Parallel Computing of Edwards—Anderson Model

A scheme for parallel computation of the two-dimensional Edwards—Anderson model based on the transfer matrix approach is proposed. Free boundary conditions are considered. The method may find application in calculations related to spin glasses and in quantum simulators. Performance data are given. The scheme of parallelisation for various numbers of threads is tested. Application to a quantum computer simulator is considered in detail. In particular, a parallelisation scheme of work of quantum computer simulator.

Parallel computing of Edwards–Anderson model

An algorithm for parallel exact calculation of the ground state of a two-dimensional Edwards–Anderson model with free boundary conditions is given. The running time of the algorithm grows exponentially as the side of the lattice square increases. If one side of the lattice is fixed, the running time grows polynomially with increasing size of the other side. The method may find application in the theory of spin glasses, in the field of quantum computing. Performance data for the bimodal distribution is given. The distribution of spin bonds can be either bimodal or Gaussian. The method makes it possible to compute systems up to a size of 40x40.

Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equation

In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove u ( t , x ) ∈ D ∞ u(t,x)\in \mathbb {D}^\infty for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model. We have extended their criteria to a localized version. Our general framework includes the parabolic Anderson model as a special case.