arXiv subjects

Sunder Sethuraman

Publications and source records attributed to Sunder Sethuraman.

At least 37 records · Page 2Linked to original sources

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.

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arXiv2021

Stationarity and inference in multistate promoter models of stochastic gene expression via stick-breaking measures

In a general stochastic multistate promoter model of dynamic mRNA/protein interactions, we identify the stationary joint distribution of the promoter state, mRNA, and protein levels through an explicit `stick-breaking' construction of interest in itself. This derivation is a constructive advance over previous work where the stationary distribution is solved only in restricted cases. Moreover, the stick-breaking construction allows to sample directly from the stationary distribution, permitting inference procedures and model selection. In this context, we discuss numerical Bayesian experiments to illustrate the results.

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arXiv2020

On hydrodynamic limits in Sinai-type random environments

We investigate the hydrodynamical behavior of a system of random walks with zero-range interactions moving in a common `Sinai-type' random environment on a one dimensional torus. The hydrodynamic equation found is a quasilinear SPDE with a `rough' random drift term coming from a scaling of the random environment and a homogenization of the particle interaction. Part of the motivation for this work is to understand how the space-time limit of the particle mass relates to that of the known single particle Brox diffusion limit. In this respect, given the hydrodynamic limit shown, we describe formal connections through a two scale limit.

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arXiv2020

Asymptotics of PDE in random environment by paracontrolled calculus

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.

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arXiv2020

Remarks on the range and multiple range of random walk up to the time of exit

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 ${\mathbb Z}^d$, for dimensions $d\geq 2$, up to time of exit from a domain $D_N$ of the form $D_N = ND$ where $D\subset {\mathbb R}^d$, 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.

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arXiv2019

Derivation of coupled KPZ-Burgers equation from multi-species zero-range processes

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.

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arXiv2019

Stick-breaking processes, clumping, and Markov chain occupation laws

We consider the connections among `clumped' residual allocation models (RAMs), a general class of stick-breaking processes including Dirichlet processes, and the occupation laws of certain discrete space time-inhomogeneous Markov chains related to simulated annealing and other applications. An intermediate structure is introduced in a given RAM, where proportions between successive indices in a list are added or clumped together to form another RAM. In particular, when the initial RAM is a Griffiths-Engen-McCloskey (GEM) sequence and the indices are given by the random times that an auxiliary Markov chain jumps away from its current state, the joint law of the intermediate RAM and the locations visited in the sojourns is given in terms of a `disordered' GEM sequence, and an induced Markov chain. Through this joint law, we identify a large class of `stick breaking' processes as the limits of empirical occupation measures for associated time-inhomogeneous Markov chains.

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arXiv2018

On Hydrodynamic Limits of Young Diagrams

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.

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arXiv2018

Hydrodynamic limits for long-range asymmetric interacting particle systems

We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on ${\mathbb Z}^n$, 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 $α\geq 1$, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.

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arXiv2017

Approximating geodesics via random points

Given a `cost' functional $F$ on paths $γ$ in a domain $D\subset\mathbb{R}^d$, in the form $F(γ) = \int_0^1 f(γ(t),\dotγ(t))dt$, it is of interest to approximate its minimum cost and geodesic paths. Let $X_1,\ldots, X_n$ 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 $0<|X_i - X_j|<ε$, and the length scale $ε=ε_n$ vanishes at a suitable rate. For a general class of functionals $F$, associated to Finsler and other distances on $D$, using a probabilistic form of Gamma convergence, we show that the minimum costs and geodesic paths, with respect to types of approximating discrete `cost' functionals, built from the random geometric graph, converge almost surely in various senses to those corresponding to the continuum cost $F$, as the number of sample points diverges. In particular, the geodesic path convergence shown appears to be among the first results of its kind.

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arXiv2017

On the growth of a superlinear preferential attachment scheme

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, we establish a.s. law of large numbers type limits and fluctuation results, as $n\uparrow\infty$, for the counts of the number of nodes with degree $k\geq 1$ at time $n\geq 1$. These limits rigorously verify and extend a physical picture of Krapivisky, Redner and Leyvraz (2000) on how the condensation arises with respect to the degree distribution.

