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Register a new userGeneralized Hammersley Process and Phase Transition for Activated Random Walk Models
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Leonardo T. Rolla
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* ACTIVATED RANDOM WALK MODEL * This is a conservative particle system on the lattice, with a Markovian continuous-time evolution. Active particles perform random walks without interaction, and they may as well change their state to passive, then stopping to jump. When particles of both types occupy the same site, they all become active. This model exhibits phase transition in the sense that for low initial densities the system locally fixates and for high densities it keeps active. Though extensively studied in the physics literature, the matter of giving a mathematical proof of such phase transition remained as an open problem for several years. In this work we identify some variables that are sufficient to characterize fixation and at the same time are stochastically monotone in the models parameters. We employ an explicit graphical representation in order to obtain the monotonicity. With this method we prove that there is a unique phase transition for the one-dimensional finite-range random walk. Joint with V. Sidoravicius. * BROKEN LINE PROCESS * We introduce the broken line process and derive some of its properties. Its discrete version is presented first and a natural generalization to the continuum is then proposed and studied. The broken lines are related to the Young diagram and the Hammersley process and are useful for computing last passage percolation values and finding maximal oriented paths. For a class of passage time distributions there is a family of boundary conditions that make the process stationary and reversible. One application is a simple proof of the explicit law of large numbers for last passage percolation with exponential and geometric distributions. Joint with V. Sidoravicius, D. Surgailis, and M. E. Vares.
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We consider the Activated Random Walk model on $mathbb{Z}$. In this model, each particle performs a continuous-time simple symmetric random walk, and falls asleep at rate $lambda$. A sleeping particle does not move but it is reactivated in the presence of another particle. We show that for any sleep rate $lambda < infty$ if the density $ zeta $ is close enough to $1$ then the system stays active.
We consider Activated Random Walks on $Z$ with totally asymmetric jumps and critical particle density, with different time scales for the progressive release of particles and the dissipation dynamics. We show that the cumulative flow of particles through the origin rescales to a pure-jump self-similar process which we describe explicitly.
Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. In these lecture notes we present the existing results and reproduce the techniques developed so far.
We consider symmetric activated random walks on $mathbb{Z}$, and show that the critical density $zeta_c$ satisfies $csqrt{lambda} leq zeta_c(lambda) leq C sqrt{lambda}$ where $lambda$ denotes the sleep rate.
This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of $Ntimes N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances $mathbb E |H_{ij}|^2$ form a band matrix with typical band width $1ll Wll N$. We consider the generalized resolvent of $H$ defined as $G(Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix such that $Z_{ij}=left(z 1_{1leq i leq W}+widetilde z 1_{ i > W} right) delta_{ij}$, with two distinct spectral parameters $zin mathbb C_+:={zin mathbb C:{rm Im} z>0}$ and $widetilde zin mathbb C_+cup mathbb R$. In this paper, we prove a sharp bound for the local law of the generalized resolvent $G$ for $Wgg N^{3/4}$. This bound is a key input for the proof of delocalization and bulk universality of random band matrices in cite{PartI}. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in cite{PartIII}.
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