Events Archive

Jakob Björnberg: Random permutations and the Heisenberg model

Abstract: We discuss models for random permutations which are closely linked to quantum spin systems from statistical physics. The cycle structure of the random permutations is intimately connected with the correlation structure in the spin-system, and it is expected that this cycle structure converges to a distribution known as Poisson--Dirichlet, in the limit of large systems.  This problem is still open but we present some partial progress.

Dr. Piotr Mitos: Phase transition for the interchange and quantum Heisenberg models on the Hamming graph

Abstract: In my talk I will present a family of random permutation models on the 2-dimensional Hamming graph H(2,n), containing the interchange process and the cycle-weighted interchange process with parameter θ>0. This family contains the random representation of the quantum Heisenberg ferromagnet. The main result is that in these models the cycle structure of permutations undergoes a phase transition -- when the number of transpositions defining the permutation is <cn^2, for small enough c>0, all cycles are microscopic, while for more than >Cn^2 transpositions, for large enough C>0, macroscopic cycles emerge with high probability. For the quantum Heisenberg ferromagnet on H(2,n) this implies that for low enough temperatures spontaneous magnetization occurs, while it is not the case for high temperatures. At the core of our approach is a novel application of the cyclic random walk, which might be of independent interest. By analyzing explorations of the cyclic random walk, we show that sufficiently long cycles of a random permutation are uniformly spread on the graph, which makes it possible to compare our models to the mean-field case, i.e., the interchange process on the complete graph, extending the approach used earlier by Schramm (joint work with Radosław Adamczak, Michał Kotowski).