Searcharxiv⌕ Search

arXiv subjects

Hindy Drillick

Publications and source records attributed to Hindy Drillick.

14 recordsLinked to original sources
arXiv2026

A universality result for the critical 2d stochastic heat flow

For a large class of discrete- and continuous-time models satisfying a linear flow property, we prove convergence to the critical $2d$ stochastic heat flow under diffusive scaling of space and time. Convergence is in the sense of finite-dimensional distributions in the space of measure-valued stochastic flows. The class of discrete models we consider includes random walks in space-time random environments with finite-range jumps, and directed polymer models with finite-range spatial correlations. The class of continuum models we consider includes several distinct types of linear-multiplicative stochastic PDEs whose driving noise is Gaussian with a smooth and compactly supported covariance kernel.

math.PR↗
arXiv2026

The stochastic six-vertex model speed process

For the stochastic six-vertex model on the quadrant $\mathbb Z_{\geq0}\times\mathbb Z_{\geq0}$ with step initial conditions and a single second-class particle at the origin, we show almost sure convergence of the speed of the second-class particle to a random limit. This allows us to define the stochastic six-vertex model speed process, whose law we show to be ergodic and stationary for the dynamics of the multi-class stochastic six-vertex process. To prove this, we develop precise bounds on the fluctuations of the height function of the stochastic six-vertex model around its limit shape using methods from integrable probability. As part of the proof, we also obtain a novel geometric stochastic domination result that states that a second-class particle to the right of any number of third-class particles will at any fixed time be overtaken by at most a geometric number of third-class particles, which is of independent interest.

math.PR↗
arXiv2026

The critical KPZ scale for the Averaging Process

KPZ-type extremal fluctuations have recently been proved for several models of random walks in space-time random environments (RWRE) in $1+1$ dimensions. A general moment criterion predicts the spatial scale at which this behavior should occur, but does not by itself guarantee non-trivial fluctuations at that scale. In this paper, we show that the averaging process provides an instance in which this criterion is not sharp: because of a degeneracy in the update mechanism, the actual KPZ limit appears only beyond the predicted scale. The relevant critical contribution arises from the interplay of two distinct fluctuation mechanisms, a phenomenon that appears to be rather special within the RWRE setting. Our proof builds on Dobrushin-type local-time limit theorems for zero-sum additive functionals, refined estimates for tilted $k$-point motions, and the recent moment-based axiomatic characterization of Cole--Hopf solutions to the one-dimensional KPZ equation. We also identify the behavior on the two sides of the critical scale, thereby sharply separating the subcritical, critical, and supercritical regimes of the model.

math.PR↗
arXiv2026

The censored stochastic six-vertex model and parabolic Kazhdan--Lusztig $R$-polynomials

We introduce a censored version of the stochastic six-vertex model. We show that for parameters $b_1 < b_2$, this model started from the initial condition ${1}_{x>0}$ is stochastically dominated at any time by the blocking measure. This is a partial analog of the censoring inequality for monotone spin systems. In particular, this result allows us to control the behavior of second-class particles. The proof uses parabolic Kazhdan--Lusztig $R$-polynomials, whose appearance is explained using a connection between the stochastic six-vertex model and the Iwahori--Hecke algebras of symmetric groups. Furthermore, we find an intertwining relation for this process using normalized parabolic Kazhdan--Lusztig $R$-polynomials as an intertwining kernel.

math.PR↗
arXiv2025

Random walks in space-time random media in all spatial dimensions: the full subcritical fluctuation regime

In arbitrary spatial dimension $d\ge 1$, we study a generalized model of random walks in a time-varying random environment (RWRE) defined by a stochastic flow of kernels. We consider the quenched probability distribution of the random walker under a scaling where the time is of order $N$ and the spatial window is of size $N^{1/2}$. This spatial window may not necessarily be centered close to the origin. We show that as $N\to \infty$ there are Gaussian fluctuations up to a certain specific spatial centering radius $ψ_N$ in the tail of the quenched probability distribution, which we call the critical scale. This critical scale depends on the spatial dimension of the underlying random walk, specifically $ψ_N = O(N^{3/4})$ when $d=1$, $ψ_N = O( N/\sqrt{\log N})$ when $d=2$, and $ψ_N = O(N)$ when $d\ge 3$. In the particular case of centering the fluctuation window at the origin, our results recover and generalize some known fluctuation results for related models. However, farther from the origin, the previous literature is more sparse. The noise coefficient in the limiting Gaussian field is nontrivial and depends on the invariant measure of the two-point motion of the underlying RWRE model. We furthermore reconcile some of these coefficient formulas with previous works. As part of the proof, we introduce a general class of Markov chains with short-range interactions that admit nice estimates and limit formulas. One of the key technical results for such Markov chains is that in $d\ge 2$, one can propagate test functions backwards in time to obtain precise limiting moment formulas.

math.PR↗
arXiv2025

Universal KPZ Fluctuations for Moderate Deviations of Random Walks in Random Environments

The theory of diffusion seeks to describe the motion of particles in a chaotic environment. Classical theory models individual particles as independent random walkers, effectively forgetting that particles evolve together in the same environment. Random Walks in a Random Environment (RWRE) models treat the environment as a random space-time field that biases the motion of particles based on where they are in the environment. We provide a universality result for the moderate deviations of the transition probability of this model over a wide class of choices of random environments. In particular, we show the convergence of moments to those of the multiplicative noise stochastic heat equation (SHE), whose logarithm is the Kardar-Parisi-Zhang (KPZ) equation. The environment only filters into the scaling limit through one parameter, which depends explicitly on the statistical description of the environment. This forms the basis for our introduction, in arXiv:2406.17733, of the extreme diffusion coefficient.

