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Markov Chain Algorithms for Planar Lattice Structures
Abstract
Consider the following Markov chain, whose states are all domino tilings of a 2n Theta 2n chessboard: starting from some arbitrary tiling, pick a 2 Theta 2 window uniformly at random. If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes 90 o in place. Repeat many times. This process is used in practice to generate a random tiling, and is a widely used tool in the study of the combinatorics of tilings and the behavior of dimer systems in statistical physics. Analogous Markov chains are used to randomly generate other structures on various two-dimensional lattices. This paper presents techniques which prove for the first time that, in many interesting cases, a small number of random moves suffice to obtain a uniform distribution. 1 Introduction This paper is concerned with algorithmic problems of the following type: given a simply connected region S of the two-dimensional Cartesian lattice (e.g., an n Theta n chessboard),...
... The particle interlacement constraints imply that the equilibrium measures are far from being product Bernoulli: particle correlations decay like the inverse distance squared and interface height fluctuations behave on large scales like a massless Gaussian field. We consider a particular choice of the transition rates, originally proposed in [15]: in terms of interlaced particles, a particle jump of length n that preserves the interlacement constraints has rate 1/(2n). This dynamics presents special features: the average mutual volume between two interface configurations decreases with time [15] and a certain one-dimensional projection of the dynamics is described by the heat equation [21]. ...
... The long-jump dynamics has an interesting story. It was originally introduced in [15] with the goal of providing a Markov chain that approaches the uniform measure on tilings, in variation distance, in a time that is polynomial in the system size L. In fact, the key point is that the mutual volume between two interface configurations is a super-martingale which, together with attractiveness of the dynamics, allows one to deduce polynomial mixing via coupling arguments. Later, in [21] it was proven that the total variation mixing time of the long-jump dynamics is actually of order L 2 log L, and that (in special domains) a particular one-dimensional projection of the height function evolves according to the one-dimensional heat equation. ...
... Later, in [21] it was proven that the total variation mixing time of the long-jump dynamics is actually of order L 2 log L, and that (in special domains) a particular one-dimensional projection of the height function evolves according to the one-dimensional heat equation. The results of [15,21] were used as a building block in [1,12] to prove that, under some conditions on the geometry of the domain, the mixing time of the single-flip Glauber dynamics is of order L 2+o (1) , like that of the long-jump dynamics. Finally, in [13] we discovered that, in contrast with the single-flip dynamics, the long-jump one satisfies certain identities, that allow to conjecture an explicit form for the hydrodynamic equation, cf. ...
We study a reversible continuous-time Markov dynamics of a discrete (2+1)-dimensional interface. This can be alternatively viewed as a dynamics of lozenge tilings of the torus, or as a conservative dynamics for a two-dimensional system of interlaced particles. The particle interlacement constraints imply that the equilibrium measures are far from being product Bernoulli: particle correlations decay like the inverse distance squared and interface height fluctuations behave on large scales like a massless Gaussian field. We consider a particular choice of the transition rates, originally proposed in [Luby-Randall-Sinclair]: in terms of interlaced particles, a particle jump of length n that preserves the interlacement constraints has rate 1/(2n). This dynamics presents special features: the average mutual volume between two interface configurations decreases with time and a certain one-dimensional projection of the dynamics is described by the heat equation. In this work we prove a hydrodynamic limit: after a diffusive rescaling of time and space, the height function evolution tends as to the solution of a non-linear parabolic PDE. The initial profile is assumed to be differentiable and to contain no "frozen region". The explicit form of the PDE was recently conjectured on the basis of local equilibrium considerations. In contrast with the hydrodynamic equation for the Langevin dynamics of the Ginzburg-Landau model [Funaki-Spohn,Nishikawa], here the mobility coefficient turns out to be a non-trivial function of the interface slope.
... Our present interest in these polynomials is due to a general bijection [14] between domino tilings 1 of regions R on the square lattice and families of non-intersecting Schröder-like paths contained in R (see Section 2.3). Motivated by this bijection, we define below a region R λ whose domino tilings correspond to λ-families of Schröder paths in the set S λ . ...
... In general, there is a bijection [14] between domino tilings of a region R on the square lattice and families of nonintersecting paths contained in R. In particular, domino tilings of R λ are in bijection with λ-families of Schröder paths. This bijection is described below. ...
We study domino tilings of certain regions , indexed by partitions , weighted according to generalized area and dinv statistics. These statistics arise from the q,t-Catalan combinatorics and Macdonald polynomials. We present a formula for the generating polynomial of these domino tilings in terms of the Bergeron--Garsia nabla operator. When is a square shape, domino tilings of are equivalent to those of the Aztec diamond of order n. In this case, we give a new product formula for the resulting polynomials by domino shuffling and its connection with alternating sign matrices. In particular, we obtain a combinatorial proof of the joint symmetry of the generalized area and dinv statistics.
... A remarkable tool in the study of tilings has been the use of various computer programs to generate and visualize random tilings. A variety of powerful algorithms and techniques for sampling tilings have been developed and studied (see for example [LRS01,W04,AR05,SZ04,PW]), and their standard implementations are widely available and well-utilized by researchers in the field. ...
