arXiv subjects

Sunder Sethuraman

Publications and source records attributed to Sunder Sethuraman.

At least 19 recordsLinked to original sources

arXiv2026

Notes on Hydrodynamic Limits and Related Topics

In these lecture notes, we discuss various `hydrodynamic LLN' and `CLT' scaling limits, among others, in types of stochastic interacting particle systems, connecting `microscopic' behaviors to continuum laws. Via `short stories', the aim is to present some of the `basics' for students and those entering the field, as a complement to books such as Kipnis-Landim 1999, Komorowski-Landim-Olla 2012, Liggett 1985, Liggett 1999. To be concrete, attention is restricted to a few `mass conservative' systems on discrete spaces, namely exclusion and zero-range processes, that have proved robust in the study of different phenomena. After preliminaries, we discuss the `entropy' and `relative entropy' methods to prove hydrodynamic limits of the bulk mass in finite volume, as well as other items such as construction of systems in infinite volume and the structure of their invariant measures, and scaling limits of local functionals, such as occupation times of sites and the motion of a tagged particle. In the last part, we also discuss equilibrium fluctuations of the bulk mass when the process starts from an invariant measure.

math.PR↗

arXiv2026

Crystal Growth on Locally Finite Partially Ordered Sets

We consider a Markovian growth process on a partially ordered set $\Lambda$, equivalent to last passage percolation (LPP) with independent (not necessarily identical) exponentially distributed weights on the elements of $\Lambda$. Such a process includes inhomogeneous exponential LPP on the Euclidean lattice $\mathbb{N}_0^d$. We give non-asymptotic bounds on the mean and variance, as well as higher, central, and exponential moments of the passage time $\tau_A$ to grow any set $A \subseteq \Lambda$ in terms of characteristics of $A$. We also give a limit shape theorem when $\Lambda$ is equipped with a monoid structure. Methods involve making use of the backward equation associated to the Markovian evolution and comparison inequalities with respect to the time-reversed generator.

math.PR↗

arXiv2025

Coupled KPZ equations and their decoupleability

We discuss characterizations of the decoupleability, partial and full, of trilinear or completely symmetric real $n\times n\times n$ tensors, which inform on the structure of certain coupled KPZ equations. Informally, when the tensor is partially decoupleable, one of the components in the coupled KPZ equation splits off from the others, while when the tensor is fully decoupleable, each of the $n$ components splits off from the others. Such a characterization is recast as a problem of membership of trilinear tensors in $O(n)$ orbits of subsets of fully decoupleable and partially decoupleable tensors. When $n=2$, we show these subsets are the same, and in this case give a single criterion in terms of the entries of a tensor for membership in the orbits of these subsets. When $n\geq 3$, the subsets are different. For $n\geq 3$, we characterize full decoupleability in terms of several abstract relations, which when $n=3$ are made explicit. When $n=3$, we also explicitly characterize partial decoupleability. The methods involve notions in applied invariant theory, relating $O(n)$ invariant subsets to stabilizer subgroup actions on smaller sets. When $n=3$ make use of the explicit basis of invariants found by Olive and Auffray. When $n=2$, we also supply two other more direct arguments.

math.PR↗

arXiv2025

Point process convergence of extremes in $K$-symmetric exclusion

We consider the behavior of extremal particles in $K$-symmetric exclusion on $\mathbb{Z}$ when the process starts from certain infinite-particle step configurations where there are no particles to the right of a maximal one. In such a system, the occupancy of a site is limited to at most $K \geq 1$. Let $X^{(0)}_t\geq X^{(1)}_t\geq \cdots$ denote the order statistics of the particles in the system. We show that the point process $\sum_{m=0}^\infty δ_{v_t(X_{t/K}^{(m)})}$ converges in distribution as $t \to \infty$ to a Poisson random measure on $\mathbb{R}$ with intensity proportional to $e^{-x}\,dx$, where $v_t(x) = (σb_t)^{-1}x - a_t$, $a_t = \log(t/ (\sqrt{2π} \log t))$, $b_t = (t/\log t)^{1/2}$, and $σ$ is the standard deviation of the random walk jump probabilities. From this limit, we further deduce the asymptotic joint distributions for the extreme statistics and the spacings between them. Moreover, to probe effects of the number of particles on the behavior of the extremes, we consider an array of truncated step profiles supported on blocks of $L(t)$ sites at times $t\geq 0$. Letting $L(t) \to \infty$ with $t \to \infty$, we obtain Poisson random measure limits in different scaling regimes determined by $L(t)$. These results show robustness of both previously known and newly introduced superdiffusive scaling limits for the extremes in the symmetric exclusion process ($K=1$) by extending them to the larger class of $K\geq 2$ exclusion. Furthermore, proofs are more general than previously known techniques, relying on moment bounds and a semigroup monotonicity estimate to control particle correlations.

