Results 1  10
of
13
Exact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics
, 1996
"... 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 ..."
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Cited by 549 (13 self)
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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 al...
Matching polytopes, toric geometry, and the nonnegative part of the Grassmannian
, 2007
"... In this paper we use toric geometry to investigate the topology of the totally nonnegative part of the Grassmannian (Grkn)≥0. (Grkn)≥0 is a cell complex whose cells ∆G can be parameterized in terms of the combinatorics of bicolored planar graphs G. To each cell ∆G we associate a complete fanFG whi ..."
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Cited by 16 (5 self)
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In this paper we use toric geometry to investigate the topology of the totally nonnegative part of the Grassmannian (Grkn)≥0. (Grkn)≥0 is a cell complex whose cells ∆G can be parameterized in terms of the combinatorics of bicolored planar graphs G. To each cell ∆G we associate a complete fanFG which is normal to a certain polytope P(G). The combinatorial structure of the polytopes P(G) is reminiscent of the wellknown Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We then use the data ofFG and P(G) to define an associated toric variety XG. We use our technology to prove that the cell decomposition of (Grkn)≥0 is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Grkn)≥0 is 1.
Sandpile Models and Lattices: A Comprehensive Survey
, 2001
"... Starting from some studies of (linear) integer partitions, we noticed that the lattice structure is strongly related to a large variety of discrete dynamical models, in particular sandpile models and chip firing games. After giving an historical survey of the main results which appeared about this, ..."
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Cited by 15 (0 self)
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Starting from some studies of (linear) integer partitions, we noticed that the lattice structure is strongly related to a large variety of discrete dynamical models, in particular sandpile models and chip firing games. After giving an historical survey of the main results which appeared about this, we propose a unified framework to explain the strong relationship between these models and lattices. In particular, we show that the apparent complexity of these models can be reduced, by showing the possibility of symplifying them, and we show how the known lattice properties can be deduced from this.
Trees and matchings
 ELECTRON. J. COMBIN. 7, RESEARCH PAPER
, 2000
"... In this article, Temperley’s bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weight ..."
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Cited by 11 (1 self)
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In this article, Temperley’s bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spanning trees (often called arborescences) of any planar graph G can be put into a onetoone weightpreserving correspondence with the perfect matchings of a related planar graph H. One special case of this result is a bijection between perfect matchings of the hexagonal honeycomb lattice and directed spanning trees of a triangular lattice. Another special case gives a correspondence between perfect matchings of the “squareoctagon” lattice and directed weighted spanning trees on a directed weighted version of the cartesian lattice. In conjunction with results of Kenyon (1997b), our main theorem allows us to compute the measures of all cylinder events for random spanning trees on any (directed, weighted) planar graph. Conversely, in cases where the perfect matching model arises from a tree model, Wilson’s algorithm allows us to quickly generate random samples of perfect matchings.
MATCHING POLYTOPES, TORIC GEOMETRY, AND THE TOTALLY NONNEGATIVE GRASSMANNIAN
, 2008
"... In this paper we use toric geometry to investigate the topology of the totally nonnegative part of the Grassmannian, denoted (Grk,n)≥0. This is a cell complex whose cells ∆G can be parameterized in terms of the combinatorics of planebipartite graphs G. To each cell ∆G we associate a certain polyto ..."
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Cited by 5 (1 self)
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In this paper we use toric geometry to investigate the topology of the totally nonnegative part of the Grassmannian, denoted (Grk,n)≥0. This is a cell complex whose cells ∆G can be parameterized in terms of the combinatorics of planebipartite graphs G. To each cell ∆G we associate a certain polytope P(G). The polytopes P(G) are analogous to the wellknown Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We use the data of P(G) to define an associated toric variety XG. We use our technology to prove that the cell decomposition of (Grk,n)≥0 is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Grk,n)≥0 is 1.
Coding Distributive Lattices with Edge Firing Games
"... In this note, we show that any distributive lattice is isomorphic to the set of reachable configurations of an Edge Firing Game. Together with the result of James Propp, saying that the set of reachable configurations of any Edge Firing Game is always a distributive lattice, this shows that the two ..."