... In reality, the Markov chain is run for a finite but large time determined by the rate of convergence or mixing time. Although mixing times of Markov chains on tilings have been studied by many, see for example [LRS01,W04,LT15], and upper bounds rigorously established in many particular settings, very little is known in general. In practice, the mixing times can often be estimated empirically by heuristic techniques such as a self-consistent analysis of autocorrelation times. ...
We present GPU accelerated implementations of Markov chain algorithms to sample random tilings, dimers, and the six-vertex model.
... CKP01]), polymer models, random tiling models in particular lozenge tilings (e.g. [Des98,LRS01,Wil04]), Gibbs models (e.g. [She05]), the Ising model (e.g. ...
We develop a new robust technique to deduce variance principles for non-integrable discrete systems. To illustrate this technique, we show the existence of a variational principle for graph homomorphisms from to a d-regular tree. This seems to be the first non-trivial example of a variational principle in a non-integrable model. Instead of relying on integrability, the technique is based on a discrete Kirszbraun theorem and a concentration inequality obtained through the dynamic of the model. As a consequence of this result, we obtain the existence of a continuum of shift-invariant ergodic gradient Gibbs measures for graph homomorphisms from to a regular tree.
... Let the upper-most edge of the double diamond A ∪ B have height h = 0. Then, regardless of the covering by dominos, the height function along the bound- These level lines are the same as the DR lattice paths [21]. ...
We study random domino tilings of a Double Aztec diamond, a region consisting of two overlapping Aztec diamonds. The random tilings give rise to two discrete determinantal point processes called the K-and L-particle processes. The correlation kernel of the K-particles was derived in Adler, Johansson and van Moerbeke (2011), who used it to study the limit process of the K-particles with different weights for horizontal and vertical dominos. Let the size of both, the Double Aztec diamond and the overlap, tend to infinity such that the two arctic ellipses just touch; then they show that the fluctuations of the K-particles near the tangency point tend to the tacnode process. In this paper, we find the limiting point process of the L-particles in the overlap when the weights of the horizontal and vertical dominos are equal, or asymptotically equal, as the Double Aztec diamond grows, while keeping the overlap finite. In this case the two limiting arctic circles are tangent in the overlap and the behavior of the L-particles in the vicinity of the point of tangency can then be viewed as two colliding GUE-minor process, which we call the tacnode GUE minor process. As part of the derivation of the kernel for the L-particles we find the inverse Kasteleyn matrix for the dimer model version of Double Aztec diamond.
... Following ideas originating from [LRS95] we define a tower Markov chain M 2T that extends M 2 . A single step of M 2T can combine several steps of M 2 . ...
We study Markov chains for -orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function . The set of -orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the -orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the -orientations of these graphs is slowly mixing.
... staircase walks) (see Figure 1 and, e.g. [16]). Notice each linear arrangement of 1's and 0's can be mapped bijectively to a lattice path in Z 2 by sending 1's to steps down and 0's to steps to the right. ...
In this paper, we study a biased version of the nearest-neighbor transposition Markov chain on the set of permutations where neighboring elements i and j are placed in order (i,j) with probability . Our goal is to identify the class of parameter sets for which this Markov chain is rapidly mixing. Specifically, we consider the open conjecture of Jim Fill that all monotone, positively biased distributions are rapidly mixing. We resolve Fill's conjecture in the affirmative for distributions arising from k-class particle processes, where the elements are divided into k classes and the probability of exchanging neighboring elements depends on the particular classes the elements are in. We further require that k is a constant, and all probabilities between elements in different classes are bounded away from 1/2. These particle processes arise in the context of self-organizing lists and our result also applies beyond permutations to the setting where all particles in a class are indistinguishable. Additionally we show that a broader class of distributions based on trees is also rapidly mixing, which generalizes a class analyzed by Bhakta et. al. (SODA '13). Our work generalizes recent work by Haddadan and Winkler (STACS '17) studying 3-class particle processes. Our proof involves analyzing a generalized biased exclusion process, which is a nearest-neighbor transposition chain applied to a 2-particle system. Biased exclusion processes are of independent interest, with applications in self-assembly. We generalize the results of Greenberg et al. (SODA '09) and Benjamini et. al (Trans. AMS '05) on biased exclusion processes to allow the probability of swapping neighboring elements to depend on the entire system, as long as the minimum bias is bounded away from 1.
... One important result was obtained for domino tilings, which are tilings of an n × n square by rectangles of dimensions 1 × 2 or 2 × 1. In this case, the edge-flip Markov chain is known to mix in time polynomial in the number of dominoes, a result that heavily relies on the connection between domino tilings and random lattice paths [13,17]. ...
We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] \times [b2^{-t}, (b+1)2^{-t}] for non-negative integers a,b,s,t. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least \Omega(n^{1.38}), improving upon the previously best lower bound of \Omega(n\log n) coming from the diameter of the chain.
