q-TASEP with position-dependent slowing

Abstract

We introduce a new interacting particle system on Z, slowed t-TASEP. It may be viewed as a q-TASEP with additional position-dependent slowing of jump rates, depending on a parameter t, which leads to discrete asymptotic fluctuations at large time. If on the other hand t → 1 as time → ∞, we prove 1. A law of large numbers for particle positions, 2. A central limit theorem, with convergence to the fixed-time Gaussian marginal of a stationary solution to SDEs derived from the particle jump rates, and 3. A bulk limit to a certain explicit stationary Gaussian process on R, with scaling exponents characteristic of the Edwards-Wilkinson universality class in (1 + 1)dimensions.The proofs relate slowed t-TASEP to a certain Hall-Littlewood process, and use contour integral formulas for observables of this process. Unlike most previously studied Macdonald processes, this one involves only local interactions, resulting in asymptotics characteristic of (1 + 1)-dimensional rather than (2 + 1)-dimensional systems.

Hall–Littlewood Polynomials, Boundaries, and p-Adic Random Matrices

“…Another interesting feature is that S (N ) is a so-called Hall-Littlewood process, which was shown in [65,Proposition 3.4] in the context of interacting particle systems, before the random matrix context in the present work was understood. In addition to motivating S (N ) as a natural object, this allows us to analyze its asymptotics using the Macdonald process tools of Borodin-Corwin [14], which we do in a separate work [61].…”

Limits and fluctuations of p-adic random matrix products

“…Proof. The result is stated in [78,Proposition 3.4] in the case of trivial initial condition and proven by verifying equality of Markov generators, which also shows the case of general initial condition automatically. □…”

“…In our setting, unfortunately, the resulting moment problem is indeterminate, as we saw above for L 1,t,χ . This was discussed at the end of [78,Section 5], where the limiting t-moments…”

“…It is often convenient to consider symmetric polynomials in an arbitrarily large or infinite number of variables, which we formalize as follows, heavily borrowing from the introductory material in [78]. One has a chain of maps…”

“…, κ ′ κ 1 . The notation λ(τ ) is, however, the same in both [78] and the present work. Note also that there is a small typo on the left hand side of [78, (5.1)], j should be i (or vice versa).…”