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Register a new userExtracting subsets maximizing capacity and Folding of Random Walks
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Authors
Amine Asselah
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We prove that in any finite set of $mathbb Z^d$ with $dge 3$, there is a subset whose capacity and volume are both of the same order as the capacity of the initial set. As an application we obtain estimates on the probability of {it covering uniformly} a finite set, and characterize some {it folding} events, under optimal hypotheses. For instance, knowing that a region of space has an {it atypically high occupation density} by some random walk, we show that this random region is most likely ball-like
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We identify a fundamental phenomenon of heterogeneous one dimensional random walks: the escape (traversal) time is maximized when the heterogeneity in transition probabilities forms a pyramid-like potential barrier. This barrier corresponds to a distinct arrangement of transition probabilities, sometimes referred to as the pendulum arrangement. We reduce this problem to a sum over products, combinatorial optimization problem, proving that this unique structure always maximizes the escape time. This general property may influence studies in epidemiology, biology, and computer science to better understand escape time behavior and construct intruder-resilient networks.
The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter $lambda$. There is a threshold for $lambda$, which is called $lambda_w$, that separates almost sure global extinction from global survival. Analogously, there exists another threshold $lambda_s$ below which any site is visited almost surely a finite number of times (i.e.~local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter $lambda_s$ is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter $lambda_w$ is the inverse of a certain function of the reproduction rates, which we denote by $K_w$. We provide here new sufficient conditions which guarantee that the global critical parameter equals $1/K_w$. This result extends previously known results for branching random walks on multigraphs and general branching random walks. We show that these sufficient conditions are satisfied by periodic tree-like branching random walks. We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment. So far, only examples where $lambda_w=1/K_w$ were known; here we provide an example where $lambda_w>1/K_w$.
Lecture Notes. Minicourse given at the workshop Activated Random Walks, DLA, and related topics at IMeRA-Marseille, March 2015.
We study random walks on the giant component of the ErdH{o}s-Renyi random graph ${cal G}(n,p)$ where $p=lambda/n$ for $lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $( u {bf d})^{-1}log n pm (log n)^{1/2+o(1)}$, where $ u$ and ${bf d}$ are the speed of random walk and dimension of harmonic measure on a ${rm Poisson}(lambda)$-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.
We study survival of nearest-neighbour branching random walks in random environment (BRWRE) on ${mathbb Z}$. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. $2times 2$ random matrices.
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