Sunder Sethuraman

Interests

Research

Probability and random processes, and its connections to analysis, mathematical physics, and statistics and data science

Teaching

Probability and stochastic processes, Analysis and its applications, Mathematical Physics, Statistics and data science

Courses

Scholarly Contributions

Chapters

• Venkataramani, S. C., & Sethuraman, S. (2019). On the Growth of a Superlinear Preferential Attachment Scheme.. In Probability and Analysis in Interacting Physical Systems(pp 243-265). Cham: Springer. doi:https://doi.org/10.1007/978-3-030-15338-0_9
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  We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the graph evolution condenses, that is a.s. in the limit graph there will be a single random node with infinite degree while all others have finite degree.In this note, among other results, we establish law of large numbers type limits in probability, as n\to \infty$, for the counts Z_k(n) of the number of nodes with degree k\geq 1 at time n\geq 1. These limits rigorously verify a physical picture of Krapivisky, Redner and Leyvraz (2000) on how the condensation arises with respect to the degree distribution.
• Bernardin, C., Goncalves, P., & Sethuraman, S. (2014). Equilibrium fluctuations of additive functionals of zero-range models.. In From Particle Systems to Partial Differential Equations(pp 143--169). Berlin: Springer-Verlag.
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  We discuss the fluctuations on occupation time and other functionals of zero-range particle systems. The main point of the paper is to understand how mixing rate, given by the type of interaction, connects with the fluctuations.

Journals/Publications

• Conroy, M., & Sethuraman, S. (2023). Gumbel laws in the symmetric exclusion process. Commun. Math. Phys., 402, 723-764.
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  We consider the symmetric exclusion particle system on $\mathbb{Z}$ startingfrom an infinite particle step configuration in which there are no particles tothe right of a maximal one. We show that the scaled position $X_t/(\sigma b_t)- a_t$ of the right-most particle at time $t$ converges to a Gumbel limit law,where $b_t = \sqrt{t/\log t}$, $a_t = \log(t/(\sqrt{2\pi}\log t))$, and$\sigma$ is the standard deviation of the random walk jump probabilities. Thiswork solves a problem left open in Arratia (1983).Moreover, to investigate the influence of the mass of particles behind theleading one, we consider initial profiles consisting of a block of $L$particles, where $L \to \infty$ as $t \to \infty$. Gumbel limit laws, underappropriate scaling, are obtained for $X_t$ when $L$ diverges in $t$. Inparticular, there is a transition when $L$ is of order $b_t$, above which thedisplacement of $X_t$ is similar to that under a infinite particle stepprofile, and below which it is of order $\sqrt{t\log L}$.Proofs are based on recently developed negative dependence properties of thesymmetric exclusion system. Remarks are also made on the behavior of theright-most particle starting from a step profile in asymmetric nearest-neighborexclusion, which complement known results.
• El Kettani, P., Funaki, T., Hilhorst, D., Park, H., & Sethuraman, S. (2023). Singular limit of an Allen-Cahn equation with nonlinear diffusion. Tunisian J. Math., 4(4), 719-754.
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  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.
• Funaki, T., van Meurs, P., Sethuraman, S., & Tsunoda, K. (2023). Constant-speed interface flow from unbalanced Glauber-Kawasaki dynamics. Ensaios Matemáticos, 38, 223-248.
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  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. [Journal_ref: ]
• Funaki, T., van Meurs, P., Sethuraman, S., & Tsunoda, K. (2023). Motion by mean-curvature from Glauber-Kawasaki with speed change. J. Stat. Phys., 190(3), 45.
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  We consider a system of particles interacting via Glauber-Kawasaki rates with speed change. In a certain scale, we derive an interface flow moving by mean-curvature. Unlike when the Kawasaki rates do not depend on the spatial configuration, in this model with speed change, the mean-curvature flow includes a mobility/surface tension parameter, reflecting the nonlinear rates.
• Landim, C., Pacheco, C. G., Sethuraman, S., & Xue, J. (2023). On a nonlinear SPDE derived from a hydrodynamic limit in a Sinai-type random environment. Ann Appl Probab, 33(1), 38pgs.
