362 research outputs found
Lower Bounds on the Ground State Entropy of the Potts Antiferromagnet on Slabs of the Simple Cubic Lattice
We calculate rigorous lower bounds for the ground state degeneracy per site,
, of the -state Potts antiferromagnet on slabs of the simple cubic
lattice that are infinite in two directions and finite in the third and that
thus interpolate between the square (sq) and simple cubic (sc) lattices. We
give a comparison with large- series expansions for the sq and sc lattices
and also present numerical comparisons.Comment: 7 pages, late
Survival of interacting Brownian particles in crowded 1D environment
We investigate a diffusive motion of a system of interacting Brownian
particles in quasi-one-dimensional micropores. In particular, we consider a
semi-infinite 1D geometry with a partially absorbing boundary and the hard-core
inter-particle interaction. Due to the absorbing boundary the number of
particles in the pore gradually decreases. We present the exact analytical
solution of the problem. Our procedure merely requires the knowledge of the
corresponding single-particle problem. First, we calculate the simultaneous
probability density of having still a definite number of surviving
particles at definite coordinates. Focusing on an arbitrary tagged particle, we
derive the exact probability density of its coordinate. Secondly, we present a
complete probabilistic description of the emerging escape process. The survival
probabilities for the individual particles are calculated, the first and the
second moments of the exit times are discussed. Generally speaking, although
the original inter-particle interaction possesses a point-like character, it
induces entropic repulsive forces which, e.g., push the leftmost (rightmost)
particle towards (opposite) the absorbing boundary thereby accelerating
(decelerating) its escape. More importantly, as compared to the reference
problem for the non-interacting particles, the interaction changes the
dynamical exponents which characterize the long-time asymptotic dynamics.
Interesting new insights emerge after we interpret our model in terms of a)
diffusion of a single particle in a -dimensional space, and b) order
statistics defined on a system of independent, identically distributed
random variables
Asymptotic Expansions for lambda_d of the Dimer and Monomer-Dimer Problems
In the past few years we have derived asymptotic expansions for lambda_d of
the dimer problem and lambda_d(p) of the monomer-dimer problem. The many
expansions so far computed are collected herein. We shine a light on results in
two dimensions inspired by the work of M. E. Fisher. Much of the work reported
here was joint with Shmuel Friedland.Comment: 4 page
Two-dimensional symmetric and antisymmetric generalizations of exponential and cosine functions
Properties of the four families of recently introduced special functions of
two real variables, denoted here by , and , are studied. The
superscripts and refer to the symmetric and antisymmetric functions
respectively. The functions are considered in all details required for their
exploitation in Fourier expansions of digital data, sampled on square grids of
any density and for general position of the grid in the real plane relative to
the lattice defined by the underlying group theory. Quality of continuous
interpolation, resulting from the discrete expansions, is studied, exemplified
and compared for some model functions.Comment: 22 pages, 10 figure
Three variable exponential functions of the alternating group
New class of special functions of three real variables, based on the
alternating subgroup of the permutation group , is studied. These
functions are used for Fourier-like expansion of digital data given on lattice
of any density and general position. Such functions have only trivial analogs
in one and two variables; a connection to the functions of is shown.
Continuous interpolation of the three dimensional data is studied and
exemplified.Comment: 10 pages, 3 figure
Probabilities in the inflationary multiverse
Inflationary cosmology leads to the picture of a "multiverse," involving an
infinite number of (spatially infinite) post-inflationary thermalized regions,
called pocket universes. In the context of theories with many vacua, such as
the landscape of string theory, the effective constants of Nature are
randomized by quantum processes during inflation. We discuss an analytic
estimate for the volume distribution of the constants within each pocket
universe. This is based on the conjecture that the field distribution is
approximately ergodic in the diffusion regime, when the dynamics of the fields
is dominated by quantum fluctuations (rather than by the classical drift). We
then propose a method for determining the relative abundances of different
types of pocket universes. Both ingredients are combined into an expression for
the distribution of the constants in pocket universes of all types.Comment: 18 pages, RevTeX 4, 2 figures. Discussion of the full probability in
Sec.VI is sharpened; the conclusions are strengthened. Note added explaining
the relation to recent work by Easther, Lim and Martin. Some references adde
The Portevin-Le Chatelier effect in the Continuous Time Random Walk framework
We present a continuous time random walk model for the Portevin-Le Chatelier
(PLC) effect. From our result it is shown that the dynamics of the PLC band can
be explained in terms of the Levy Walk
An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem
Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer
lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower
bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q).
The upper bound is based on a conjecture claiming that the p monomer-dimer
entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We
compute the first three terms in the formal asymptotic expansion of
(lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching
conjecture is satisfied for (lambda_d)(p).Comment: 15 pages, much more about d=1,2,
Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process
Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d.
complex Gaussian coefficients a_n. We show that these zeros form a
determinantal process: more precisely, their joint intensity can be written as
a minor of the Bergman kernel. We show that the number of zeros of f in a disk
of radius r about the origin has the same distribution as the sum of
independent {0,1}-valued random variables X_k, where P(X_k=1)=r^{2k}. Moreover,
the set of absolute values of the zeros of f has the same distribution as the
set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in [0,1].
The repulsion between zeros can be studied via a dynamic version where the
coefficients perform Brownian motion; we show that this dynamics is conformally
invariant.Comment: 37 pages, 2 figures, updated proof
Ground State Entropy of Potts Antiferromagnets: Bounds, Series, and Monte Carlo Measurements
We report several results concerning , the
exponent of the ground state entropy of the Potts antiferromagnet on a lattice
. First, we improve our previous rigorous lower bound on for
the honeycomb (hc) lattice and find that it is extremely accurate; it agrees to
the first eleven terms with the large- series for . Second, we
investigate the heteropolygonal Archimedean lattice, derive a
rigorous lower bound, on , and calculate the large- series
for this function to where . Remarkably, these agree
exactly to all thirteen terms calculated. We also report Monte Carlo
measurements, and find that these are very close to our lower bound and series.
Third, we study the effect of non-nearest-neighbor couplings, focusing on the
square lattice with next-nearest-neighbor bonds.Comment: 13 pages, Latex, to appear in Phys. Rev.
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