The probability of long cycles in interchange processes

Abstract

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. This allows to study formation of long cycles in the interchange process, including a precise formula for the probability that the permutation is one long cycle at a given time t, and estimates for the cases of shorter cycles.

14 pages, 3rd version adds more detailed description of Toth's conjecture

References in corpus (2)

• Proof of Aldous' spectral gap conjecture
• The mixing time for simple exclusion

Cited by in corpus (9)

• Random loop representations for quantum spin systems
• The free energy in a class of quantum spin systems and interchange processes
• Coxeter factorizations with generalized Jucys-Murphy weights and Matrix Tree theorems for reflection groups
• Quantum spins and random loops on the complete graph
• Phase transition for the interchange and quantum Heisenberg models on the Hamming graph
• Critical parameter of random loop model on trees
• Large cycles in random permutations related to the Heisenberg model
• On a class of orthogonal-invariant quantum spin systems on the complete graph
• Mixing of fast random walks on dynamic random permutations