math.PR↗

arXiv2025

Poisson statistics, vanishing correlations, and extremal particle limits for symmetric exclusion in d > 1

We consider the symmetric simple exclusion system on $\mathbb{Z}^d$, $d \ge 2$, starting from a class of ``step'' initial conditions in which particles are constrained within a half-space. One may count the number $N_t$ of particles that have moved beyond a distance $z = z(t)$ into the initially-empty half of $\mathbb{Z}^d$ at time $t$. We show in large generality that when $\lim_{t\to\infty} E[N_t]$ exists, correlations between particles beyond $z$ vanish as $t \to \infty$ so as to allow convergence of $N_t$ to the same Poisson distribution one would get were the particles allowed to move independently. When the initial condition constrains a region of polynomial growth, we identify $z(t)$ and the limit of $E[N_t]$ explicitly. As a consequence of the limit, we obtain a Gumbel limit distribution for the extremal particle position, as well as the limiting distributions of all order statistics.

math.PR↗

arXiv2025

Singular diffusion limit of a tagged particle in zero range processes with Sinai-type random environment

We derive a singular diffusion limit for the position of a tagged particle in zero range interacting particle processes on a one dimensional torus with a Sinai-type random environment via two steps. In the first step, a regularization is introduced by averaging the random environment over an $\varepsilon N$-neighborhood. With respect to such an environment, the microscopic drift of the tagged particle is in form $\frac{1}{N}W_\varepsilon'$, where $W_\varepsilon'$ is a regularized White noise. Scaling diffusively, we find the nonequilibrium limit of the tagged particle $x^\varepsilon_t$ is the unique weak solution of $d x_t^{\varepsilon} = 2\frac{Φ(ρ^{\varepsilon}(t, x_t^{\varepsilon}))}{ρ^{\varepsilon}(t, x_t^\varepsilon)} \,W_{\varepsilon}'(x_t^\varepsilon) + \sqrt{\frac{Φ(ρ^{\varepsilon}(t, x_t^\varepsilon))}{ρ^{\varepsilon}(t, x_t^\varepsilon)}} \,dB_t$, in terms of the hydrodynamic mass density $ρ^\varepsilon$ recently identified and homogenized interaction rate $Φ$. In the second step, we show that $x^\varepsilon$, as $\varepsilon$ vanishes, converges in law to the diffusion $x^0$ described informally by $d x_t^0 = 2\frac{Φ(ρ^{0}(t, x_t^{0}))}{ρ^{0}(t, x_t^0)} \,W'(x_t^0) + \sqrt{\frac{Φ(ρ^{0}(t, x_t^0))}{ρ^{0}(t, x_t^0)}} \,dB_t$, where $W'$ is a spatial White noise and $ρ^0$ is the para-controlled limit of $ρ^\varepsilon$ also recently identified, solving the singular PDE $ \partial_t ρ^0 = \frac{1}{2}ΔΦ(ρ^0) - 2\nabla \big(W' Φ(ρ^0)\big)$.