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  With the recent developments on nonlinear SPDE's, where smoothing of rough noises is needed, one is naturally led to study interacting particle systems whose macroscopic evolution is described by these equations and which possess an in-built smoothing. In this article, our main results are to derive regularized versions of the ill-posed one dimensional SPDE $$\partial_t \rho = \frac{1}{2}\Delta \Phi(\rho) - 2 \nabla \big(W'\Phi(\rho)\big),$$ where the spatial white noise $W'$ is replaced by a regularization $W'_\varepsilon$, as quenched and annealed hydrodynamic limits of zero-range interacting particle systems in $\varepsilon$-regularized Sinai-type random environments. Some computations are also made about annealed mean hydrodynamic limits in unregularized Sinai-type random environments with respect to independent particles.
• Doehrmann, T., Sethuraman, S., & Venkataramani, S. (2021). Remarks on the range and multiple range of random walk up to the time of exit. Rocky Mountain Journal of Mathematics, 51(5), 1603--1614. doi:DOI: 10.1216/rmj.2021.51.1603
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  We consider the scaling behavior of the range and $p$-multiple range, that is the number of points visited and the number of points visited exactly $p\geq 1$ times, of simple random walk on $\Z^d$, for dimensions $d\geq 2$, up to time of exit from a domain $D_N$ of the form $D_N = ND$ where $D\subset \R^d$, as $N\uparrow\infty$. Recent papers have discussed connections of the range and related statistics with the Gaussian free field, identifying in particular that the distributional scaling limit for the range, in the case $D$ is a cube in $d\geq 3$, is proportional to the exit time of Brownian motion. The purpose of this note is to give a concise, different argument that the scaled range and multiple range, in a general setting in $d\geq 2$, both weakly converge to proportional exit times of Brownian motion from $D$, and that the corresponding limit moments are `polyharmonic', solving a hierarchy of Poisson equations.
• El Kettani, P., Funaki, T., Hilhorst, D., Park, H., & Sethuraman, S. (2022). Mean curvature interface limit from Glauber+Zero-range interacting particles. Commun. Math. Phys., 394(3), 1173--1223. doi:https://doi.org/10.1007/s00220-022-04424-8
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  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 favoring 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 generation and propagation of the interface with respect to this `discretized. PDE.
• Lippitt, W., Sethuraman, S., & Tang, X. (2022). Stationarity and Inference in Multistate Promoter Models of Stochastic Gene Expression via Stick-Breaking Measures. SIAM Journal on Applied Mathematics, 82(6), 1953-1986. doi:10.1137/21m1440876
• Lippitt, W., Sethuraman, S., & Tang, X. (2022). Stick-breaking Stationary Measures in Multistate Promoter Stochastic Gene Expression Networks. SIAM Journal of Applied Mathematics, 82(6), 35pgs.
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  We show that the stationary distribution of a multistate promoter mRNA model can be characterized in terms of `stick breaking' measures, and also consider statistical inference procedures using this characterization.
• Bernardin, C., Sethuraman, S., & Funaki, T. (2021). Derivation of coupled KPZ-Burgers equation from multi-species zero-range processes.. Ann. Appl. Probab..
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  We consider the fluctuation fields of multi-species weakly-asymmetric zero-range interacting particle systems in one dimension, where the mass density of each species is conserved. Although such fields have been studied in systems with a single species, the multi-species setting is much less understood. Among other results, we show that, when the system starts from stationary states, with a particular property, the scaling limits of the multi-species fluctuation fields, seen in a characteristic traveling frame, solve a coupled Burgers SPDE, which is a formal spatial gradient of a coupled KPZ equation.
• Dietz, Z., Lippitt, W., & Sethuraman, S. (2021). STICK-BREAKING PROCESSES, CLUMPING, AND MARKOV CHAIN OCCUPATION LAWS. Sankhya A, 43pgs.
