Sunder Sethuraman

A shock-tracking algorithm for surface evolution under reactive-ion etching

A new algorithm that determines the evolution of a surface eroding under reactive‐ion etching is presented. The surface motion is governed by both the Hamilton–Jacobi equation and the entropy condition for a given etch rate. The trajectories of ‘‘shocks’’ and ‘‘rarefaction waves’’ are then directly tracked, and thus this method may be regarded as a generalization of the method of characteristics. This allows slope discontinuities to be accurately calculated without artificial diffusion. The algorithm is compared with ‘‘geometric’’ surface evolution methods, such as the line‐segment method.

Central limit theorems for additive functionals of the simple exclusion process

Some invariance principles for additive functionals of simple exclusion with finite-range translation-invariant jump rates p i j = p j − i in dimensions d ≥ 1 are established. A previous investigation concentrated on the case of p symmetric. The principal tools to take care of nonreversibility, when p is asymmetric, are invariance principles for associated random variables and a “local balance” estimate on the asymmetric generator of the process. As a by-product, we provide upper and lower bounds on some transition probabilities for mean-zero asymmetric second-class particles, which are not Markovian, that show they behave like their symmetric Markovian counterparts. Also some estimates with respect to second-class particles with drift are discussed. In addition, a dichotomy between the occupation time process limits in d = 1 and d ≥ 2 for symmetric exclusion is shown. In the former, the limit is fractional Brownian motion with parameter 3/4, and in the latter, the usual Brownian motion.

A stochastic Burgers equation from a class of microscopic interactions

We consider a class of nearest-neighbor weakly asymmetric mass conservative particle systems evolving on Z, which includes zero-range and types of exclusion processes, starting from a perturbation of a stationary state. When the weak asymmetry is of order O(n−γ) for 1/2<γ≤1, we show that the scaling limit of the fluctuation field, as seen across process characteristics, is a generalized Ornstein–Uhlenbeck process. However, at the critical weak asymmetry when γ=1/2, we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic Burgers equation derived from taking the gradient of the KPZ equation. The proofs make use of a sharp “Boltzmann–Gibbs” estimate which improves on earlier bounds.

On extremal measures for conservative particle systems

Abstract It is well known that the exclusion, zero-range and misanthrope particle systems possess families of invariant measures due to the mass conservation property. Although these families have been classified a great deal, a full characterization of their extreme points is not available. In this article, we consider an approach to the study of this classification. One of the results in this note is that the zero-range product invariant measures, ∏i∈Sμα(·), for an infinite countable set S, under mild conditions, are identified as extremal for α(·)∈HZR where μα(i)(k)=Z(α(i))−1α(i)k/g(1)⋯g(k) with g and Z the rate function and normalization respectively, and HZR is the set of invariant measures for the transition probability p.

A study of counts of Bernoulli strings via conditional Poisson processes

A sequence of random variables, each taking values 0 or 1, is called a Bernoulli sequence. We say that a string of length d occurs, in a Bernoulli sequence, if a success is followed by exactly (d 1) failures before the next success. The counts of such d-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic d- cycle counts in random permutations. In this note, we give a new framework, in terms of conditional Poisson processes, which allows for a quick characterization of the joint distribution of the counts of all d-strings, in a general class of Bernoulli sequences, as certain mixtures of the product of Poisson measures. In particular, this general class includes all Bernoulli sequences considered in the literature, as well as a host of new sequences.

Large deviations for the current and tagged particle in 1D nearest-neighbor symmetric simple exclusion

Laws of large numbers, starting from certain nonequilibrium measures, have been shown for the integrated current across a bond, and a tagged particle in one-dimensional symmetric nearest-neighbor simple exclusion [Ann. Inst. Henri Poincare Probab. Stat. 42 (2006) 567–577]. In this article, we prove corresponding large deviation principles and evaluate the rate functions, showing different growth behaviors near and far from their zeroes which connect with results in [J. Stat. Phys. 136 (2009) 1–15].

On the asymptotics of discrete order statistics

Let X1,X2,...,Xn be a sequence of independent, identically distributed positive integer random variables. We study the asymptotics of the likelihood that the sample maximum is achieved k times and in its spacing relative to the second highest value. Earlier and other results are discussed in context. Also, some investigation is made when the sample is Markovian. Different results emerge in this case.

Random walk distances in data clustering and applications

In this paper, we develop a family of data clustering algorithms that combine the strengths of existing spectral approaches to clustering with various desirable properties of fuzzy methods. In particular, we show that the developed method “Fuzzy-RW,” outperforms other frequently used algorithms in data sets with different geometries. As applications, we discuss data clustering of biological and face recognition benchmarks such as the IRIS and YALE face data sets.

A remark on the maximum eigenvalue for circulant matrices

We point out that the method of Davis-Mikosch (Ann. Probab. 27 (1999) 522-536) gives for a symmetric circulant n × n matrix composed of i.i.d. entries with mean 0 and finite (2 + δ)-moments in the first half-row that the maximum eigenvalue is on the order � 2n log n, and the fluctuations are Gumbel.

Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear zero-range processes

Nonequilibrium fluctuations of a tagged, or distinguished particle in a class of one dimensional mean-zero zero-range systems with sublinear, increasing rates are derived. In Jara–Landim–Sethuraman (Probab. Theory Related Fields 145 (2009) 565–590), processes with at least linear rates are considered. A different approach to establish a main “local replacement” limit is required for sublinear rate systems, given that their mixing properties are much different. The method discussed also allows to capture the fluctuations of a “second-class” particle in unit rate, symmetric zero-range models. Resume. Nous demontrons les fluctuations hors d’equilibre d’une particule marquee pour une classe de systemes de particules a portee nulle uni-dimensionels de moyenne nulle dont le taux de sauts croit de maniere sous-lineaire. Dans Jara–Landim– Sethuraman (Probab. Theory Related Fields 145 (2009) 565–590), ce resutat a ete demontre pour des processus dont le taux croit au moins lineairement. La demonstration du lemme de remplacement dans le cas sous-lineaire exige une nouvelle approche en consequence des differences entre les proprietes de melanges des deux processus. La methode presentee permet egalement de demontrer les fluctuations d’une particule de deuxieme classe dans le modele a portee nulle symmetrique dont le taux de sauts est egal a 1.