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Cited by 4 (0 self)
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In this note, we show that any distributive lattice is isomorphic to the set of reachable configurations of an Edge Firing Game. Together with the result of James Propp, saying that the set of reachable configurations of any Edge Firing Game is always a distributive lattice, this shows that the two concepts are equivalent.
Regular Labelings and Geometric Structures
, 2010
"... Three types of geometric structure—grid triangulations, rectangular subdivisions, and orthogonal polyhedra— can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an associated maximal or nearmaximal planar graph. We briefly survey the ..."
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Cited by 4 (1 self)
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Three types of geometric structure—grid triangulations, rectangular subdivisions, and orthogonal polyhedra— can each be described combinatorially by a regular labeling: an assignment of colors and orientations to the edges of an associated maximal or nearmaximal planar graph. We briefly survey the connections and analogies between these three kinds of labelings, and their uses in designing efficient geometric algorithms.
Exact Sampling with Markov Chains
, 1996
"... Random sampling has found numerous applications in computer science, statistics, and physics. The most widely applicable method of random sampling is to use a Markov chain whose steady state distribution is the probability distribution ß from which we wish to sample. After the Markov chain has been ..."
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Cited by 3 (0 self)
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Random sampling has found numerous applications in computer science, statistics, and physics. The most widely applicable method of random sampling is to use a Markov chain whose steady state distribution is the probability distribution ß from which we wish to sample. After the Markov chain has been run for long enough, its state is approximately distributed according to ß. The principal problem with this approach is that it is often difficult to determine how long to run the Markov chain. In this thesis we present several algorithms that use Markov chains to return samples distributed exactly according to ß. The algorithms determine on their own how long to run the Markov chain. Two of the algorithms may be used with any Markov chain, but are useful only if the state space is not too large. Nonetheless, a spinoff of these two algorithms is a procedure for sampling random spanning trees of a directed graph that runs more quickly than the Aldous/Broder algorithm. Another of the exact sa...
Distributive Lattices, Polyhedra, and Generalized Flow
"... A Dpolyhedron is a polyhedron P such that if x, y are in P then so are their componentwise max and min. In other words, the point set of a Dpolyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of Dpolyhedra. Aside from be ..."
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Cited by 3 (2 self)
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A Dpolyhedron is a polyhedron P such that if x, y are in P then so are their componentwise max and min. In other words, the point set of a Dpolyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of Dpolyhedra. Aside from being a nice combination of geometric and order theoretic concepts, Dpolyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact every Dpolyhedron corresponds to a directed graph with arcparameters, such that every point in the polyhedron corresponds to a vertex potential on the graph. Alternatively, an edgebased description of the point set can be given. The objects in this model are dual to generalized flows, i.e., dual to flows with gains and losses. These models can be specialized to yield some cases of distributive lattices that have been studied previously. Particular specializations are: lattices of flows of planar digraphs (Khuller, Naor and Klein), of αorientations of planar graphs (Felsner), of corientations (Propp) and of ∆bonds of digraphs (Felsner and Knauer). As an additional application we exhibit a distributive lattice structure on generalized flow of breakeven planar digraphs.
Generalized Tilings with Height Functions
, 2001
"... In this paper, we introduce a generalization of a class of tilings which appear in the literature: the tilings over which a height function can be dened (for example, the famous tilings of polyominoes with dominoes). We show that many properties of these tilings can be seen as the consequences of pr ..."
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Cited by 3 (2 self)
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In this paper, we introduce a generalization of a class of tilings which appear in the literature: the tilings over which a height function can be dened (for example, the famous tilings of polyominoes with dominoes). We show that many properties of these tilings can be seen as the consequences of properties of the generalized tilings we introduce. In particular, we show that any tiling problem which can be modelized in our generalized framework has the following properties: the tilability of a region can be constructively decided in polynomial time, the number of connected components in the undirected ipaccessibility graph can be determined, and the directed ipaccessibility graph induces a distributive lattice structure. Finally, we give a few examples of known tiling problems which can be viewed as particular cases of the new notions we introduce.