math.PR↗

arXiv2024

Linear fluctuation of interfaces in Glauber-Kawasaki dynamics

In this article, we find a scaling limit of the space-time mass fluctuation field of Glauber + Kawasaki particle dynamics around its hydrodynamic mean curvature interface limit. Here, the Glauber rates are scaled by $K=K_N$, the Kawasaki rates by $N^2$ and space by $1/N$. We start the process so that the interface $Γ_t$ formed is stationary that is, $Γ_t$ is `flat'. When the Glauber rates are balanced on $T^d$, $Γ_t=Γ=\{x: x_1=0\}$ is immobile and the hydrodynamic limit is given by $ρ(t,v) = ρ_+$ for $v_1\in (0,1/2)$ and $ρ(t,v)= ρ_-$ for $v_1\in (-1/2,0)$ for all $t\ge 0$, where $v=(v_1,\ldots,v_d)\in T^d$ identified with $[-1/2,1/2)^d$. Since in the formation the boundary region about the interface has width $O(1/\sqrt{K_N})$, we will scale the $v_1$ coordinate in the fluctuation field by $\sqrt{K_N}$ so that the scaling limit will capture information `near' the interface. We identify the fluctuation limit as a Gaussian field when $K_N\uparrow \infty$ and $K_N= O(\sqrt{\log(N)})$ in $d\leq 2$. In the one dimensional case, the field limit is given by ${\bf e}(v_1) B_t$ where $B_t$ is a Brownian motion and ${\bf e}$ is the normalized derivative of a decreasing `standing wave' solution $ϕ$ of $\partial^2_{v_1} ϕ- V'(ϕ)=0$ on $R$, where $V'$ is the homogenization of the Glauber rates. In two dimensions, the limit is ${\bf e}(v_1)Z_t(v_2)$ where $Z_t$ is the solution of a one dimensional stochastic heat equation. The appearance of the function ${\bf e}(\cdot)$ in the limit field indicates that the interface fluctuation retains the shape of the transition layer $ϕ$.

math.PR↗

arXiv2024

Higher energy state approximations in the `Many Interacting Worlds' model

In the `Many Interacting Worlds' (MIW) discrete Hamiltonian system approximation of Schrödinger's wave equation, introduced in \cite{hall_2014}, convergence of ground states to the Normal ground state of the quantum harmonic oscillator, via Stein's method, in Wasserstein-$1$ distance with rate $\mathcal{O}(\sqrt{\log N}/N)$ has been shown in McKeague-Levin (2016), Chen-Thanh (2023), McKeague-Swan (2023). In this context, we construct approximate higher energy states of the MIW system, and show their convergence with the same rate in Wasserstein-$1$ distance to higher energy states of the quantum harmonic oscillator. In terms of techniques, we apply the `differential equation' approach to Stein's method, which allows to handle behavior near zeros of the higher energy states.

math-ph↗

arXiv2023

Atypical behaviors of a tagged particle in asymmetric simple exclusion

Consider the asymmetric nearest-neighbor exclusion process (ASEP) on ${\mathbb Z}$ with single particle drift $γ>0$, starting from a Bernoulli product invariant measure $ν_ρ$ with density $ρ$. It is known that the position $X_{N}$ of a tagged particle, say initially at the origin, at time $N$ satisfies an a.s. law of large numbers $\frac{1}{N}X_N \rightarrow γ(1-ρ)$ as $N\uparrow\infty$. In this context, we study the `typical' behavior of the tagged particle and `bulk' density evolution subject to `atypical' events $\{X_N\geq AN\}$ or $\{X_N\leq AN\}$ for $A\neq γ(1-ρ)$. We detail different structures, depending on whether $A<0$, $0\leq A< γ(1-ρ)$, $γ(1-ρ)<A< γ$, or $A\geq γ$, under which these atypical events are achieved, and compute associated large deviation costs. Among our results is an `upper tail' large deviation principle in scale $N$ for $\frac{1}{N}X_N$.

math.PR↗

arXiv2022

Gumbel laws in the symmetric exclusion process

We consider the symmetric exclusion particle system on $\mathbb{Z}$ starting from an infinite particle step configuration in which there are no particles to the right of a maximal one. We show that the scaled position $X_t/(σb_t) - a_t$ of the right-most particle at time $t$ converges to a Gumbel limit law, where $b_t = \sqrt{t/\log t}$, $a_t = \log(t/(\sqrt{2π}\log t))$, and $σ$ is the standard deviation of the random walk jump probabilities. This work solves a problem left open in Arratia (1983). Moreover, to investigate the influence of the mass of particles behind the leading one, we consider initial profiles consisting of a block of $L$ particles, where $L \to \infty$ as $t \to \infty$. Gumbel limit laws, under appropriate scaling, are obtained for $X_t$ when $L$ diverges in $t$. In particular, there is a transition when $L$ is of order $b_t$, above which the displacement of $X_t$ is similar to that under a infinite particle step profile, and below which it is of order $\sqrt{t\log L}$. Proofs are based on recently developed negative dependence properties of the symmetric exclusion system. Remarks are also made on the behavior of the right-most particle starting from a step profile in asymmetric nearest-neighbor exclusion, which complement known results.

