3 research outputs found
Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon
The raise and peel model of a one-dimensional fluctuating interface (model A)
is extended by considering one source (model B) or two sources (model C) at the
boundaries. The Hamiltonians describing the three processes have, in the
thermodynamic limit, spectra given by conformal field theory. The probability
of the different configurations in the stationary states of the three models
are not only related but have interesting combinatorial properties. We show
that by extending Pascal's triangle (which gives solutions to linear relations
in terms of integer numbers), to an hexagon, one obtains integer solutions of
bilinear relations. These solutions give not only the weights of the various
configurations in the three models but also give an insight to the connections
between the probability distributions in the stationary states of the three
models. Interestingly enough, Pascal's hexagon also gives solutions to a
Hirota's difference equation.Comment: 33 pages, an abstract and an introduction are rewritten, few
references are adde
Refined Razumov-Stroganov conjectures for open boundaries
Recently it has been conjectured that the ground-state of a Markovian
Hamiltonian, with one boundary operator, acting in a link pattern space is
related to vertically and horizontally symmetric alternating-sign matrices
(equivalently fully-packed loop configurations (FPL) on a grid with special
boundaries).We extend this conjecture by introducing an arbitrary boundary
parameter. We show that the parameter dependent ground state is related to
refined vertically symmetric alternating-sign matrices i.e. with prescribed
configurations (respectively, prescribed FPL configurations) in the next to
central row.
We also conjecture a relation between the ground-state of a Markovian
Hamiltonian with two boundary operators and arbitrary coefficients and some
doubly refined (dependence on two parameters) FPL configurations. Our
conjectures might be useful in the study of ground-states of the O(1) and XXZ
models, as well as the stationary states of Raise and Peel models.Comment: 11 pages LaTeX, 8 postscript figure
The Grand-Canonical Asymmetric Exclusion Process and the One-Transit Walk
The one-dimensional Asymmetric Exclusion Process (ASEP) is a paradigm for
nonequilibrium dynamics, in particular driven diffusive processes. It is
usually considered in a canonical ensemble in which the number of sites is
fixed. We observe that the grand-canonical partition function for the ASEP is
remarkably simple. It allows a simple direct derivation of the asymptotics of
the canonical normalization in various phases and of the correspondence with
One-Transit Walks recently observed by Brak et.al.Comment: Published versio