We introduce discrete time Markov chains that preserve uniform measures on boxed plane partitions. Elementary Markov steps change the size of the box from (a x b x c) to ((a-1) x (b+1) x c) or ((a+1) x (b-1) x c). Algorithmic realization of each step involves O((a+b)c) operations. One application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions. Trajectories of our Markov chains can be viewed as random point configurations in the three-dimensional lattice. We compute the bulk limits of the correlation functions of the resulting random point process on suitable two-dimensional sections. The limiting correlation functions define a two-dimensional determinantal point processes with certain Gibbs properties.
For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately, it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop and that outputs samples in exact accordance with the desired distribution. The method uses couplings which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the algorithm itself. If the state space has a partial order that is preserved under the moves of the Markov chain, then the coupling is often particularly efficient. Using our approach, one can sample from the Gibbs distributions associated with various statistical mechanics models (including Ising, random-cluster, ice, and dimer) or choose uniformly at random from the elements of a finite distributive lattice. © 1996 John Wiley & Sons, Inc.
When can a given finite region consisting of cells in a regular lattice (triangular, square, or hexagonal) in 2 be perfectly tiled by tiles drawn from a finite set of tile shapes? This paper gives necessary conditions for the existence of such tilings using boundary invariants, which are combinatorial group-theoretic invariants associated to the boundaries of the tile shapes and the regions to be tiled. Boundary invariants are used to solve problems concerning the tiling of triangular-shaped regions of hexagons in the hexagonal lattice with certain tiles consisting of three hexagons. Boundary invariants give stronger conditions for nonexistence of tilings than those obtainable by weighting or coloring arguments. This is shown by considering whether or not a region has a signed tiling, which is a placement of tiles assigned weights 1 or −1, such that all cells in the region are covered with total weight 1 and all cells outside with total weight 0. Any coloring (or weighting) argument that proves nonexistence of a tiling of a region also proves nonexistence of any signed tiling of the region as well. A partial converse holds: if a simply connected region has no signed tiling by simply connected tiles, then there is a generalized coloring argument proving that no signed tiling exists. There exist regions possessing a signed tiling which can be shown to have no perfect tiling using boundary invariants.
We give a combinatorial interpretation for any minor (or binomial determinant) of the matrix of binomial coefficients. This interpretation involves configurations of nonintersecting paths, and is related to Young tableaux and hook length formulae.
A randomised approximation scheme for the permanent of a 0-1 matrix is presented. The task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph; the latter is accomplished by simulating a Markov chain whose states are the matchings in the graph. For a wide class of 0-1 matrices the approximation scheme is fully-polynomial, i.e., runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output. This class includes all dense matrices (those that contain sufficiently many 1's) and almost all sparse matrices in some reasonable probabilistic model for 0-1 matrices of given density. For the approach sketched above to be computationally efficient, the Markov chain must be rapidly mixing: informally, it must converge in a short time to its stationary distribution. A major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing, apparently the first such result for a Markov chain with genuinely complex structure. The techniques used seem to have general applicability, and are applied again in the paper to validate a fully-polynomial randomised approximation scheme for the partition function of an arbitrary monomer-dimer system.
A survey of relevant papers is given, and five simple and simply handleable algorithms of low complexity based on results contained in these papers are described (without proofs):
Algorithms A and C enable the number of Kekul structures and Pauling's bond orders to be determined and the characteristic polynomial and the eigenspaces (eigenvalues, eigenvectors) to be calculated for all benzenoid systems.
Algorithms B and D enable the same to be done, in a more efficient way, for those benzenoid systems whose dualist graph is a tree (representing catacondensed benzenoid hydrocarbons).
Algorithm E enables, in a particularly efficient way, the number of spanning trees of any benzenoid system to be determined.
All of these algorithms are variants of a simple summation procedure following the edges in a cycle-free directed graph.
We give efficient randomized schemes to sample and approximately count Eulerian orientations of any Eulerian graph. Eulerian orientations are natural flow-like structures, and Welsh has pointed out that computing their number (i)corresponds to evaluating the Tutte polynomial at the point (0, –2) [8,19] and (ii) is equivalent to evaluating “ice-type partition functions” in statistical physics [20].Our algorithms are based on a reduction to sampling and approximately counting perfect matchings for a class of graphs for which the methods of Broder [3, 10] and others [4, 6] apply. A crucial step of the reduction is the “Monotonicity Lemma” (Lemma 3.3) which is of independent combinatorial interest. Roughly speaking, the Monotonicity Lemma establishes the intuitive fact that “increasing the number of constraints applied on a flow problem can only decrease the number of solutions”. In turn, the proof of the lemma involves a new decomposition technique which decouples problematically overlapping structures (a recurrent obstacle in handling large combinatorial populations) and allows detailed enumeration arguments. As a byproduct, (i) we exhibit a class of graphs for which perfect and near-perfect matchings are polynomially related, and hence the permanent can be approximated, for reasons other than “short augmenting paths” (previously the only known approach); and (ii) we obtain a further direct sampling scheme for Eulerian orientations which is faster than the one suggested by the reduction to perfect matchings.Finally, with respect to our approximate counting algorithm, we give the complementary hardness result, namely, that counting exactly Eulerian orientations is #P-complete, and provide some connections with Eulerian tours.