math.PR↗

arXiv2022

Motion by mean curvature from Glauber-Kawasaki dynamics with speed change

We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of Glauber-Kawasaki dynamics with speed change. The Kawasaki part describes the movement of particles through particle interactions. It is speeded up in a diffusive space-time scaling. The Glauber part governs the creation and annihilation of particles. The Glauber part is set to favor two levels of particle density. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, a mean-curvature interface flow emerges, with a homogenized `surface tension-mobility' parameter reflecting microscopic rates. The interface separates the two levels of particle density. Similar hydrodynamic limits have been derived in two recent papers; one where the Kawasaki part describes simple nearest neighbor interactions, and one where the Kawasaki part is replaced by a zero-range process. We extend the main results of these two papers beyond nearest-neighbor interactions. The main novelty of our proof is the derivation of a `Boltzmann-Gibbs' principle which covers a class of local particle interactions.

math.PR↗

arXiv2022

Constant-speed interface flow from unbalanced Glauber-Kawasaki dynamics

We derive the hydrodynamic limit of Glauber-Kawasaki dynamics. The Kawasaki part is simple and describes independent movement of the particles with hard core exclusive interactions. It is speeded up in a diffusive space-time scaling. The Glauber part describes the birth and death of particles. It is set to favor two levels of particle density with a preference for one of the two. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, the limiting particle density instantly takes either of the two favored density values. The interface which separates these two values evolves with constant speed (Huygens' principle). Similar hydrodynamic limits have been derived in four recent papers. The crucial difference with these papers is that we consider Glauber dynamics which has a preferences for one of the two favored density values. As a result, we observe limiting dynamics on a shorter time scale, and the evolution is different from the mean curvature flow obtained in the four previous papers. While several steps in our proof can be adopted from these papers, the proof for the propagation of the interface is new.

math.PR↗

arXiv2022

Condensation, boundary conditions, and effects of slow sites in zero-range systems

We consider the space-time scaling limit of the particle mass in zero-range particle systems on a $1$D discrete torus $\mathbb{Z}/N\mathbb{Z}$ with a finite number of defects. We focus on two classes of increasing jump rates $g$, when $g(n)\sim n^α$, for $0<α\leq 1$, and when $g$ is a bounded function. In such a model, a particle at a regular site $k$ jumps equally likely to a neighbor with rate $g(n)$, depending only on the number of particles $n$ at $k$. At a defect site $k_{j,N}$, however, the jump rate is slowed down to $λ_j^{-1}N^{-β_j}g(n)$ when $g(n)\sim n^α$, and to $λ_j^{-1}g(n)$ when $g$ is bounded. Here, $N$ is a scaling parameter where the grid spacing is seen as $1/N$ and time is speeded up by $N^2$. Starting from initial measures with $O(N)$ relative entropy with respect to an invariant measure, we show the hydrodynamic limit and characterize boundary behaviors at the macroscopic defect sites $x_j = \lim_{N\uparrow \infty} k_{j, N}/N$, for all defect strengths. For rates $g(n)\sim n^α$, at critical or super-critical slow sites ($β_j=α$ or $β_j>α$), associated Dirichlet boundary conditions arise as a result of interactions with evolving atom masses or condensation at the defects. Differently, when $g$ is bounded, at any slow site ($λ_j>1$), we find the hydrodynamic density must be bounded above by a threshold value reflecting the strength of the defect. Moreover, due to interactions with masses of atoms stored at the slow sites, the associated boundary conditions bounce between being periodic and Dirichlet.