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arXiv2017

On collisions times of self-sorting interacting particles in one-dimension with random initial positions and velocities

We investigate a one-dimensional system of $N$ particles, initially distributed with random positions and velocities, interacting through binary collisions. The collision rule is such that there is a time after which the $N$ particles do not interact and become sorted according to their velocities. When the collisions are elastic, we derive asymptotic distributions for the final collision time of a single particle and the final collision time of the system as the number of particles approaches infinity, under different assumptions for the initial distributions of the particles' positions and velocities. For comparison, a numerical investigation is carried out to determine how a non-elastic collision rule, which conserves neither momentum nor energy, affects the median collision time of a particle and the median final collision time of the system.

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arXiv2016

Consistency of modularity clustering on random geometric graphs

We consider a large class of random geometric graphs constructed from samples $\mathcal{X}_n = \{X_1,X_2,\ldots,X_n\}$ of independent, identically distributed observations of an underlying probability measure $ν$ on a bounded domain $D\subset \mathbb{R}^d$. The popular `modularity' clustering method specifies a partition $\mathcal{U}_n$ of the set $\mathcal{X}_n$ as the solution of an optimization problem. In this paper, under conditions on $ν$ and $D$, we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions $\mathcal{U}_n$ converge in a certain sense to a continuum partition $\mathcal{U}$ of the underlying domain $D$, characterized as the solution of a type of Kelvin's shape optimization problem.

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arXiv2015

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 $\mathbb{Z}$, 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 $1/2<γ\leq1$, 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.

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arXiv2014

On microscopic derivation of a fractional stochastic Burgers equation

We derive from a class of microscopic asymmetric interacting particle systems on ${\mathbb Z}$, with long range jump rates of order $|\cdot|^{-(1+α)}$ for $0<α<2$, different continuum fractional SPDEs. More specifically, we show the equilibrium fluctuations of the hydrodynamics mass density field of zero-range processes, depending on the stucture of the asymmetry, and whether the field is translated with process characteristics velocity, is governed in various senses by types of fractional stochastic heat or Burgers equations. The main result: Suppose the jump rate is such that its symmetrization is long range but its (weak) asymmetry is nearest-neighbor. Then, when $α<3/2$, the fluctuation field in space-time scale $1/α:1$, translated with process characteristic velocity, irrespective of the strength of the asymmetry, converges to a fractional stochastic heat equation, the limit also for the symmetric process. However, when $α\geq 3/2$ and the strength of the weak asymmetry is tuned in scale $1-3/2α$, the associated limit points satisfy a martingale formulation of a fractional stochastic Burgers equation.

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arXiv2014

Occupation times of long-range exclusion and connections to KPZ class exponents

With respect to a class of long-range exclusion processes on $\mathbb{Z}^d$, with single particle transition rates of order $|\cdot|^{-(d+α)}$, starting under Bernoulli invariant measure $ν_ρ$ with density $ρ$, we consider the fluctuation behavior of occupation times at a vertex and more general additive functionals. Part of our motivation is to investigate the dependence on $α$, $d$ and $ρ$ with respect to the variance of these functionals and associated scaling limits. In the case the rates are symmetric, among other results, we find the scaling limits exhaust a range of fractional Brownian motions with Hurst parameter $H\in [1/2,3/4]$. However, in the asymmetric case, we study the asymptotics of the variances, which when $d=1$ and $ρ=1/2$ points to a curious dichotomy between long-range strength parameters $0<α\leq 3/2$ and $α>3/2$. In the former case, the order of the occupation time variance is the same as under the process with symmetrized transition rates, which are calculated exactly. In the latter situation, we provide consistent lower and upper bounds and other motivations that this variance order is the same as under the asymmetric short-range model, which is connected to KPZ class scalings of the space-time bulk mass density fluctuations.

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arXiv2014

A scaling limit for the degree distribution in sublinear preferential attachment schemes

We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures. In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to `non-tree' evolutions where cycles may develop in the network. A main part of the argument is to analyze an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, in terms of $C_0$-semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003).

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arXiv2013

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].

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