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  We connect the empirical or ‘occupation’ laws of certain discrete space time-inhomogeneous Markov chains, related to simulated annealing, toa novel class of ‘stick-breaking’ processes, a ‘nonexchangeable’ generalizationof the Dirichlet process used in nonparametric Bayesian statistics. To make this unexpected correspondence, we examine an intermediate ‘clumped’ structure in both the time-inhomogeneous Markov chains and the stick-breaking processes, perhaps of its own interest, which records the sequence of different states visited and the scaled proportions of time spent on them. By matching the associated intermediate structures, we identify the limits of the empirical measures of the time-inhomogeneous Markov chains as types of stick-breaking processes.
• Funaki, T., Hoshino, M., Sethuraman, S., & Xie, B. (2021). Asymptotics of PDE in random environment by paracontrolled calculus. Ann Inst Henri Poincare Prob Stat, 34.
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  We apply the paracontrolled calculus to study the asymptotic behavior of a certain quasilinear PDE with smeared mild noise, which originally appears as the space-time scaling limit of a particle system in random environment on one dimensional discrete lattice. We establish the convergence result and show a local in time well-posedness of the limit stochastic PDE with spatial white noise. It turns out that our limit stochastic PDE does not require any renormalization. We also show a comparison theorem for the limit equation
• Lippitt, W., & Sethuraman, S. (2021). ON THE USE OF MARKOVIAN STICK-BREAKING PRIORS. AMS Contemporary Mathematics Volume in Honor of MM Rao, 18pgs.
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  In Dietz-Lippitt-Sethuraman (2020), a ‘Markovian stick-breaking’ process which generalizes the Dirichlet process(μ,θ) with respect to a discrete base space X was introduced. In particular, a sample from from the ‘Markovian stick-breaking’ processs may be represented in stick-breaking form ∑P_iδ_(T_i) where {T_i} is a stationary, irreducible Markov chain on X with stationary distributionμ, 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 Dietz-Lippitt-Sethuraman (2020) 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.
• Sethuraman, S. (2021). My friend and mentor, Thomas Liggett. Celabratio Mathematica.
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  A tribute to Thomas Liggett online in Celebratio Mathematica (2 pgs).
• Fatkullin, I., Sethuraman, S., & Xue, J. (2020). On Hydrodynamic Limits of Young Diagrams. Elec. J. Probab., 25, Paper no. 58, 44pgs. doi:doi:10.1214/20-EJP455
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  We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. ‘Static’ scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding ‘dynamical’ limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types parabolic PDEs, depending on the energy structure.
• Davis, E., & Sethuraman, S. (2019). Approximating geodesics via random points. Ann. Appl. Probab., 29, 1446--1486.
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  Given a ‘cost’ functional F on paths \gamma in a domain D⊂R^d, in the form F(\gamma) = \int_0^1 f(\dot \gamma, \gamma)dt, it is of interest to approximate its minimumcost and geodesic paths. Let X_1, ..., X_n be points drawn independently from D according to a distribution with a density. Form a random geometric graph on the points where X_i and X_j are connected when 0
• Davis, E., & Sethuraman, S. (2016). Consistency of modularity clustering of random geometric graphs. Ann. Appl. Probab..
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  We consider samples X_1,X_2,...,X_n of independent, identically distributed observations of an underlying probability measure \nu on a bounded domain D\subset R^d. The popular `modularity' clustering method specifies a partition U of the data set as the output of an optimization problem. In this paper, under conditions on \nu and D, we show a geometric form of consistency of versions of this method, among other limit results. That is, we show via Gamma convergence arguments, as n \to \infty, that the discrete optimal partitions U_n converge in a certain graphical sense to a continuum optimal partition U of the domain D, characterized as the output of a type of a Kelvin's shape optimization problem.
• Davis, E., & Sethuraman, S. (2017). Consistency of modularity clustering of random geometric graphs. Ann. Appl. Probab., 28, 2003--2062.
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  We consider samples X_1,X_2,...,X_n of independent, identically distributed observations of an underlying probability measure \nu on a bounded domain D\subset R^d. The popular `modularity' clustering method specifies a partition U of the data set as the output of an optimization problem. In this paper, under conditions on \nu and D, we show a geometric form of consistency of versions of this method, among other limit results. That is, we show via Gamma convergence arguments, as n \to \infty, that the discrete optimal partitions U_n converge in a certain graphical sense to a continuum optimal partition U of the domain D, characterized as the output of a type of a Kelvin's shape optimization problem.
