A reading group devoted to random permutations with spatial structure, such as those inspired by the Feynman-Kac representation of quantum bosons, or by the random stirring model, or...

Contact: Alan Hammond and James Martin

Past meetings

27.11.2009 Alan Hammond  •  2:30pm at Oxford-Man Institute

Random permutations have been an object of a lot of mathematical study. For example, if transpositions are applied independently and successively from the starting point of the identity element, the random permutation that results experiences a phase transition as time evolves that mirrors the emergence of a giant component as edges are randomly opened in the complete graph on $n$ elements. A physcially important and mathematically very interesting extension of such models occurs if we consider random permutations arising from some spatial stucture.
An example: scatter a collection of $\lambda n$ points uniformly and independent in a $d$-dimensional torus of diameter $n$. The quantity $lambda > 0$ is a parameter, as is $\beta > 0$ in what follows. From each point, an independent Brownian motion on the torus is run for time $\beta$.
Condition this system - the randomness of the initial locations and of the Brownian paths - so that the set of the locations of the $n$ points coincides at time $\beta$ with the initial (time $0$) set of locations. The resulting measure induces a probability on random permutations of $n$ points: in a realization, we define a permutation that maps any particle location at time $0$ to the location at which the corresponding Brownian path arrives at time $\beta$. It is known that this model experiences a phase transition at a critical value of $\beta_c$, above which, in the high $n$ limit, the distribution on permutations has cycles of size $\Theta(n)$, and below which, all cycles are bounded, uniformly in $n$.
Physically, the model corresponds to a non-interacting Bose gas, and the question of existence of macroscopic cycles, to the occurrence of Bose-Einstein condensation. A closely related concept to condensation is superfluidity, a phenomenon in which a substance such as helium may, at very low temperatures, form a single-particle layer that spreads around the walls of a container in which it is placed, permitting it to escape. Of great physical significance is the corresponding question for an interacting Bose gas. Now, the Brownian particles are again run for time $\beta$, but each pair of particles is liable to interact when the pair comes within unit distance during the time interval $[0,\beta]$. To the previous conditioning is added the conditioning that no pair interacting occurs. Condensation is characterised as before by the formation of large cycles. It is a major open problem to show that it occurs for high $\beta$ in dimension $3$. The extra conditioning effectively introduces an avoidance condition between the particles in a way reminiscent of Dyson's diffusion.
In 1953, after introducing the path-integral formulation of quantum mechanics, Feynman used a version of it to discuss the $\lambda$-transition in liquid helium, along the above lines. The reading class could include his original paper, recent mathematical work by Daniel Uelschi and Robert Seiringer, and try to draw paralells with non-spatial models of random permutations, such as Oded Schramm's article "random tranpositions". The topic seems to be an exciting extension of existing mathematical work on measures on $S_n$ and has a lot of physical relevance.
02.12.2009 Daniel Ueltschi  •  3:30pm at Oxford-Man Institute

Daniel Ueltschi will talk about "Bose-Einstein condensation and spatial random permutations"

Abstract: Following Alan's excellent presentation of Feynman's approach to the quantum Bose gas, I will discuss the physical background by explaining the standard Bose-Einstein condensation in the ideal (i.e. non-interacting) Bose gas. Then I will discuss the Feynman-Kac representation, which naturally leads to spatial random permutations. The occurrence of infinite cycles was proved by Suto; I will explain the ideas of the proof, and also discuss some generalization that I obtained recently with Volker Betz. The main goal of the talk is to describe several models that should be very attractive to probabilists, with many open questions.
If time permits, I would also discuss lattice models. The random stirring model of Balint Toth, in particular, appears in certain representations of lattice boson or quantum spin models.
20.01.2010 Alan Hammond  •  2:30pm at Oxford-Man Institute

I'm planning to speak an an informal way, about the random stirring model on a tree, and about a vague plan for proving infinite cycles in the Gibbs measure model.
03.02.2010 James Martin & Daniel Ueltschi  •  2:00pm at Oxford-Man Institute

James Martin may be speaking, probably about the random transposition model on the complete graph,
Daniel Ueltschi about his recent work with Robert Seiringer on absence of infinite cycles at high temperature in an interacting Bose gas, I may speak for a while about open problems regarding Gibbs measures and infinite volume limits ... anyway, some mix of these ingredients!
17.02.2010 James Martin  •  2:00pm at Oxford-Man Institute

James Martin is going to speak about Oded Schramm's article "random transpositions", which establishes the macroscopic structure of cycles in the random stirring model on the complete graph after the onset of large cycles.