math.PR↗

arXiv2022

Norm-Agnostic Linear Bandits

Linear bandits have a wide variety of applications including recommendation systems yet they make one strong assumption: the algorithms must know an upper bound $S$ on the norm of the unknown parameter $θ^*$ that governs the reward generation. Such an assumption forces the practitioner to guess $S$ involved in the confidence bound, leaving no choice but to wish that $\|θ^*\|\le S$ is true to guarantee that the regret will be low. In this paper, we propose novel algorithms that do not require such knowledge for the first time. Specifically, we propose two algorithms and analyze their regret bounds: one for the changing arm set setting and the other for the fixed arm set setting. Our regret bound for the former shows that the price of not knowing $S$ does not affect the leading term in the regret bound and inflates only the lower order term. For the latter, we do not pay any price in the regret for now knowing $S$. Our numerical experiments show standard algorithms assuming knowledge of $S$ can fail catastrophically when $\|θ^*\|\le S$ is not true whereas our algorithms enjoy low regret.

stat.ML↗

arXiv2021

Mean curvature interface limit from Glauber+Zero-range interacting particles

We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of a class of Glauber+Zero-range particle systems. The Zero-range part moves particles while preserving particle numbers, and the Glauber part governs the creation and annihilation of particles and is set to favor two levels of particle density. When the two parts are simultaneously seen in certain different time-scales, the Zero-range part being diffusively scaled while the Glauber part is speeded up at a lesser rate, a mean-curvature interface flow emerges, with a homogenized `surface tension-mobility' parameter reflecting microscopic rates, between the two levels of particle density. We use relative entropy methods, along with a suitable `Boltzmann-Gibbs' principle, to show that the random microscopic system may be approximated by a `discretized' Allen-Cahn PDE with nonlinear diffusion. In turn, we show the behavior, especially generation and propagation of interface properties, of this `discretized' PDE.

math.PR↗

arXiv2021

Schauder estimate for quasilinear discrete PDEs of parabolic type

We investigate quasilinear discrete PDEs $\partial_t u = Δ^N φ(u)+ Kf(u)$ of reaction-diffusion type with nonlinear diffusion term defined on an $n$-dimensional unit torus discretized with mesh size $\tfrac1N$ for $N\in {\mathbb N}$, where $Δ^N$ is the discrete Laplacian, $φ$ is a strictly increasing $C^5$ function and $f$ is a $C^1$ function. We establish $L^\infty$ bounds and space-time Hölder estimates, both uniform in $N$, of the first and second spatial discrete derivatives of the solutions. In the equation, $K>0$ is a large constant and we show how these estimates depend on $K$. The motivation for this work stems originally from the study of hydrodynamic scaling limits of interacting particle systems. Our method is a two steps approach in terms of the Hölder estimate and Schauder estimate, which is known for continuous parabolic PDEs. We first show the discrete Hölder estimate uniform in $N$ for the solutions of the associated linear discrete PDEs with continuous coefficients, based on the Nash estimate. We next establish the discrete Schauder estimate for linear discrete PDEs with uniform Hölder coefficients. The link between discrete and continuous settings is given by the polylinear interpolations. Since this operation has a non-local nature, the method requires proper modifications. We also discuss another method based on the study of the corresponding fundamental solutions.

math.AP↗

arXiv2021

Singular limit of an Allen-Cahn equation with nonlinear diffusion

We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scaling limit of certain interacting particle systems. We investigate its singular limit and show the generation and propagation of an interface in the limit. The evolution of this limit interface is governed by mean curvature flow with a novel, homogenized speed in terms of a surface tension-mobility parameter emerging from the nonlinearity in our equation.

math.AP↗

arXiv2021

On the use of Markovian stick-breaking priors

In [10], a `Markovian stick-breaking' process which generalizes the Dirichlet process $(μ, θ)$ with respect to a discrete base space ${\mathfrak X}$ was introduced. In particular, a sample from from the `Markovian stick-breaking' processs may be represented in stick-breaking form $\sum_{i\geq 1} P_i δ_{T_i}$ where $\{T_i\}$ is a stationary, irreducible Markov chain on ${\mathfrak X}$ with stationary distribution $μ$, instead of i.i.d. $\{T_i\}$ each distributed as $μ$ as in the Dirichlet case, and $\{P_i\}$ is a GEM$(θ)$ residual allocation sequence. Although the motivation in [10] was to relate these Markovian stick-breaking processes to empirical distributional limits of types of simulated annealing chains, these processes may also be thought of as a class of priors in statistical problems. The aim of this work in this context is to identify the posterior distribution and to explore the role of the Markovian structure of $\{T_i\}$ in some inference test cases.

math.ST↗