• Davis, E., & Sethuraman, S. (2018). CONSISTENCY OF MODULARITY CLUSTERING ON RANDOM GEOMETRIC GRAPHS. ANNALS OF APPLIED PROBABILITY, 28(4), 2003-2062.
• Lega, J., Sethuraman, S., & Young, A. (2017). On collisions times of elastic and inelastic particles in one-dimension with random initial positions and velocities. J. Stat. Phys., 170, 1088--1122.
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  We investigate a one-dimensional system of N particles, initially distributed with random positions and velocities, interacting through binary collisions. The collisions rule is such that there is a time after which the N particles do not interact and sorted according to their velocities. In such a system, the distribution of the collision times is of interest. When the collisions are elastic, we derive asymptotic distributions for the final collision time for a single particle and the final collision time of the system as the number of particles approaches infinity under different assumptions of the initial distributions of the particles’ positions and velocities. Additionally, a numerical investigation is carried out to determine how an inelastic collision rule which conserves neither momentum nor energy affects the distribution of collision times, the final collision time of a particle, and the final collision time of the system.
• Lega, J., Sethuraman, S., & Young, A. L. (2018). On Collisions Times of 'Self-Sorting' Interacting Particles in One-Dimension with Random Initial Positions and Velocities. JOURNAL OF STATISTICAL PHYSICS, 170(6), 1088-1122.
• Sethuraman, S., & Shahar, D. (2018). Hydrodynamic limits for long-range asymmetric interacting particle systems. ELECTRONIC JOURNAL OF PROBABILITY, 23.
• Sethuraman, S., & Shahar, D. (2018). Hydrodynamics for asymmetric long-range interacting particle systems. Elec. J. Probab., 23, 54pgs.
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  We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on Zn, where the jump rates are asymmetric and long-range of order |x|^{n+α} for a particle displacement of order |x|. Two types of evolution equations are identified depending on the strength of the long-range asymmetry. When 0 < α < 1, we find a new integro-partial differential hydrodynamic equation, in an anomalous space-time scale. On the other hand, when α ≥ 1, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.
• Young, A. L., Young, A. L., Sethuraman, S., Sethuraman, S., Lega, J. C., & Lega, J. C. (2018). On collisions times of `self-sorting' interacting particles in one-dimension with random initial positions and velocities. Journal of Statistical Physics, 170, 1088-1122. doi:http://dx.doi.org/10.1007/s10955-018-1974-4
• Bernardin, C., Goncalves, P., & Sethuraman, S. (2016). Occupation times of long-range exclusion and connections to KPZ class exponents. Probability Theory and Related Fields, 166, 365-428.
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  It is known that the fluctuations of the occupation time in nearest-neighbor asymmetric simple exclusion processes, in some regime of parameters, reflects the fluctuations of the height function of certain interfaces related to KPZ class phenomena. Here, we consider asymmetries which are long-range and find phase transitions in the occupation time fluctuations in terms of a parameter governing the strength of the long-range transitions. We propose an extension of the nearest-neighbor KPZ class to these long-range models.
• Choi, J., Sethuraman, S., & Venkatasubramani, S. (2016). A scaling limit for the degree distribution in sublinear preferential attachment. Random Structures and Algorithms, 48, 703-731.
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  This paper establishes a rigorous mean-field theory for the degree distribution in a growth model of random graphs which grow by a sublinear reinforcement rule, the so-called sublinear preferential attachment graph. The new technique merges probability and dynamical system arguments.
• Sethuraman, S. (2016). On microscopic derivation of a fractional stochastic Burgers equation. Communications in Mathematical Physics, 341, 625-665.
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  We derive a fractional Burgers stochastic partial differential equations as a limit of the fluctuation field from certain long-range weakly asymmetric interacting particle systems. This work introduces these SPDEs and also physically motivates them.
• Choi, J., & Sethuraman, S. (2013). Large deviations for the degree structure in preferential attachment schemes. Annals of Applied Probability, 23(2), 722-763.
