42 research outputs found
A max-type recursive model: some properties and open questions
We consider a simple max-type recursive model which was introduced in the
study of depinning transition in presence of strong disorder, by Derrida and
Retaux. Our interest is focused on the critical regime, for which we study the
extinction probability, the first moment and the moment generating function.
Several stronger assertions are stated as conjectures.Comment: A version accepted to Charles Newman Festschrift (to appear by
Springer
The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics
We consider the continuous time version of the Random Walk Pinning Model
(RWPM), studied in [5,6,7]. Given a fixed realization of a random walk Y$ on
Z^d with jump rate rho (that plays the role of the random medium), we modify
the law of a random walk X on Z^d with jump rate 1 by reweighting the paths,
giving an energy reward proportional to the intersection time L_t(X,Y)=\int_0^t
\ind_{X_s=Y_s}\dd s: the weight of the path under the new measure is exp(beta
L_t(X,Y)), beta in R. As beta increases, the system exhibits a
delocalization/localization transition: there is a critical value beta_c, such
that if beta>beta_c the two walks stick together for almost-all Y realizations.
A natural question is that of disorder relevance, that is whether the quenched
and annealed systems have the same behavior. In this paper we investigate how
the disorder modifies the shape of the free energy curve: (1) We prove that, in
dimension d larger or equal to three 3, the presence of disorder makes the
phase transition at least of second order. This, in dimension larger or equal
to 4, contrasts with the fact that the phase transition of the annealed system
is of first order. (2) In any dimension, we prove that disorder modifies the
low temperature asymptotic of the free energy.Comment: 18 page
Self-avoiding walks and connective constants
The connective constant of a quasi-transitive graph is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph .
We present upper and lower bounds for in terms of the
vertex-degree and girth of a transitive graph.
We discuss the question of whether for transitive
cubic graphs (where denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
We present strict inequalities for the connective constants
of transitive graphs , as varies.
As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with
arXiv:1304.721
The free energy in the Derrida--Retaux recursive model
We are interested in a simple max-type recursive model studied by Derrida and
Retaux (2014) in the context of a physics problem, and find a wide range for
the exponent in the free energy in the nearly supercritical regime
Stretched Polymers in Random Environment
We survey recent results and open questions on the ballistic phase of
stretched polymers in both annealed and quenched random environments.Comment: Dedicated to Erwin Bolthausen on the occasion of his 65th birthda