20 research outputs found
Dynamical Freezing in a Spin Glass System with Logarithmic Correlations
We consider a continuous time random walk on the two-dimensional discrete
torus, whose motion is governed by the discrete Gaussian free field on the
corresponding box acting as a potential. More precisely, at any vertex the walk
waits an exponentially distributed time with mean given by the exponential of
the field and then jumps to one of its neighbors, chosen uniformly at random.
We prove that throughout the low-temperature regime and at in-equilibrium
timescales, the process admits a scaling limit as a spatial K-process driven by
a random trapping landscape, which is explicitly related to the limiting
extremal process of the field. Alternatively, the limiting process is a
supercritical Liouville Brownian motion with respect to the continuum Gaussian
free field on the box. This demonstrates rigorously and for the first time, as
far as we know, a dynamical freezing in a spin glass system with
logarithmically correlated energy levels.Comment: Final version available at Electron. J. Proba
Finite connections for supercritical Bernoulli bond percolation in 2D
Two vertices are said to be finitely connected if they belong to the same
cluster and this cluster is finite. We derive sharp asymptotics for finite
connection probabilities for supercritical Bernoulli bond percolation on Z^2
Exploring an Infinite Space with Finite Memory Scouts
Consider a small number of scouts exploring the infinite -dimensional grid
with the aim of hitting a hidden target point. Each scout is controlled by a
probabilistic finite automaton that determines its movement (to a neighboring
grid point) based on its current state. The scouts, that operate under a fully
synchronous schedule, communicate with each other (in a way that affects their
respective states) when they share the same grid point and operate
independently otherwise. Our main research question is: How many scouts are
required to guarantee that the target admits a finite mean hitting time?
Recently, it was shown that is an upper bound on the answer to this
question for any dimension and the main contribution of this paper
comes in the form of proving that this bound is tight for .Comment: Added (forgotten) acknowledgement