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  Abstract: Preferential attachment schemes, where the selection mechanism is linear and possibly time-dependent, are considered, and an infinite-dimensional large deviation principle for the sample path evolution of the empirical degree distribution is found by Dupuis-Ellis-type methods. Interestingly, the rate function, which can be evaluated, contains a term which accounts for the cost of assigning a fraction of the total degree to an "infinite" degree component, that is, when an atypical "condensation" effect occurs with respect to the degree structure. As a consequence of the large deviation results, a sample path a.s. law of large numbers for the degree distribution is deduced in terms of a coupled system of ODEs from which power law bounds for the limiting degree distribution are given. However, by analyzing the rate function, one can see that the process can deviate to a variety of atypical nonpower law distributions with finite cost, including distributions typically associated with sub and superlinear selection models. © 2013 Institute of Mathematical Statistics.
• Jara, M., Landim, C., & Sethuraman, S. (2013). Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear zero-range processes. Annales de l'institut Henri Poincare (B) Probability and Statistics, 49(3), 611-637.
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  Abstract: 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. © Association des Publications de l'Institut Henri Poincaré, 2013.
• Liu, S., Matzavinos, A., & Sethuraman, S. (2013). Random walk distances in data clustering and applications. Advances in Data Analysis and Classification, 7(1), 83-108.
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  Abstract: 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. © 2013 Springer-Verlag Berlin Heidelberg.
• Sethuraman, S., & Varadhan, S. R. (2013). Large deviations for the current and tagged particle in 1D nearest-neighbor symmetric simple exclusion. Annals of Probability, 41(3 A), 1461-1512.
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  Abstract: 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 Poincaré 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. © Institute of Mathematical Statistics, 2013.
• Athreya, S., Sethuraman, S., & Tóth, B. (2011). On the range, local times and periodicity of random walk on an interval. Alea, 8(1), 269-284.
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  Abstract: The range, local times, and periodicity of symmetric, weakly asymmetric and asymmetric random walks at the time of exit from a strip with N locations are considered. Several results on asymptotic distributions are obtained.
• Sethuraman, J., & Sethuraman, S. (2010). Connections between bernoulli strings and random permutations. tThe Legacy of Alladi Ramakrishnan in the Mathematical Sciences, 389-399.
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  Abstract: A sequence of random variables, each taking only two values 0 or 1, is called a Bernoulli sequence. Consider the counts of occurrences of strings of the form {11}, {101}, {1001}, ⋯ in Bernoulli sequences. Counts of such Bernoulli strings arise in the study of the cycle structure of random permutations, Bayesian nonparametrics, record values etc. The joint distribution of such counts is a problem worked on by several researchers. In this paper, we summarize the recent technique of using conditional marked Poisson processes which allows to treat all cases studied previously. We also give some related open problems. © 2010 Springer Science+Business Media, LLC.
• Bryc, W., Minda, D., & Sethuraman, S. (2009). Large deviations for the leaves in some random trees. Advances in Applied Probability, 41(3), 845-873.
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  Abstract: Large deviation principles and related results are given for a class of Markov chains associated to the leaves.' in random recursive trees and preferential attachment random graphs, as well as the 'cherries' in Yule trees. In particular, the method of proof, combining analytic and Dupuis-Ellis-type path arguments, allows for an explicit computation of the large deviation pressure. © Applied Probability Trust 2009.
• Huffer, F. W., Sethuraman, J., & Sethuraman, S. (2009). A study of counts of bernoulli strings via conditional poisson processes. Proceedings of the American Mathematical Society, 137(6), 2125-2134.
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  Abstract: 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 paper, 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. © 2008 American Mathematical Society.
• Jara, M. D., Landim, C., & Sethuraman, S. (2009). Nonequilibrium fluctuations for a tagged particle in mean-zero one-dimensional zero-range processes. Probability Theory and Related Fields, 145(4), 565-590.
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  Abstract: We prove a non-equilibrium functional central limit theorem for the position of a tagged particle in mean-zero one-dimensional zero-range process. The asymptotic behavior of the tagged particle is described by a stochastic differential equation governed by the solution of the hydrodynamic equation. © Springer-Verlag 2008.
