948 research outputs found
The number of lattice paths below a cyclically shifting boundary
We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result can be viewed as an extension of well-known enumerative formulae concerning lattice paths dominated by lines of integer slope (e.g. the generalized ballot theorem). Its proof is bijective, involving a classical “reflection” argument. Moreover, a straightforward refinement of our bijection allows for the counting of paths with a specified number of corners. We also show how the result can be applied to give elegant derivations for the number of lattice walks under certain periodic boundaries. In particular, we recover known expressions concerning paths dominated by a line of half-integer slope, and some new and old formulae for paths lying under special “staircases.
Dimers and cluster integrable systems
We show that the dimer model on a bipartite graph on a torus gives rise to a
quantum integrable system of special type - a cluster integrable system. The
phase space of the classical system contains, as an open dense subset, the
moduli space of line bundles with connections on the graph. The sum of
Hamiltonians is essentially the partition function of the dimer model. Any
graph on a torus gives rise to a bipartite graph on the torus. We show that the
phase space of the latter has a Lagrangian subvariety. We identify it with the
space parametrizing resistor networks on the original graph.We construct
several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci.
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Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
We consider the level 1 solution of quantum Knizhnik-Zamolodchikov equation
with reflecting boundary conditions which is relevant to the Temperley--Lieb
model of loops on a strip. By use of integral formulae we prove conjectures
relating it to the weighted enumeration of Cyclically Symmetric Transpose
Complement Plane Partitions and related combinatorial objects
Triggering waves in nonlinear lattices: Quest for anharmonic phonons and corresponding mean free paths
Guided by a stylized experiment we develop a self-consistent anharmonic
phonon concept for nonlinear lattices which allows for explicit
"visualization." The idea uses a small external driving force which excites the
front particles in a nonlinear lattice slab and subsequently one monitors the
excited wave evolution using molecular dynamics simulations. This allows for a
simultaneous, direct determination of the existence of the phonon mean free
path with its corresponding anharmonic phonon wavenumber as a function of
temperature. The concept for the mean free path is very distinct from known
prior approaches: the latter evaluate the mean free path only indirectly, via
using both, a scale for the phonon relaxation time and yet another one for the
phonon velocity. Notably, the concept here is neither limited to small lattice
nonlinearities nor to small frequencies. The scheme is tested for three
strongly nonlinear lattices of timely current interest which either exhibit
normal or anomalous heat transport
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