the model under consideration in this article with the specialization of $T=N$ and $u=a_i=b_j$ for all $i,j$, while on the right is the general model.
  • Figure 3: A depiction of the paths we consider near the boundary between different big squares. The two solid green paths go from the topmost site to the bottommost site in their respective big squares, where the environment is homogeneous. We do not consider the dotted green path needed to connect them, which is valid for proving an upper bound on the lower tail since including its weight will only increase the overall weight.

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5: Effective range of $x$
  • Remark 1.6: Effective range of $q$
  • Remark 1.7: Tail exponent of $3/2$
  • Remark 1.8: Comparison to exponential LPP
  • Theorem 1.9
  • Remark 1.10: An argument for the lower tail of $x_N(T)$
  • ...and 53 more