• Athreya, K. B., Ghosh, A. P., & Sethuraman, S. (2008). Growth of preferential attachment random graphs via continuous-time branching processes. Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 118(3), 473-494.
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  Abstract: Some growth asymptotics of a version of 'preferential attachment' random graphs are studied through an embedding into a continuous-time branching scheme. These results complement and extend previous work in the literature.
• Balázs, M., Rassoul-Agha, F., Seppäläinen, T., & Sethuraman, S. (2007). Existence of the zero range process and a deposition model with superlinear growth rates. Annals of Probability, 35(4), 1201-1249.
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  Abstract: We give a construction of the zero range and bricklayers' processes in the totally asymmetric, attractive case. The novelty is mat we allow jump rates to grow exponentially. Earlier constructions have permitted at most linearly growing rates. We also show the invariance and extremality of a natural family of i.i.d. product measures indexed by particle density. Extremality is proved with an approach mat is simpler than existing ergodicity proofs. © Institute of Mathematical Statistics, 2007.
• Dietz, Z., & Sethuraman, S. (2007). Occupation laws for some time-nonhomogeneous Markov chains. Electronic Journal of Probability, 12, 661-683.
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  Abstract: We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time n is I + G/nζ where G is a "generator" matrix, that is G(i, j) > 0 for i, j distinct, and G(i, i) = -∑k≠i G(i, k), and ζ > 0 is a strength parameter. In these chains, as time grows, the positions are less and less likely to change, and so form simple models of age-dependent time-reinforcing schemes. These chains, however, exhibit a trichotomy of occupation behaviors depending on parameters. We show that the average occupation or empirical distribution vector up to time n, when variously 0 < ζ < 1, ζ > 1 or ζ = 1, converges in probability to a unique "stationary" vector V G, converges in law to a nontrivial mixture of point measures, or converges in law to a distribution μG with no atoms and full support on a simplex respectively, as n ↑ ∞. This last type of limit can be interpreted as a sort of "spreading" between the cases 0 < ζ 1 and ζ > 1. In particular, when G is appropriately chosen, μG is a Dirichlet distribution, reminiscent of results in Pólya urns.
• Sethuraman, S. (2007). On diffusivity of a tagged particle in asymmetric zero-range dynamics. Annales de l'institut Henri Poincare (B) Probability and Statistics, 43(2), 215-232.
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  Abstract: Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑ j p (j) ≠ 0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d ≥ 1, and at most order t in d = 1 and d ≥ 3 for a wide class of rates g. Also, in d = 1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g (k) increases, and g (k) / k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given. © 2006 Elsevier Masson SAS. All rights reserved.
• Sethuraman, S. (2006). Erratum: Central limit theorems for additive functionals of the simple exclusion process (Annals of Probability (2006) 34:1 (427-428)). Annals of Probability, 34(1), 427-428.
• Sethuraman, S. (2006). Superdiffusivity of occupation-time variance in 2-dimensional asymmetric exclusion processes with density ρ = 1/2. Journal of Statistical Physics, 123(4), 787-802.
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  Abstract: We compute that the growth of the occupation-time variance at the origin up to time t in dimension d = 2 with respect to asymmetric simple exclusion in equilibrium with density ρ = 1/2 is in a certain sense at least tlog (log t) for general rates, and at least t(log t) 1/2 for rates which are asymmetric only in the direction of one of the axes. These estimates give a complement to bounds in the literature when d = 1, and are consistent with an important conjecture with respect to the transition function and variance of "second-class" particles. © 2006 Springer Science+Business Media, Inc.
• Dietz, Z., & Sethuraman, S. (2005). Large deviations for a class of nonhomogeneous Markov chains. Annals of Applied Probability, 15(1 A), 421-486.
