238 research outputs found
Cumulants, lattice paths, and orthogonal polynomials
A formula expressing free cumulants in terms of the Jacobi parameters of the
corresponding orthogonal polynomials is derived. It combines Flajolet's theory
of continued fractions and Lagrange inversion. For the converse we discuss
Gessel-Viennot theory to express Hankel determinants in terms of various
cumulants.Comment: 11 pages, AMS LaTeX, uses pstricks; revised according to referee's
suggestions, in particular cut down last section and corrected some wrong
attribution
Matrix models and stochastic growth in Donaldson-Thomas theory
We show that the partition functions which enumerate Donaldson-Thomas
invariants of local toric Calabi-Yau threefolds without compact divisors can be
expressed in terms of specializations of the Schur measure. We also discuss the
relevance of the Hall-Littlewood and Jack measures in the context of BPS state
counting and study the partition functions at arbitrary points of the Kaehler
moduli space. This rewriting in terms of symmetric functions leads to a unitary
one-matrix model representation for Donaldson-Thomas theory. We describe
explicitly how this result is related to the unitary matrix model description
of Chern-Simons gauge theory. This representation is used to show that the
generating functions for Donaldson-Thomas invariants are related to
tau-functions of the integrable Toda and Toeplitz lattice hierarchies. The
matrix model also leads to an interpretation of Donaldson-Thomas theory in
terms of non-intersecting paths in the lock-step model of vicious walkers. We
further show that these generating functions can be interpreted as
normalization constants of a corner growth/last-passage stochastic model.Comment: 31 pages; v2: comments and references added; v3: presentation
improved, comments added; final version to appear in Journal of Mathematical
Physic
On General-n Coefficients in Series Expansions for Row Spin-Spin Correlation Functions in the Two-Dimensional Ising Model
We consider spin-spin correlation functions for spins along a row, , in the two-dimensional Ising model.
We discuss a method for calculating general- expressions for coefficients in
high-temperature and low-temperature series expansions of and apply it to
obtain such expressions for several higher-order coefficients. In addition to
their intrinsic interest, these results could be useful in the continuing quest
for an ordinary differential equation whose solution would determine ,
analogous to the known ordinary differential equation whose solution determines
the diagonal correlation function in
this model.Comment: 21 pages, late
Gibbs and Quantum Discrete Spaces
Gibbs measure is one of the central objects of the modern probability,
mathematical statistical physics and euclidean quantum field theory. Here we
define and study its natural generalization for the case when the space, where
the random field is defined is itself random. Moreover, this randomness is not
given apriori and independently of the configuration, but rather they depend on
each other, and both are given by Gibbs procedure; We call the resulting object
a Gibbs family because it parametrizes Gibbs fields on different graphs in the
support of the distribution. We study also quantum (KMS) analog of Gibbs
families. Various applications to discrete quantum gravity are given.Comment: 37 pages, 2 figure
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