712 research outputs found
On the functions counting walks with small steps in the quarter plane
Models of spatially homogeneous walks in the quarter plane
with steps taken from a subset of the set of jumps to the eight
nearest neighbors are considered. The generating function of the numbers of such walks starting at the origin and
ending at after steps is studied. For all
non-singular models of walks, the functions and are continued as multi-valued functions on having
infinitely many meromorphic branches, of which the set of poles is identified.
The nature of these functions is derived from this result: namely, for all the
51 walks which admit a certain infinite group of birational transformations of
, the interval of variation of splits into
two dense subsets such that the functions and are shown to be holonomic for any from the one of them and
non-holonomic for any from the other. This entails the non-holonomy of
, and therefore proves a conjecture of
Bousquet-M\'elou and Mishna.Comment: 40 pages, 17 figure
-Martin boundary of killed random walks in the quadrant
We compute the -Martin boundary of two-dimensional small steps random
walks killed at the boundary of the quarter plane. We further provide explicit
expressions for the (generating functions of the) discrete -harmonic
functions. Our approach is uniform in , and shows that there are three
regimes for the Martin boundary.Comment: 18 pages, 2 figures, to appear in S\'eminaire de Probabilit\'e
Phase diagram of a generalized ABC model on the interval
We study the equilibrium phase diagram of a generalized ABC model on an
interval of the one-dimensional lattice: each site is occupied by a
particle of type \a=A,B,C, with the average density of each particle species
N_\a/N=r_\a fixed. These particles interact via a mean field
non-reflection-symmetric pair interaction. The interaction need not be
invariant under cyclic permutation of the particle species as in the standard
ABC model studied earlier. We prove in some cases and conjecture in others that
the scaled infinite system N\rw\infty, i/N\rw x\in[0,1] has a unique
density profile \p_\a(x) except for some special values of the r_\a for
which the system undergoes a second order phase transition from a uniform to a
nonuniform periodic profile at a critical temperature .Comment: 25 pages, 6 figure
On the dynamical behavior of the ABC model
We consider the ABC dynamics, with equal density of the three species, on the
discrete ring with sites. In this case, the process is reversible with
respect to a Gibbs measure with a mean field interaction that undergoes a
second order phase transition. We analyze the relaxation time of the dynamics
and show that at high temperature it grows at most as while it grows at
least as at low temperature
An approximate analysis of a bernoulli alternating service model
We consider a discrete-time queueing system with one server
and two types of customers, say type-1 and type-2 customers. The server
serves customers of either type alternately according to a Bernoulli pro-
cess. The service times of the customers are deterministically equal to
1 time slot. For this queueing system, we derive a functional equation
for the joint probability generating function of the number of type-1 and
type-2 customers. The functional equation contains two unknown partial
generating functions which complicates the analysis. We investigate the
dominant singularity of these two unknown functions and propose an
approximation for the coefficients of the Maclaurin series expansion of
these functions. This approximation provides a fast method to compute
approximations of various performance measures of interest
Micromechanical based model for predicting aged rubber fracture properties
Environmental aging induces a slow and irreversible alteration of the rubber material’s macromolecular network. This alteration is triggered by two mechanisms which act at the microscale: crosslinking and chain scission. While crosslinking induces an early hardening of the material, chain scission leads to the occurrence of dangling chains responsible of the damage at the macromolecular scale. Consequently, the mechanical behavior as well as the fracture properties are affected. In this work, the effect of aging on the mechanical behavior up to fracture of elastomeric materials and the evolution of their fracture properties are first experimentally investigated. Further, a modeling attempt using an approach based upon a micro-mechanical but physical description of the aging mechanisms is proposed to predict the mechanical and fracture properties evolution of aged elastomeric materials. The proposed micro-mechanical model incorporates the concepts of residual stretch associated with the crosslinking mechanism and a so-called “healthy” elastic active chain (EAC) density associated with chain scission mechanism. The validity of the proposed approach is assessed using a wide set of experimental data either generated by the authors or available in the literature
Phase fluctuations in the ABC model
We analyze the fluctuations of the steady state profiles in the modulated
phase of the ABC model. For a system of sites, the steady state profiles
move on a microscopic time scale of order . The variance of their
displacement is computed in terms of the macroscopic steady state profiles by
using fluctuating hydrodynamics and large deviations. Our analytical prediction
for this variance is confirmed by the results of numerical simulations
Phase diagram of the ABC model on an interval
The three species asymmetric ABC model was initially defined on a ring by
Evans, Kafri, Koduvely, and Mukamel, and the weakly asymmetric version was
later studied by Clincy, Derrida, and Evans. Here the latter model is studied
on a one-dimensional lattice of N sites with closed (zero flux) boundaries. In
this geometry the local particle conserving dynamics satisfies detailed balance
with respect to a canonical Gibbs measure with long range asymmetric pair
interactions. This generalizes results for the ring case, where detailed
balance holds, and in fact the steady state measure is known only for the case
of equal densities of the different species: in the latter case the stationary
states of the system on a ring and on an interval are the same. We prove that
in the N to infinity limit the scaled density profiles are given by (pieces of)
the periodic trajectory of a particle moving in a quartic confining potential.
We further prove uniqueness of the profiles, i.e., the existence of a single
phase, in all regions of the parameter space (of average densities and
temperature) except at low temperature with all densities equal; in this case a
continuum of phases, differing by translation, coexist. The results for the
equal density case apply also to the system on the ring, and there extend
results of Clincy et al.Comment: 52 pages, AMS-LaTeX, 8 figures from 10 eps figure files. Revision:
minor changes in response to referee reports; paper to appear in J. Stat.
Phy
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