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  Abstract: Large deviation results are given for a class of perturbed nonhomogeneous Markov chains on finite state space which formally includes some stochastic optimization algorithms. Specifically, let {P n} be a sequence of transition matrices on a finite state space which converge to a limit transition matrix P. Let [X n] be the associated nonhomogeneous Markov chain where P n controls movement from time n - 1 to n. The main statements are a large deviation principle and bounds for additive functionals of the nonhomogeneous process under some regularity conditions. In particular, when P is reducible, three regimes that depend on the decay of certain "connection" P n probabilities are identified. Roughly, if the decay is too slow, too fast or in an intermediate range, the large deviation behavior is trivial, the same as the time-homogeneous chain run with P or nontrivial and involving the decay rates. Examples of anomalous behaviors are also given when the approach P n → P is irregular. Results in the intermediate regime apply to geometrically fast running optimizations, and to some issues in glassy physics. © Institute of Mathematical Statistics, 2005.
• Sethuraman, S., & Varadhan, S. R. (2005). A martingale proof of dobrushin's theorem for non-homogeneous Markov Chains. Electronic Journal of Probability, 10.
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  Abstract: In 1956, Dobrushin proved an important central limit theorem for non-homogeneous Markov chains. In this note, a shorter and different proof elucidating more the assumptions is given through martingale approximation.
• Seppäläinen, T., & Sethuraman, S. (2003). Transience of second-class particles and diffusive bounds for additive functionals in one-dimensional asymmetric exclusion processes. Annals of Probability, 31(1), 148-169.
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  Abstract: Consider a one-dimensional exclusion process with finite-range translation-invariant jump rates with nonzero drift. Let the process be stationary with product Bernoulli invariant distribution at density ρ. Place a second-class particle initially at the origin. For the case ≠ 1/2 we show that the time spent by the second-class particle at the origin has finite expectation. This strong transience is then used to prove that variances of additive functionals of local mean-zero functions are diffusive when ≠ 1/2. As a corollary to previous work, we deduce the invariance principle for these functionals. The main arguments are comparisons of H-1 norms, a large deviation estimate for second-class particles and a relation between occupation times of second-class particles, and additive functional variances.
• Sethuraman, S. (2003). A clustering law for some discrete order statistics. Journal of Applied Probability, 40(1), 226-241.
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  Abstract: Let X1, X2, ..., Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a 'clustering' condition, limn → ∞[1 -F(n + 1)]/[1 - F(n)] = 0. In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if limn → ∞ [1 - F(n + r + 1)]/[1 - F(n)] = 0. Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.
• Sethuraman, S. (2003). An equivalence of H_-1 norms for the simple exclusion process. Annals of Probability, 31(1), 35-62.
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  Abstract: Resolvent H-1 norms with respect to simple exclusion processes play an important role in many problems with respect to additive functionals, tagged particles, and hydrodynamics, among other concerns. Here, general translation-invariant finite-range simple exclusion processes with and without a distinguished particle are considered. For the standard system of indistinguishable particles, it is proved that the corresponding H-1 norms are equivalent, in a sense, to the H-1 norms of a nearest-neighbor system. The same result holds for systems with a distinguished particle in dimensions d ≥ 2. However, in dimension d = 1, this equivalence does not hold. An application of the H-1 norm equivalence to additive functional variances is also given.
• Sethuraman, S. (2003). Conditional survival distributions of Brownian trajectories in a one dimensional Piossonian environment. Stochastic Processes and their Applications, 103(2), 169-209.
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  Abstract: Large time annealed path measure limits for a one-dimensional Brownian motion, with possibly a small drift, moving among "soft" Poissonian traps are considered. Limits with respect to both scaled and unscaled motions are derived. The results in both cases considered here agree with those shown before for the related model with "hard" traps. The proofs follow by generalizing previous techniques which identify a large clearing empty of traps in which typically the Brownian motion is confined. What is understood then, in the cases studied, is that under the annealed measure the soft traps organize to act in effect as their hard counterparts. © 2002 Elsevier Science B.V. All rights reserved.
• Athreya, J. S., & Sethuraman, S. (2001). On the asymptotics of discrete order statistics. Statistics and Probability Letters, 54(3), 243-249.
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  Abstract: 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. © 2001 Elsevier Science B.V.
• Sethuraman, S. (2001). On extremal measures for conservative particle systems. Annales de l'institut Henri Poincare (B) Probability and Statistics, 37(2), 139-154.
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  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. © 2001 Éditions scientifiques et médicales Elsevier SAS.
