1 2 Heisenberg models with possibly anisotropic spin interactions and certain spin 1 models
with SU (2)-invariance. Quantum spin correlations are given by loop correlations. Decay of …

We study the length of cycles of random permutations drawn from the Mallows distribution.
Under this distribution, the probability of a permutation π∈S_n is proportional to q^inv(π) …

These notes give a mathematical introduction to two seemingly unrelated topics:(i) quantum
spin systems and their cycle and loop representations, due to Tóth and Aizenman …

We establish a complete picture of condensation in the inclusion process in the
thermodynamic limit with vanishing diffusion, covering all scaling regimes of the diffusion …

Spatial random permutations and Poisson-Dirichlet law of cycle lengths Page 1 E lectron i c
J o u r nal o f P r o ba bility Vol.16 (2011), Paper no. 41, pages 1173–1192. Journal URL http://www.math.washington.edu/~ejpecp …

We study the length of cycles in the model of spatial random permutations in Euclidean
space. In this model, for given length L, density ρ ρ, dimension d and jump density φ φ, one …

We examine the number of cycles of length k in a permutation as a function on the symmetric
group. We write it explicitly as a combination of characters of irreducible representations …

Let x 1,…, xn be a fixed sequence of real numbers. At each stage, pick two indices I and J
uniformly at random, and replace x I, x J by (x I+ x J)/2,(x I+ x J)/2. Clearly, all the coordinates …

We establish that the phase transition for infinite cycles in the random stirring model on an
infinite regular tree of high degree is sharp. That is, we prove that there exists d_0 d 0 such …

We study random spatial permutations on ℤ 3 where each jump x↦ π (x) is penalized by a
factor e^-T‖x-π(x)‖^2. The system is known to exhibit a phase transition for low enough T …