79 research outputs found

### Tunneling behavior of Ising and Potts models in the low-temperature regime

We consider the ferromagnetic $q$-state Potts model with zero external field
in a finite volume and assume that the stochastic evolution of this system is
described by a Glauber-type dynamics parametrized by the inverse temperature
$\beta$. Our analysis concerns the low-temperature regime $\beta \to \infty$,
in which this multi-spin system has $q$ stable equilibria, corresponding to the
configurations where all spins are equal. Focusing on grid graphs with various
boundary conditions, we study the tunneling phenomena of the $q$-state Potts
model. More specifically, we describe the asymptotic behavior of the first
hitting times between stable equilibria as $\beta \to \infty$ in probability,
in expectation, and in distribution and obtain tight bounds on the mixing time
as side-result. In the special case $q=2$, our results characterize the
tunneling behavior of the Ising model on grid graphs.Comment: 13 figure

### Competitive nucleation in metastable systems

Metastability is observed when a physical system is close to a first order
phase transition. In this paper the metastable behavior of a two state
reversible probabilistic cellular automaton with self-interaction is discussed.
Depending on the self-interaction, competing metastable states arise and a
behavior very similar to that of the three state Blume-Capel spin model is
found

### Basic Ideas to Approach Metastability in Probabilistic Cellular Automata

Cellular Automata are discrete--time dynamical systems on a spatially
extended discrete space which provide paradigmatic examples of nonlinear
phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular
Automata, are discrete time Markov chains on lattice with finite single--cell
states whose distinguishing feature is the \textit{parallel} character of the
updating rule. We review some of the results obtained about the metastable
behavior of Probabilistic Cellular Automata and we try to point out
difficulties and peculiarities with respect to standard Statistical Mechanics
Lattice models.Comment: arXiv admin note: text overlap with arXiv:1307.823

### A comparison between different cycle decompositions for Metropolis dynamics

In the last decades the problem of metastability has been attacked on
rigorous grounds via many different approaches and techniques which are briefly
reviewed in this paper. It is then useful to understand connections between
different point of views. In view of this we consider irreducible, aperiodic
and reversible Markov chains with exponentially small transition probabilities
in the framework of Metropolis dynamics. We compare two different cycle
decompositions and prove their equivalence

### Short paths for first passage percolation on the complete graph

We study the complete graph equipped with a topology induced by independent
and identically distributed edge weights. The focus of our analysis is on the
weight W_n and the number of edges H_n of the minimal weight path between two
distinct vertices in the weak disorder regime. We establish novel and simple
first and second moment methods using path counting to derive first order
asymptotics for the considered quantities. Our results are stated in terms of a
sequence of parameters (s_n) that quantifies the extreme-value behaviour of the
edge weights, and that describes different universality classes for first
passage percolation on the complete graph. These classes contain both
n-independent and n-dependent edge weight distributions. The method is most
effective for the universality class containing the edge weights E^{s_n}, where
E is an exponential(1) random variable and s_n log n -> infty, s_n^2 log n ->
0. We discuss two types of examples from this class in detail. In addition, the
class where s_n log n stays finite is studied. This article is a contribution
to the program initiated in \cite{BhaHof12}.Comment: 31 pages, 4 figure

### Delay performance in random-access grid networks

We examine the impact of torpid mixing and meta-stability issues on the delay
performance in wireless random-access networks. Focusing on regular meshes as
prototypical scenarios, we show that the mean delays in an $L\times L$ toric
grid with normalized load $\rho$ are of the order $(\frac{1}{1-\rho})^L$. This
superlinear delay scaling is to be contrasted with the usual linear growth of
the order $\frac{1}{1-\rho}$ in conventional queueing networks. The intuitive
explanation for the poor delay characteristics is that (i) high load requires a
high activity factor, (ii) a high activity factor implies extremely slow
transitions between dominant activity states, and (iii) slow transitions cause
starvation and hence excessively long queues and delays. Our proof method
combines both renewal and conductance arguments. A critical ingredient in
quantifying the long transition times is the derivation of the communication
height of the uniformized Markov chain associated with the activity process. We
also discuss connections with Glauber dynamics, conductance and mixing times.
Our proof framework can be applied to other topologies as well, and is also
relevant for the hard-core model in statistical physics and the sampling from
independent sets using single-site update Markov chains

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