• Sethuraman, S. (2000). Central limit theorems for additive functionals of the simple exclusion process. Annals of Probability, 28(1), 277-302.
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  Abstract: 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.
• Sethuraman, S., Varadhan, S. R., & Yau, H. (2000). Diffusive limit of a tagged particle in asymmetric simple exclusion processes. Communications on Pure and Applied Mathematics, 53(8), 972-1006.
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  Abstract: Invariance principles are proved under diffusive scaling for the centered position of a tagged particle in the simple exclusion process with asymmetric nonzero drift jump probabilities in dimensions d ≥ 3. The method of proof is by martingale techniques which rely on the fact that symmetric random walks are transient in high dimensions. © 2000 John Wiley & Sons, Inc.
• Landim, C., Sethuraman, S., & Varadhan, S. (1996). Spectral gap for zero-range dynamics. Annals of Probability, 24(4), 1871-1902.
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  Abstract: We give a lower bound on the spectral gap for symmetric zero-range processes. Under some conditions on the rate function, we show that the gap shrinks as n-2, independent of the density, for the dynamics localized on a cube of size nd. We follow the method outlined by Lu and Yau, where a similar spectral gap is proved for Kawasaki dynamics.
• Sethuraman, S., & Lin, X. u. (1996). A central limit theorem for reversible exclusion and zero-range particle systems. Annals of Probability, 24(4), 1842-1870.
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  Abstract: We give easily verifiable conditions under which a functional central limit theorem holds for additive functionals of symmetric simple exclusion and symmetric zero-range processes. Also a reversible exclusion model with speed change is considered. Let η(t) be the configuration of the process at time t and let f(η) be a function on the state space. The question is: For which functions f does λ -1/2∫λt0 f(η(s)) ds converge to a Brownian motion? A general but often intractable answer is given by Kipnis and Varadhan. In this article we determine what conditions beyond a mean-zero condition on f(η) are required for the diffusive limit above. Specifically, we characterize the H-1 space in an applicable way. Our method of proof relies primarily on a sharp estimate on the "spectral gap" of the process and weak regularity properties for the invariant measures.
• Sethuraman, S., & Lin, X. u. (1996). Spin depolarization decay rates in α-symmetric stable fields on cubic lattices. Communications on Pure and Applied Mathematics, 49(12), 1281-1298.
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  Abstract: We study the asymptotic, long-time behavior of the energy function E(t; λ; f) = 1/t ln E exp { -tf [1/t ∑cursive Greek chi∈Zd (λ ∫t0 δ{cursive Greek chi} (Xs) ds)α]} where {Xs : 0 ≤ s < ∞} is the standard random walk on the d-dimensional lattice ℤd, 1 < α ≤ 2, and f : ℝ+ → ℝ+ is any nondecreasing concave function. In the special case f(cursive Greek chi) = cursive Greek chi, our setting represents a lattice model for the study of transverse magnetization of spins diffusing in a homogeneous, α-stable, i.i.d., random, longitudinal field {λV(cursive Greek chi) : cursive Greek chi ∈ ℤd} with common marginal distribution, the standard α-symmetric stable distribution; the parameter λ describes the intensity of the field. Using large-deviation techniques, we show that Sc(λ, α, f) = limt→∞ E(t; λ; f) exists. Moreover, we obtain a variational formula for this decay rate Sc. Finally, we analyze the behavior Sc(λ, α, f) as λ → 0 when f(cursive Greek chi) = cursive Greek chiβ for all 1 ≥ β > 0. Consequently, several physical conjectures with respect to lattice models of transverse magnetization are resolved by setting β = 1 in our results. We show that Sc(λ, α, 1) ∼ λα for d ≥ 3, λα(ln 1/λ)α-1 in d = 2, and λ2α/α+1 in d = 1. © 1996 John Wiley & Sons, Inc.

Proceedings Publications

• Gales, S., Sethuraman, S., & Jun, K. S. (2022, Summer). Norm-Agnostic linear bandits. In AISTATS 2022, 73--91.
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  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∗| ≤ S is true to guarantee that the regret will be low. In this paper, we propose novel algorithms that donot 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∗| ≤ S is not true whereas our algorithms enjoy low regret.