Mixing times of the biased card shuffling and the asymmetric exclusion process

Abstract

Abstract. Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N . Fix a parameter p between 0 and 1. In this model a "shuffle" consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin comes up heads, then we arrange the two cards so that the lower-numbered card comes before the higher-numbered card. If the coin comes up tails, then we arrange the cards with the higher-numbered card first. In this paper we prove that for all p = 1/2, the mixing time of this card shuffling is O(N 2 ), as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process.

Sampling biased monotonic surfaces using exponential metrics

“…In two dimensions we show that the biased chain is rapidly mixing for any uniform bias on a large family of simply connected regions, even when the bias is arbitrarily close to one. Our proof is significantly simpler than the arguments of [2], while achieving the same optimal bounds on the mixing time for square regions when the bias is constant. In fact, we get optimal bounds for all rectangular h × w regions.…”

“…In two dimensions, monotonic surfaces, called staircase walks, are paths within a finite region of the lattice Z 2 that step to the right or down at every edge (see Figure 1(a)). Markov chains for sampling staircase walks have been used to analyse card-shuffling algorithms by associating to a permutation a set of staircase walks [2,18]. One simple Markov chain M U for sampling uniformly from the set of staircase walks, known as the 'mountain / valley chain' , tries to invert a mountain that moves to the right and then down to a valley that goes down and then to the right, or vice versa.…”

“…The three-dimensional analogue has also been used to study self-assembly, where tiles are now shaped like cubes (as in Figure 1(b)) and complementary sequences are used to encourage specified pairs of faces to attach. The problem of generating biased surfaces in two dimensions was independently studied in the context of biased card shuffling, where we allow nearest-neighbour transpositions but favour putting each pair in order at each step [2].…”

“…Previous work has focused primarily on the case where the biases are uniform; that is, λx = λ for everyx, for which the stationary probability will be proportional to λ |σ | , where |σ | is the number of unit cubes lying below the surface σ . In two dimensions, the uniform bias Markov chain is equivalent to an asymmetric exclusion process, which Benjamini, Berger, Hoffman and Mossel [2] studied in order to analyse a biased card shuffling algorithm that favours putting each pair of cards in the lexicographically correct order. They give a bound of (n 2 ) on the mixing rate of the biased chain on h × w regions of Z 2 (where h + w = n) for any uniform bias λ > 1 that is a constant.…”

Scaling behavior of the contact process in networks with long-range connections

“…in sociophysics [11] or in the context of conductive properties of linear polymers with crosslinks that connect remote monomers [12]. In general, a pair of nodes separated by the distance l is connected by an edge with a probability p l ≃ βl −s for large l [13,14,15,16,17,18,19,20,21,22,23,24]. We mention that the case s = 0 corresponds to the WattsStrogatz graph [25], that displays the small-world phenomenon, although that model is constructed by rewiring edges rather than adding new ones therefore the resulting graph may be disconnected.…”