2,736 research outputs found
A product formula and combinatorial field theory
We treat the problem of normally ordering expressions involving the standard boson operators a, ay where [a; ay] = 1. We show that a simple product formula for formal power series | essentially an extension of the Taylor expansion | leads to a double exponential formula which enables a powerful graphical description of the generating functions of the combinatorial sequences associated with such functions | in essence, a combinatorial eld theory. We apply these techniques to some examples related to specic physical Hamiltonians
Partially directed paths in a wedge
The enumeration of lattice paths in wedges poses unique mathematical
challenges. These models are not translationally invariant, and the absence of
this symmetry complicates both the derivation of a functional recurrence for
the generating function, and solving for it. In this paper we consider a model
of partially directed walks from the origin in the square lattice confined to
both a symmetric wedge defined by , and an asymmetric wedge defined
by the lines and Y=0, where is an integer. We prove that the
growth constant for all these models is equal to , independent of
the angle of the wedge. We derive functional recursions for both models, and
obtain explicit expressions for the generating functions when . From these
we find asymptotic formulas for the number of partially directed paths of
length in a wedge when .
The functional recurrences are solved by a variation of the kernel method,
which we call the ``iterated kernel method''. This method appears to be similar
to the obstinate kernel method used by Bousquet-Melou. This method requires us
to consider iterated compositions of the roots of the kernel. These
compositions turn out to be surprisingly tractable, and we are able to find
simple explicit expressions for them. However, in spite of this, the generating
functions turn out to be similar in form to Jacobi -functions, and have
natural boundaries on the unit circle.Comment: 26 pages, 5 figures. Submitted to JCT
Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles
We derive and study two different formalisms used for non-equilibrium
processes: The coherent-state path integral, and an effective, coarse-grained
stochastic equation of motion. We first study the coherent-state path integral
and the corresponding field theory, using the annihilation process
as an example. The field theory contains counter-intuitive quartic vertices. We
show how they can be interpreted in terms of a first-passage problem.
Reformulating the coherent-state path integral as a stochastic equation of
motion, the noise generically becomes imaginary. This renders it not only
difficult to interpret, but leads to convergence problems at finite times. We
then show how alternatively an effective coarse-grained stochastic equation of
motion with real noise can be constructed. The procedure is similar in spirit
to the derivation of the mean-field approximation for the Ising model, and the
ensuing construction of its effective field theory. We finally apply our
findings to stochastic Manna sandpiles. We show that the coherent-state path
integral is inappropriate, or at least inconvenient. As an alternative, we
derive and solve its mean-field approximation, which we then use to construct a
coarse-grained stochastic equation of motion with real noise.Comment: 29 pages, 33 figures. This is a pedagogic introduction to stochastic
processes, their modeling, and effective field theory. Version 2: writing
improved + a new appendi
History of Catalan numbers
We give a brief history of Catalan numbers, from their first discovery in the
18th century to modern times. This note will appear as an appendix in Richard
Stanley's forthcoming book on Catalan numbers.Comment: 10 page
- …