Results 1  10
of
94
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...
A p* primer: logit models for social networks
 SOCIAL NETWORKS
, 1999
"... A major criticism of the statistical models for analyzing social networks developed by Holland, Leinhardt, and others wHolland, P.W., Leinhardt, S., 1977. Notes on the statistical analysis of social network data; Holland, P.W., Leinhardt, S., 1981. An exponential family of probability distributions ..."
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Cited by 78 (1 self)
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A major criticism of the statistical models for analyzing social networks developed by Holland, Leinhardt, and others wHolland, P.W., Leinhardt, S., 1977. Notes on the statistical analysis of social network data; Holland, P.W., Leinhardt, S., 1981. An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association. 76, pp. 33–65 Ž with discussion.; Fienberg, S.E., Wasserman,
Coalescing random walks and voter model consensus times on the torus in Zd
 THE ANNALS OF PROBABILITY
, 1989
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Stochastic spatial models
 SIAM Rev
, 1999
"... Abstract. In the models we will consider, space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites. Our main aim here is to explain an idea of Durrett and Levin (1994): the behavior of these ..."
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Cited by 55 (2 self)
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Abstract. In the models we will consider, space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites. Our main aim here is to explain an idea of Durrett and Levin (1994): the behavior of these models can be predicted from the properties of the mean field ODE, i.e., the equations for the densities of the various types that result from pretending that all sites are always independent. We will illustrate this picture through a discussion of eight families of examples from statistical mechanics, genetics, population biology, epidemiology, and ecology. Some of our findings are only conjectures based on simulation, but in a number of cases we are able to prove results for systems with “fast stirring ” by exploiting connections between the spatial model and an associated reaction diffusion equation.
On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted
, 1997
"... This paper deals with rates of convergence in the CLT for certain types of dependency. The main idea is to combine a modification of a theorem of Stein, requiring a coupling construction, with a dynamic setup provided by a Markov structure that suggests natural coupling variables. More specifically ..."
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Cited by 49 (1 self)
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This paper deals with rates of convergence in the CLT for certain types of dependency. The main idea is to combine a modification of a theorem of Stein, requiring a coupling construction, with a dynamic setup provided by a Markov structure that suggests natural coupling variables. More specifically, given a stationary Markov chain X�t � , and a function U = U�X�t��, we propose a way to study the proximity of U to a normal random variable when the state space is large. We apply the general method to the study of two problems. In the first, we consider the antivoter chain X�t � =�X �t� i �i∈ � � t = 0 � 1���� � where � is the vertex set of an nvertex regular graph, and X �t� i =+1or−1. The chain evolves from time t to t + 1 by choosing a random vertex i, and a random neighbor of it j, and setting X �t+1� i =−X �t� j and X�t+1� k = X �t� k for all k = i. For a stationary antivoter chain, we study the normal approximation of Un = U �t� n = ∑ i X �t� i for large n and consider some conditions on sequences of graphs such that Un is asymptotically normal, a problem posed by Aldous and Fill. The same approach may also be applied in situations where a Markov chain does not appear in the original statement of a problem but is constructed as an auxiliary device. This is illustrated by considering weighted Ustatistics. In particular we are able to unify and generalize some results on normal convergence for degenerate weighted Ustatistics and provide rates. 1. Introduction and
Hydrodynamic Scaling, Convex Duality, and Asymptotic Shapes of Growth Models
, 1996
"... . We present a technique for simultaneously deriving two related results: Hydrodynamic scaling limits for onedimensional asymmetric particle systems and asymptotic shapes for growth models. The idea is to specify the particle dynamics in terms of a microscopic LaxOleinik formula which leads direct ..."
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Cited by 40 (9 self)
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. We present a technique for simultaneously deriving two related results: Hydrodynamic scaling limits for onedimensional asymmetric particle systems and asymptotic shapes for growth models. The idea is to specify the particle dynamics in terms of a microscopic LaxOleinik formula which leads directly to the macroscopic description in terms of a nonlinear conservation law. The law of large numbers required for this link comes from the growth model that is embedded in the particle system. In the limit, the asymptotic shape of the growth model becomes the convex conjugate of the flux of the conservation law, and the latter is computable from the particle system in equilibrium. The asymptotic shape is then obtained from the duality relation. The method is illustrated with four applications. Mathematics Subject Classification: Primary 60K35, Secondary 60C05, 82C22 Keywords: Hydrodynamic scaling limit, growth model, LaxOleinik formula, convex duality Address: Department of Mathematics, Iow...
Rescaled voter models converge to superBrownian motion
 Ann. Probab
, 2000
"... We show that a sequence of voter models, suitably rescaled in space and time, converges weakly to superBrownian motion. The result includes both nearest neighbor and longer range voter models and complements a limit theorem of Mueller and Tribe in one dimension. ..."
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Cited by 33 (14 self)
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We show that a sequence of voter models, suitably rescaled in space and time, converges weakly to superBrownian motion. The result includes both nearest neighbor and longer range voter models and complements a limit theorem of Mueller and Tribe in one dimension.
Improved Lower Bound On The Thermodynamic Pressure Of The Spin 1/2 Heisenberg Ferromagnet
"... . We introduce a new stochastic representation of the partition function of the spin 1/2 Heisenberg ferromagnet. We express some of the relevant thermodynamic quantities in terms of expectations of functionals of so called random stirrings on Z d . By use of this representation we improve the lowe ..."
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Cited by 29 (0 self)
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. We introduce a new stochastic representation of the partition function of the spin 1/2 Heisenberg ferromagnet. We express some of the relevant thermodynamic quantities in terms of expectations of functionals of so called random stirrings on Z d . By use of this representation we improve the lower bound on the pressure given by Conlon and Solovej in [CS2]. AMS subject classification (1991): 82D40, 82B20 1. Introduction and Result We consider the 1 2 spin isotropic quantum Heisenberg ferromagnet (QHF) on the ddimensional hypercubic lattice. The Hamiltonian of the model is H = 1 2 X jx\Gammayj=1 h (S(x) \Gamma S(y)) 2 \Gamma 1 i (1.1) where S(x) = (S X (x); S Y (x); S Z (x)) ; x 2 , are the local spin operators and the summation runs over nearest neighbour pairs of lattice sites in the rectangular box , with periodic boundary conditions. The canonical commutation relations satisfied by the spin operators are: \Theta S ff (x); S fi (y) = iffi x;y ffl ff;fi;fl S fl ...
Voter model perturbations and reaction diffusion equations
, 2009
"... We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions d ≥ 3. Combining this result with properties of the PDE, some methods arising from a low density sup ..."
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Cited by 17 (7 self)
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We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions d ≥ 3. Combining this result with properties of the PDE, some methods arising from a low density superBrownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of nontrivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of four systems when the parameters are close to the voter model: (i) a stochastic spatial LotkaVolterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, (iii) a continuous time version of the nonlinear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin, (iv) a voter model in which opinion changes are followed by an exponentially distributed latent period during which voters will not change again. The first application confirms a conjecture of Cox and Perkins [8] and the second confirms a conjecture of Ohtsuki et al [41] in the context of certain infinite graphs. An important feature of our general results is that they do not require the process to be attractive.