55,764 research outputs found
Simple Asymmetric Exclusion Model and Lattice Paths: Bijections and Involutions
We study the combinatorics of the change of basis of three representations of
the stationary state algebra of the two parameter simple asymmetric exclusion
process. Each of the representations considered correspond to a different set
of weighted lattice paths which, when summed over, give the stationary state
probability distribution. We show that all three sets of paths are
combinatorially related via sequences of bijections and sign reversing
involutions.Comment: 28 page
Asymptotics of Selberg-like integrals by lattice path counting
We obtain explicit expressions for positive integer moments of the
probability density of eigenvalues of the Jacobi and Laguerre random matrix
ensembles, in the asymptotic regime of large dimension. These densities are
closely related to the Selberg and Selberg-like multidimensional integrals. Our
method of solution is combinatorial: it consists in the enumeration of certain
classes of lattice paths associated to the solution of recurrence relations
Continuum Nonsimple Loops and 2D Critical Percolation
Substantial progress has been made in recent years on the 2D critical
percolation scaling limit and its conformal invariance properties. In
particular, chordal SLE6 (the Stochastic Loewner Evolution with parameter k=6)
was, in the work of Schramm and of Smirnov, identified as the scaling limit of
the critical percolation ``exploration process.'' In this paper we use that and
other results to construct what we argue is the full scaling limit of the
collection of all closed contours surrounding the critical percolation clusters
on the 2D triangular lattice. This random process or gas of continuum nonsimple
loops in the plane is constructed inductively by repeated use of chordal SLE6.
These loops do not cross but do touch each other -- indeed, any two loops are
connected by a finite ``path'' of touching loops.Comment: 16 pages, 3 figure
Lattice path matroids: enumerative aspects and Tutte polynomials
Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North
steps with P never going above Q. We show that the lattice paths that go from
(0,0) to (m,r) and that remain in the region bounded by P and Q can be
identified with the bases of a particular type of transversal matroid, which we
call a lattice path matroid. We consider a variety of enumerative aspects of
these matroids and we study three important matroid invariants, namely the
Tutte polynomial and, for special types of lattice path matroids, the
characteristic polynomial and the beta invariant. In particular, we show that
the Tutte polynomial is the generating function for two basic lattice path
statistics and we show that certain sequences of lattice path matroids give
rise to sequences of Tutte polynomials for which there are relatively simple
generating functions. We show that Tutte polynomials of lattice path matroids
can be computed in polynomial time. Also, we obtain a new result about lattice
paths from an analysis of the beta invariant of certain lattice path matroids.Comment: 28 pages, 11 figure
Quantum Interference in Superconducting Wire Networks and Josephson Junction Arrays: Analytical Approach based on Multiple-Loop Aharonov-Bohm Feynman Path-Integrals
We investigate analytically and numerically the mean-field
superconducting-normal phase boundaries of two-dimensional superconducting wire
networks and Josephson junction arrays immersed in a transverse magnetic field.
The geometries we consider include square, honeycomb, triangular, and kagome'
lattices. Our approach is based on an analytical study of multiple-loop
Aharonov-Bohm effects: the quantum interference between different electron
closed paths where each one of them encloses a net magnetic flux. Specifically,
we compute exactly the sums of magnetic phase factors, i.e., the lattice path
integrals, on all closed lattice paths of different lengths. A very large
number, e.g., up to for the square lattice, exact lattice path
integrals are obtained. Analytic results of these lattice path integrals then
enable us to obtain the resistive transition temperature as a continuous
function of the field. In particular, we can analyze measurable effects on the
superconducting transition temperature, , as a function of the magnetic
filed , originating from electron trajectories over loops of various
lengths. In addition to systematically deriving previously observed features,
and understanding the physical origin of the dips in as a result of
multiple-loop quantum interference effects, we also find novel results. In
particular, we explicitly derive the self-similarity in the phase diagram of
square networks. Our approach allows us to analyze the complex structure
present in the phase boundaries from the viewpoint of quantum interference
effects due to the electron motion on the underlying lattices.Comment: 18 PRB-type pages, plus 8 large figure
Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations
Defant, Engen, and Miller defined a permutation to be uniquely sorted if it
has exactly one preimage under West's stack-sorting map. We enumerate classes
of uniquely sorted permutations that avoid a pattern of length three and a
pattern of length four by establishing bijections between these classes and
various lattice paths. This allows us to prove nine conjectures of Defant.Comment: 18 pages, 16 figures, new version with updated abstract and
reference
On FPL configurations with four sets of nested arches
The problem of counting the number of Fully Packed Loop (FPL) configurations
with four sets of a,b,c,d nested arches is addressed. It is shown that it may
be expressed as the problem of enumeration of tilings of a domain of the
triangular lattice with a conic singularity. After reexpression in terms of
non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a
formula as a sum of determinants. This is made quite explicit when
min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates
the numbers of configurations with b=d.Comment: 22 pages, TeX, 16 figures; a new formula for a generating function
adde
- …