6 research outputs found
Coupling, Concentration and Random Walks in Dynamic Random Environments
No full textIn this thesis we discuss concentration inequalities, relaxation to equilibrium of stochastic dynamics, and random walks in dynamic random environments. In stochastic systems one is interested in macroscopic and/or asymptotic properties as well as in fluctuations around typical behaviour. But the dependence structure induced by the interaction between the components of the system makes the analysis challenging. In order to overcome this in different settings a variety of methods are employed. Additive functionals of Markov processes play important roles in applications. In order to get exponential and moment estimates for their fluctuations a non-standard martingale approximation is used. The resulting general theorems do not require special properties like reversibility or a spectral gap. What is needed is some control on the expected evolution. That is, the difference of the evolution starting from two "adjacent'' configurations has to be controlled. Coupling methods are well suited to do perform this comparison. In concrete examples couplings are used to prove the conditions of the theorems. In statistical mechanics Gibbs measures and Markov random fields play important roles. The Poincaré inequality is an important property describing the regularity of the measure. We prove the Poincaré inequality via a martingale telescoping argument. To control the individual increments of the martingale we use a coupling method called disagreement percolation. If the clusters of this percolation are sufficiently small we obtain the Poincaré inequality. When interacting spin systems and their dynamics have a delicate connection to their ergodic measure(s) one has to take more care. We carefully study the graphical construction of the dynamics to understand how the influence of the measure can be preserved. An assumption is made that one can control how fast the system in equilibrium can compensate for a single spin flip. Under this assumption we obtain relaxation speed estimates for general functions. In attractive spin systems the condition can be reduced to the decay of auto-correlation of the spin at the origin. An application where this is of use is the low-temperature Ising model. Finally we look at random walks in dynamic random environments. Here a time-changing random environment drives the motion of a particle. The goal is to understand under which conditions the macroscopic behaviour of this random walk is like that of a Brownian motion. We use coupling to prove a law of large numbers as well as a functional central limit theorem for the position of the random walk. Only polynomial decay of correlations in time are needed for the environment, and the influence of the environment on the walk can be very general.Applied mathematicsElectrical Engineering, Mathematics and Computer Scienc
Mixing Properties for Random Walk in Random Scenery
Get PDFConsider the lattice , together with a stochastic black-white coloring of its points and on it a random walk that is independent of the coloring. A local scenery perceived at a given time is a pattern of colors seen by the walker in a finite box around his current position. Under weak assumptions on the probability distributions governing walk and coloring, we prove asymptotic independence of local sceneries perceived at times 0 and , in the limit as , and at times 0 and , in the limit as , where is the random th hitting time of a black point. An immediate corollary of the latter result is the convergence in distribution of the interarrival times between successive black hits, i.e., of as . The limit distribution is expressed in terms of the distribution of the first hitting time . The proof uses coupling arguments and ergodic theory.Electrical Engineering, Mathematics and Computer Scienc
Tail Triviality for Sums of Stationary Random Variables
Get PDFWe study tail -fields and loss of memory associated with sums of stationary integer-valued random variables. An application concerns convergence in distribution of interarrival times in zero-one sequences.Electrical Engineering, Mathematics and Computer Scienc
Localization transition for a copolymer in an emulsion
Get PDFIn this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The polymer is a random concatenation of monomers of two types, A and B, each occurring with density 1/2. The emulsion is a random mixture of liquids of two types, A and B, organized in large square blocks occurring with density p and 1-p, respectively, where p ¿ (0, 1). The polymer in the emulsion has an energy that is minus a times the number of AA-matches minus ß times the number of BB-matches, where a, ß ¿ R are interaction parameters. Symmetry considerations show that without loss of generality we may restrict our attention to the cone {(a, ß) ¿ R2 : a ?? |ß|}. We derive a variational expression for the quenched free energy per monomer in the limit as the length n of the polymer tends to infinity and the blocks in the emulsion have size Ln such that Ln ¿ 8 and Ln/n ¿ 0. To make the model mathematically tractable, we assume that the polymer can only enter and exit a pair of neighboring blocks at diagonally opposite corners. Although this is an unphysical restriction, it turns out that the model exhibits rich and physically relevant behavior. Let pc ˜ 0.64 be the critical probability for directed bond percolation on the square lattice. We show that for p ?? pc the free energy has a phase transition along one curve in the cone, which turns out to be independent of p. At this curve, there is a transition from a phase where the polymer is fully A-delocalized (i.e., it spends almost all of its time deep inside the A-blocks) to a phase where the polymer is partially AB-localized (i.e., it spends a positive fraction of its time near those interfaces where it diagonally crosses the A-block rather than the B-block). We show that for p <pc the free energy has a phase transition along two curves in the cone, both of which turn out to depend on p. At the first curve there is a transition from a phase where the polymer is fully A,B-delocalized (i.e., it spends almost all of its time deep inside the A-blocks and the B-blocks) to a partially BA-localized phase, while at the second curve there is a transition from a partially BA-localized phase to a phase where both partial BA-localization and partial AB-localization occur simultaneously. We derive a number of qualitative properties of the critical curves. The supercritical curve is nondecreasing and concave with a finite horizontal asymptote. Remarkably, the first subcritical curve does not share these properties and does not converge to the supercritical curve as p ¿ pc. Rather, the second subcritical curve converges to the supercritical curve as p ¿ 0
