

My research centers on Markov chain Monte Carlo methods in statistical mechanics. This includes side work on lattice percolation and selfavoiding walks. My thesis topic is critical behavior for the model of random spatial permutations. This project lies at the crossroads of probability theory, statistical mechanics, functional analysis, statistics, and numerical methods.
The randomspatialpermutation model arises in the study of the Bose gas, although it is also of intrinsic probabilistic interest; its history includes BoseEinstein, Feynman, PenroseOnsager, Sütő, and UeltschiBetz. Random permutations arise physically when one symmetrizes the Nboson Hamiltonian with pair interactions and applies a multiparticle FeynmanKac formula. System energy is now expressed in terms of point positions and permutations of positions, where permutations occur with nonuniform probability. Feynman’s claim is that BEC occurs if and only if there are infinite cycles. The central point of this approach is that the system energy has been recast in terms of permutations, which are amenable to analysis and simulation. Furthermore, interactions between permutations are recast as collision probabilities between Brownian bridges in Feynman time. These Brownian bridges are integrated out, resulting in a model which lends itself readily to simulations without the need for CPUintensive pathintegral Monte Carlo (PIMC). This permits a new perspective on the venerable question: how does the critical temperature of BoseEinstein condensation depend on interparticle interaction strength?
Obtaining a full answer to this question is a longterm project. Breaking it into manageable pieces, my coadvisor Daniel Ueltschi and I consider modified Hamiltonians as well as various point configurations. For example, points may distributed throughout the continuum or held fixed on a lattice. Through careful use of MCMC algorithms, statistical analysis, and finitesize scaling, we are able to quantify the dependence of critical temperature on interaction strength for certain models.
In the animation at the top of this page, there are 10^{3} points uniformly distributed in a 3dimensional box. There are no interactions and the temperature is subcritical. Lines between points indicate permutation cycles. Twenty permutations are displayed.
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