Sampling Biased Lattice Configurations using Exponential Metrics

Abstract

Monotonic surfaces spanning finite regions of Z d arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. We explore how we can sample these surfaces when the distribution is biased to favor higher surfaces. We show that a natural local chain is rapidly mixing with any bias for regions in Z 2 , and for bias λ > d 2 in Z d , when d > 2. Moreover, our bounds on the mixing time are optimal on d-dimensional hyper-cubic regions. The proof uses a geometric distance function and introduces a variant of path coupling in order to handle distances that are exponentially large.

Mixing Times of Markov Chains for Self-Organizing Lists and Biased Permutations

“…They relate this biased shuffling Markov chain to a chain on an asymmetric simple exclusion process (ASEP) and showed that they both converge in Θ(n 2 ) time. These bounds were matched by Greenberg et al [10] who also generalized the result on ASEPs to sampling biased surfaces in two and higher dimensions in optimal Θ(n d ) time. Note that when the bias is a constant for all i < j there are other methods for sampling from the stationary distribution, but studying the Markov chain M nn is of independent interest, partly because of the connection to ASEPs and other combinatorial structures.…”

“…We use a mapping from biased permutations to multiple particle ASEP configurations with n zeros and n ones. The resulting ASEPs are in bijection with staircase walks [10], which are sequences of n ones and n zeros, that correspond to paths on the Cartesian lattice from (0, n) to (n, 0), where each 1 represents a step to the right and each 0 represents a step down (see Figure 1b). In [10], Greenberg et al examined the Markov chain which attempts to swap a neighboring (0, 1) pair, which essentially adds or removes a unit square from the region below the walk, with probability depending on the position of that unit square.…”

“…The resulting ASEPs are in bijection with staircase walks [10], which are sequences of n ones and n zeros, that correspond to paths on the Cartesian lattice from (0, n) to (n, 0), where each 1 represents a step to the right and each 0 represents a step down (see Figure 1b). In [10], Greenberg et al examined the Markov chain which attempts to swap a neighboring (0, 1) pair, which essentially adds or removes a unit square from the region below the walk, with probability depending on the position of that unit square. The probability of each walk w is proportional to xy<w λ x,y , where the bias λ x,y ≥ 1/2 is assigned to the square at (x, y) and xy < w whenever the square at (x, y) lies underneath the walk w. We show that there are settings of the {λ x,y } which cause the chain to be slowly mixing from any starting configuration (or walk).…”

“…The Markov chain M tree (T ), when proposing a swap of the elements a and b, will only attempt to swap them if a, b correspond to some adjacent L and R in the string associated with a ∧ b. Swapping a and b does not affect any other string, so each non-leaf node v represents an independent exclusion process with L(v) L s and R(v) R s. These exclusion processes have been well-studied [4,19,2,10]. We use the following bounds on the mixing times of the symmetric and asymmetric simple exclusion processes.…”

“…If p > 1/2 + c for some positive constant c, then τ ( ) = O(k 2 log( −1 )). [10] 2. Otherwise, τ ( ) = O(k 3 log(k/ )).…”