Statistical Physics and Interdisciplinary Applications

Correlations play an important role in statistical physics. In 1967 Griffiths derived some simple inequalities between correlations in a ferromagnetic system where interacting elements tend to have the same position in the configuration space. I will introduce these inequalities along with some generalizations to see how one can apply them to other interacting systems. References: [1] R. B. Griffiths, 1967. [2] C. M. Fortuin, P. W. Kasteleyn and J. Ginibre, 1971.

The cavity method is a well established technique for solving classical spin models on sparse random graphs (mean-field models with finite connectivity). Laumann, Scardicchio and Sondhi [arXiv:0706.4391] proposed recently an extension of this method to quantum spin-1/2 models in a transverse field, using a discretized Suzuki-Trotter imaginary time formalism. Here we show how to take analytically the continuous imaginary time limit. Our main technical contribution is an explicit procedure to generate the spin trajectories in a path integral representation of the imaginary time dynamics.

Given some partial information about a system we like to reconstruct the interaction pattern of its elements. These elements can be neurons in a neural network, genes in a cell, pixels in an image or computers in the Internet. In the simplest scenario we assume that elements take only two states and interact pairwise. In this talk I will introduce some methods that people use to deal with this problem.

References:
[1] William Bialek et. al, Faster solutions of the inverse pairwise Ising problem, 2007.

Given some partial information about a system we like to reconstruct the interaction pattern of its elements. These elements can be neurons in a neural network, genes in a cell, pixels in an image or computers in the Internet. In the simplest scenario we assume that elements take only two states and interact pairwise. In this talk I will introduce some methods that people use to deal with this problem.

References:
[1] William Bialek et. al, Faster solutions of the inverse pairwise Ising problem, 2007.

Gaussian mixtures have been used in many mathematical and computational models to approximate arbitrary distributions. Nonparametric Belief Propagation (NBP) algorithms use Gaussian mixtures to approximated the message and belief distributions in cases where discretized representations may become unfeasible. The key issue in Gaussian mixture based NBP is the computation of products of Gaussian mixtures. In this talk, I will introduce the approach I developed to approximate Gaussian mixture product efficiently based on a Gaussian mixture reduction technique.