24,032 research outputs found
Two-dimensional crystal melting and D4-D2-D0 on toric Calabi-Yau singularities
We construct a two-dimensional crystal melting model which reproduces the BPS
index of D2-D0 states bound to a non-compact D4-brane on an arbitrary toric
Calabi-Yau singularity. The crystalline structure depends on the toric divisor
wrapped by the D4-brane. The molten crystals are in one-to-one correspondence
with the torus fixed points of the moduli space of the quiver gauge theory on
D-branes. The F- and D-term constraints of the gauge theory are regarded as a
generalization of the ADHM constraints on instantons. We also show in several
examples that our model is consistent with the wall-crossing formula for the
BPS index.Comment: 72 pages, 44 figure
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
Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers
We give asymptotic formulas for the multiplicities of weights and irreducible
summands in high-tensor powers of an irreducible
representation of a compact connected Lie group . The weights
are allowed to depend on , and we obtain several regimes of pointwise
asymptotics, ranging from a central limit region to a large deviations region.
We use a complex steepest descent method that applies to general asymptotic
counting problems for lattice paths with steps in a convex polytope.Comment: 38 pages, no figure
Area versus Length Distribution for Closed Random Walks
Using a connection between the -oscillator algebra and the coefficients of
the high temperature expansion of the frustrated Gaussian spin model, we derive
an exact formula for the number of closed random walks of given length and
area, on a hypercubic lattice, in the limit of infinite number of dimensions.
The formula is investigated in detail, and asymptotic behaviours are evaluated.
The area distribution in the limit of long loops is computed. As a byproduct,
we obtain also an infinite set of new, nontrivial identities.Comment: 17 page
Lattice path counting and the theory of queues
In this paper we will show how recent advances in the combinatorics of lattice paths can be applied to solve interesting and nontrivial problems in the theory of queues. The problems we discuss range from classical ones like M^a/M^b/1 systems to open tandem systems with and without global blocking and to queueing models that are related to random walks in a quarter plane like the Flatto-Hahn model or systems with preemptive priorities. (author´s abstract)Series: Research Report Series / Department of Statistics and Mathematic
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