Matrix models and stochastic growth in DonaldsonThomas theory
Abstract
We show that the partition functions which enumerate DonaldsonThomas invariants of local toric CalabiYau threefolds without compact divisors can be expressed in terms of specializations of the Schur measure. We also discuss the relevance of the HallLittlewood 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 onematrix model representation for DonaldsonThomas theory. We describe explicitly how this result is related to the unitary matrix model description of ChernSimons gauge theory. This representation is used to show that the generating functions for DonaldsonThomas invariants are related to taufunctions of the integrable Toda and Toeplitz lattice hierarchies. The matrix model also leads to an interpretation of DonaldsonThomas theory in terms of nonintersecting paths in the lockstep model of vicious walkers. We further show that these generating functions can be interpreted as normalization constants of a corner growth/lastpassage stochastic model.
 Authors:
 Department of Mathematics, HeriotWatt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, United Kingdom and Maxwell Institute for Mathematical Sciences, Edinburgh (United Kingdom)
 Grupo de Fisica Matematica, Complexo Interdisciplinar da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, PT1649003 Lisboa (Portugal)
 (Spain)
 Publication Date:
 OSTI Identifier:
 22093760
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 53; Journal Issue: 10; Other Information: (c) 2012 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; GAUGE INVARIANCE; INTEGRAL CALCULUS; MTHEORY; PARTITION FUNCTIONS; STOCHASTIC PROCESSES; SYMMETRY
Citation Formats
Szabo, Richard J., Tierz, Miguel, and Departamento de Analisis Matematico, Facultad de Ciencias Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid. Matrix models and stochastic growth in DonaldsonThomas theory. United States: N. p., 2012.
Web. doi:10.1063/1.4748525.
Szabo, Richard J., Tierz, Miguel, & Departamento de Analisis Matematico, Facultad de Ciencias Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid. Matrix models and stochastic growth in DonaldsonThomas theory. United States. doi:10.1063/1.4748525.
Szabo, Richard J., Tierz, Miguel, and Departamento de Analisis Matematico, Facultad de Ciencias Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid. 2012.
"Matrix models and stochastic growth in DonaldsonThomas theory". United States.
doi:10.1063/1.4748525.
@article{osti_22093760,
title = {Matrix models and stochastic growth in DonaldsonThomas theory},
author = {Szabo, Richard J. and Tierz, Miguel and Departamento de Analisis Matematico, Facultad de Ciencias Matematicas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid},
abstractNote = {We show that the partition functions which enumerate DonaldsonThomas invariants of local toric CalabiYau threefolds without compact divisors can be expressed in terms of specializations of the Schur measure. We also discuss the relevance of the HallLittlewood 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 onematrix model representation for DonaldsonThomas theory. We describe explicitly how this result is related to the unitary matrix model description of ChernSimons gauge theory. This representation is used to show that the generating functions for DonaldsonThomas invariants are related to taufunctions of the integrable Toda and Toeplitz lattice hierarchies. The matrix model also leads to an interpretation of DonaldsonThomas theory in terms of nonintersecting paths in the lockstep model of vicious walkers. We further show that these generating functions can be interpreted as normalization constants of a corner growth/lastpassage stochastic model.},
doi = {10.1063/1.4748525},
journal = {Journal of Mathematical Physics},
number = 10,
volume = 53,
place = {United States},
year = 2012,
month =
}

Stochastic Hamiltonians for noncritical string field theories from doublescaled matrix models
We present detailed discussions on the stochastic Hamiltonians for noncritical string field theories on the basis of matrix models. Beginning from the simplest {ital c}=0 case, we derive the explicit forms of the Hamiltonians for the higher critical case {ital k}=3 (which corresponds to {ital c}={minus}22/5) and for the case {ital c}=1/2, directly from the doublescaled matrix models. In particular, for the twomatrix case, we do not put any restrictions on the spin configurations of the string fields. The properties of the resulting infinite algebras of SchwingerDyson operators associated with the Hamiltonians and the derivation of the Virasoro and {italmore » 
Stochastic quantization of matrix and lattice gauge models
We introduce a stochastic diffusion equation and the FokkerPlanck equation for various matrix models including the U(N) x U(N) chiral model and lattice gauge theories. It is shown how to calculate various U(N) integrals using the stochastic equation. In particular, in the externalfield problem, the exact largeN result (in the weakcoupling region) is reproduced and a 1/N/sup 2/ correction is computed. Also, the order parameter is calculated up to order 1/..beta../sup 2/. In the U(N) x U(N) chiral model, the largeN reduction and quenching is done in the context of stochastic quantization, and the semiclassical results of Bars, Gunaydin, andmore » 
On implementation of EMtype algorithms in the stochastic models for a matrix computing on GPU
The paper discusses the main ideas of an implementation of EMtype algorithms for computing on the graphics processors and the application for the probabilistic models based on the Cox processes. An example of the GPU’s adapted MATLAB source code for the finite normal mixtures with the expectationmaximization matrix formulas is given. The testing of computational efficiency for GPU vs CPU is illustrated for the different sample sizes.