Aspects of Markov Chains and Particle Systems
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The thesis concerns asymptotic behavior of particle systems and the underlying Markov chains used to model various natural phenomena. The objective is to describe and analyze stochastic models involving spatial structure and evolution over time. Fundamental objects of interest in such systems include the equilibrium measure which the system converges to, the phenomenon of phase transition in the long term behavior and the time taken to converge to stationarity. In this thesis we present three examples highlighting the above aspects. In Chapter 2, we will discuss Competitive Erosion: a multi-particle system introduced by James Propp in 2003, as a generalization of a fundamental growth model known as Internal Diffusion Limited Aggregation. In this model, each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective bases and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. We consider competitive erosion on discretizations of smooth planar simply connected domains. In Chapter 2 we establish positively, a conjecture of Propp regarding conformal invariance of the the model at stationarity, by showing that, with high probability the blue and the red regions are separated by an orthogonal circular arc on the disc and by a suitable hyperbolic geodesic on a general `smooth' simply connected domain. In Chapter 3, we discuss a family of conservative stochastic processes known as Activated Random Walk (ARW) which interpolates between ordinary random walk and the Stochastic Sandpile; the latter being a canonical example of Self Organized Criticality. These processes are conjectured to exhibit a sharp change in long time behavior depending on the value of certain parameters. Informally ARW is a particle system on Z with mass conservation. One starts with a mass density mu>0 of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate lambda. Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process and show for small enough lambda, the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as lambda tends to zero. This positively answers two open questions from Dickman, Rolla, Sidoravicius (J. Stat. Phys., 2010) and Rolla, Sidoravicius (Invent. Math., 2012). In Chapter 4, we discuss a model of constrained Glauber dynamics, known as the East Process, exhibiting sharp convergence to equilibrium. The East process is a 1-D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Informally, it is a two spin (0,1) system on Z, where every site at rate one tries to randomize its spin using a fresh Bernoulli (p). However the move is suppressed unless the site to the left is in the 0 state. Thus the Glauber dynamics move is carried out only in the presence of a certain `kinetic' constraint. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on L sites has order L. Since the relaxation time is of a smaller order than the mixing time it is natural to expect a sharp convergence to equilibrium . Proving this, is the goal of this chapter, where we establish Cutoff for mixing, with an optimal window size.
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