Hahn polynomials and the Burnside process

Mixing times of a Burnside process Markov chain on set partitions

“…Twisted Burnside Process. In [11], Diaconis and Zhong introduced a generalization of the original Burnside process, called the twisted Burnside process. Let G be a finite group acting on a finite set X.…”

“…Aldous and Fill [2] considered the case where X = [k] n , the set of n-tuples with entries from [k], and G = S n is the symmetric group acting on X by permuting the positions of the coordinates, and studied the mixing time using coupling. Diaconis [9] found a connection to Bose-Einstein configurations of n balls dropped into k boxes and used minorization to obtain an upper bound on the mixing time of order k!, independent of n. More recently, Diaconis and Zhong [11] considered the special case where k = 2 and used spectral methods to get sharp bounds on the mixing time. They also introduced a variant called the twisted Burnside process, which produces a class of chains which have the Hahn polynomials as eigenfunctions.…”

“…The Stirling numbers of the second kind n k count the number of set partitions of [n] into k nonempty subsets. These numbers are connected through the sum B n = n k=0 n k .We consider a variant of the Burnside process studied by Aldous and Fill[2], Diaconis[9], and Diaconis and Zhong[11]. Let X = [k] n , where k ≥ n, and let G = S k .…”