48 research outputs found
Fast Mixing for the Low Temperature 2D Ising Model Through Irreversible Parallel Dynamics
We study tunneling and mixing time for a non-reversible probabilistic cellular automaton. With a suitable choice of the parameters, we first show that the stationary distribution is close in total variation to a low temperature Ising model. Then we prove that both the mixing time and the time to exit a metastable state grow polynomially in the size of the system, while this growth is exponential in reversible dynamics. In this model, non-reversibility, parallel updatings and a suitable choice of boundary conditions combine to produce an efficient dynamical stability
Fast mixing for the low temperature 2d Ising model through irreversible parallel dynamics
We study metastability and mixing time for a non-reversible probabilistic
cellular automaton. With a suitable choice of the parameters, we first show
that the stationary distribution is close in total variation to a low
temperature Ising model. Then we prove that both the mixing time and the time
to exit a metastable state grow polynomially in the size of the system, while
this growth is exponential in reversible dynamics. In this model,
non-reversibility, parallel updatings and a suitable choice of boundary
conditions combine to produce an efficient dynamical stability
Shaken dynamics: an easy way to parallel MCMC
We define a Markovian parallel dynamics for a class of spin systems on
general interaction graphs. In this dynamics, beside the usual set of
parameters , the strength of the interaction between the spins
and , and , the external field at site ,
there is an inertial parameter measuring the tendency of the system to
remain locally in the same state. This dynamics is reversible with an
explicitly defined stationary measure. For suitable choices of parameter this
invariant measure concentrates on the ground states of the Hamiltonian. This
implies that this dynamics can be used to solve, heuristically, difficult
problems in the context of combinatorial optimization. We also study the
dynamics on with homogeneous interaction and external field and
with arbitrary boundary conditions. We prove that for certain values of the
parameters the stationary measure is close to the related Gibbs measure. Hence
our dynamics may be a good tool to sample from Gibbs measure by means of a
parallel algorithm. Moreover we show how the parameter allow to interpolate
between spin systems defined on different regular lattices.Comment: 5 figure
On Diffusion Limited Deposition
We propose a simple model of columnar growth through
{\it diffusion limited aggregation} (DLA). Consider a graph
G_N\times\realmathbb{N}, where the basis has vertices
, and two vertices and
are adjacent if . Consider there a simple
random walk {\it coming from infinity}
which {\it deposits} on a growing cluster as follows:
the cluster is a collection of columns, and the height of
the column first hit by the walk immediately grows by one unit.
Thus, columns do not grow laterally.
We prove that there is a critical time scale
for the maximal height of the piles, i.e., there exist
constants such that the maximal pile height
at time is of order , while at time
is larger than for some positive .
This suggests that a \emph{monopolistic regime} starts at such a
time and only the highest pile goes on growing.
If we rather consider a walk whose height-component
goes down deterministically, the resulting
\emph{ballistic deposition} has maximal height of order at time
.
These two deposition models, diffusive and ballistic, are also compared
with uniform random allocation and Polya's urn
Sampling from a Gibbs measure with pair interaction by means of PCA
We consider the problem of approximate sampling from the finite volume Gibbs
measure with a general pair interaction. We exhibit a parallel dynamics
(Probabilistic Cellular Automaton) which efficiently implements the sampling.
In this dynamics the product measure that gives the new configuration in each
site contains a term that tends to favour the original value of each spin. This
is the main ingredient that allows to prove that the stationary distribution of
the PCA is close in total variation to the Gibbs measure. The presence of the
parameter that drives the "inertial" term mentioned above gives the possibility
to control the degree of parallelism of the numerical implementation of the
dynamics.Comment: 21 page
A probabilistic proof of Cooper and Frieze's "First Visit Time Lemma"
In this short note we present an alternative proof of the so-called First
Visit Time Lemma (FVTL), originally presented by Cooper and Frieze in its first
formulation in [21], and then used and refined in a list of papers by Cooper,
Frieze and coauthors. We work in the original setting, considering a growing
sequence of irreducible Markov chains on states. We assume that the chain
is rapidly mixing and with a stationary measure having no entry which is too
small nor too large. Under these assumptions, the FVTL shows the exponential
decay of the distribution of the hitting time of a given state -- for the
chain started at stationarity -- up to a small multiplicative correction. While
the proof of the FVTL presented by Cooper and Frieze is based on tools from
complex analysis, and it requires an additional assumption on a generating
function, we present a completely probabilistic proof, relying on the theory of
quasi-stationary distributions and on strong-stationary times arguments. In
addition, under the same set of assumptions, we provide some quantitative
control on the Doob's transform of the chain on the complement of the state
Phase transitions for the cavity approach to the clique problem on random graphs
We give a rigorous proof of two phase transitions for a disordered system
designed to find large cliques inside Erdos random graphs. Such a system is
associated with a conservative probabilistic cellular automaton inspired by the
cavity method originally introduced in spin glass theory.Comment: 36 pages, 4 figure
Effects of boundary conditions on irreversible dynamics
We present a simple one-dimensional Ising-type spin system on which we define
a completely asymmetric Markovian single spin-flip dynamics. We study the
system at a very low, yet non-zero, temperature and we show that for empty
boundary conditions the Gibbs measure is stationary for such dynamics, while
introducing in a single site a condition the stationary measure changes
drastically, with macroscopical effects. We achieve this result defining an
absolutely convergent series expansion of the stationary measure around the
zero temperature system. Interesting combinatorial identities are involved in
the proofs