cond-mat.stat-mech↗
arXiv2024

KPZ equation limit of sticky Brownian motion

We consider the motion of a particle under a continuum random environment whose distribution is given by the Howitt-Warren flow. In the moderate deviation regime, we establish that the quenched density of the motion of the particle (after appropriate centering and scaling) converges weakly to the $(1+1)$ dimensional stochastic heat equation driven by multiplicative space-time white noise. Our result confirms physics predictions and computations in [LDT17, BLD20] and is the first rigorous instance of such weak convergence in the moderate deviation regime. Our proof relies on a certain Girsanov transform and works for all Howitt-Warren flows with finite and nonzero characteristic measures. Our results capture universality in the sense that the limiting distribution depends on the flow only via the total mass of the characteristic measure. As a corollary of our results, we prove that the fluctuations of the maximum of an $N$-point sticky Brownian motion are given by the KPZ equation plus an independent Gumbel on timescales of order $(\log N)^2.$

math.PR↗
arXiv2024

Multiplicative SHE limit of random walks in space-time random environments

We show that under a certain moderate deviation scaling, the multiplicative-noise stochastic heat equation (SHE) arises as the fluctuations of the quenched density of a 1D random walk whose transition probabilities are iid [0,1]-valued random variables. In contrast to the case of directed polymers in the intermediate disorder regime, the variance of our weights is fixed rather than vanishing under the diffusive rescaling of space-time. Consequently, taking a naive limit of the chaos expansion fails for this model, and a nontrivial noise coefficient is observed in the limit. Rather than using chaos techniques, our proof instead uses the fact that in this regime the quenched density solves a discrete SPDE which resembles the SHE. As a byproduct of our techniques, it is shown that independent noise is generated in the limit, in the sense that the prelimiting noise field does not converge to the driving noise of the limiting SPDE.

math.PR↗
arXiv2024

Extreme Diffusion Measures Statistical Fluctuations of the Environment

We consider many-particle diffusion in one spatial dimension modeled as Random Walks in a Random Environment (RWRE). A shared short-range space-time random environment determines the jump distributions that drive the motion of the particles. We determine universal power-laws for the environment's contribution to the variance of the extreme first passage time and extreme location. We show that the prefactors rely upon a single extreme diffusion coefficient that is equal to the ensemble variance of the local drift imposed on particles by the random environment. This coefficient should be contrasted with the Einstein diffusion coefficient, which determines the prefactor in the power-law describing the variance of a single diffusing particle and is equal to the jump variance in the ensemble averaged random environment. Thus a measurement of the behavior of extremes in many-particle diffusion yields an otherwise difficult to measure statistical property of the fluctuations of the generally hidden environment in which that diffusion occurs. We verify our theory and the universal behavior numerically over many RWRE models and system sizes.

cond-mat.stat-mech↗
arXiv2022

Strong law of large numbers for the stochastic six vertex model

We consider the inhomogeneous stochastic six vertex model with periodicity starting from step initial data. We prove that it converges almost surely to a deterministic limit shape. For the proof, we map the stochastic six vertex model to a deformed version of the discrete Hammersley process. Then we construct a colored version of the model and apply Liggett's superadditive ergodic theorem. The construction of the colored model includes a new idea using a Boolean-type product, which generalizes and simplifies the method used in arXiv:2204.11158.

math.PR↗
arXiv2022

Hydrodynamics of the $t$-PNG model via a colored $t$-PNG model

The $t$-PNG model introduced in Aggarwal, Borodin, and Wheeler (2021) is a deformed version of the polynuclear growth (PNG) model. In this paper, we prove the hydrodynamic limit of the model using soft techniques. One key element of the proof is the construction of a colored version of the $t$-PNG model, which allows us to apply the superadditive ergodic theorem and obtain the hydrodynamic limit, albeit without identifying the limiting constant. We then find this constant by proving a law of large numbers for the $α$-points, which generalizes Groeneboom (2001). Along the way, we construct the stationary $t$-PNG model and prove a version of Burke's theorem for it.

math.PR↗
arXiv2021

Non-Rigid Rank-One Infinite Measures on the Circle

For a class of irrational numbers, depending on their Diophantine properties, we construct explicit rank-one transformations that are totally ergodic and not weakly mixing. We classify when the measure is finite or infinite. In the finite case they are isomorphic to irrational rotations. We also obtain rank-one nonrigid infinite invariant measures for irrational rotations, and, for each Krieger type, nonsingular measures on irrational rotations. In the third version, in the infinite case we use the constructions to provide examples of non-weakly mixing infinite measure-preserving ergodic transformations which do not have any nontrivial probability preserving factors with discrete spectrum, thereby answering a questions of Aaronson and Nakada and of Glasner and Weiss.

math.DS↗
arXiv2020

Smocked Metric Spaces and their Tangent Cones

We introduce the notion of a smocked metric spaces and explore the balls and geodesics in a collection of different smocked spaces. We find their rescaled Gromov-Hausdorff limits and prove these tangent cones at infinity exist, are unique, and are normed spaces. We close with a variety of open questions suitable for advanced undergraduates, masters students, and doctoral students.

math.MG↗