an:05775687
Zbl 1203.60145
Caputo, Pietro; Liggett, Thomas M.; Richthammer, Thomas
Proof of Aldous' spectral gap conjecture
EN
J. Am. Math. Soc. 23, No. 3, 831-851 (2010).
00266434
2010
j
60K35 60J27 05C50
random walk; weighted graph; spectral gap; interchange process; symmetric exclusion process
Two random processes are considered on an undirected graph with edge weights: the random walk process with the weights as jump rates and the random transposition (or interchange) process with the weights as transition rates. Aldous' spectral gap conjecture asserts that both processes have the same spectral gap. The recursive strategy used to prove the conjecture is a natural extension of the method already used for trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses coset decompositions of the associated matrices with rows and columns indexed by permutations. In the last section the authors study the spectral gap of other stochastic processes associated to weighted graphs. Symmetric exclusion process, cycle process and matching process are successively considered.
Dominique Lepingle (Orl??ans)