Results 1  10
of
38
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 ..."
Abstract

Cited by 406 (13 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 47 (0 self)
 Add to MetaCart
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,
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 ..."
Abstract

Cited by 31 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 20 (4 self)
 Add to MetaCart
. 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...
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 ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
. 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 ...
SuperBrownian Limits of Voter Model Clusters
 Ann. Probab. 29, 1001–1032 (2001) Zbl pre01906008 MR 2003c:60160
, 2000
"... this paper, we will study the limiting spatial structure of the voter model in d ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
this paper, we will study the limiting spatial structure of the voter model in d
Ergodicity for spin systems with stirrings
 Ann. Probab
, 1990
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
The Dynamics of Defect Ensembles in OneDimensional Cellular Automata
, 1994
"... We investigate the dynamics of ensembles of diffusive defects in onedimensional deterministic cellular automata. The work builds on earlier results on individual random walks in c.a. (5;6) . Here we give a natural condition guaranteeing diffusive behavior also in the presence of other defects. ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
We investigate the dynamics of ensembles of diffusive defects in onedimensional deterministic cellular automata. The work builds on earlier results on individual random walks in c.a. (5;6) . Here we give a natural condition guaranteeing diffusive behavior also in the presence of other defects. Simple branching and birth mechanisms are introduced and prototype classes of cellular automata exhibiting weakly interacting walks capable of annihilation and coalescence are studied. Their equilibrium behavior is also characterized. The design principles of cellular automata with desired diffusive interaction properties becomes transparent from this analysis. Keywords: Cellular automaton, permutivity, topological defect, random walk. AMS Classification: 58F08, 60K35, 82C41 1 Research partially supported by the Academy of Finland and The Finnish Cultural Foundation 1 Introduction Topological defects, Bloch walls or contours can be identified in a number of standard lattice model...
Microscopic Selection Principle for a DiffusionReaction Equation
 Journal of Statistical Physics
, 1986
"... We consider a model of stochastically interacting particles on 2~, where each site is assumed to be empty or occupied by at most one particle. Particles jump to each empty neighboring site with rate 7/2 and also create new particles with rate 1/2 at these sites. We show that as seen from the rightmo ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
We consider a model of stochastically interacting particles on 2~, where each site is assumed to be empty or occupied by at most one particle. Particles jump to each empty neighboring site with rate 7/2 and also create new particles with rate 1/2 at these sites. We show that as seen from the rightmost particle, this process has precisely one invariant distribution. The average velocity of this particle V(7,) then satisfies 7 ~/2V(Y) ~ ~/2 as y. oo. This limit corresponds to that of the macroscopic density obtained by rescaling lengths by a factor},~/2 and letting y ~ oo. This density solves the reactiondiffusion equation u, = 89 + u(1u), and under Heaviside initial data converges to a traveling wave moving at the same rate,fi. KEY WORDS: Diffusionreaction equation.
Strong Law of Large Numbers for the Interface in Ballistic Deposition
, 1999
"... We prove a hydrodynamic limit for ballistic deposition on a multidimensional lattice. In this growth model particles rain down at random and stick to the growing cluster at the first point of contact. The theorem is that if the initial random interface converges to a deterministic macroscopic funct ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
We prove a hydrodynamic limit for ballistic deposition on a multidimensional lattice. In this growth model particles rain down at random and stick to the growing cluster at the first point of contact. The theorem is that if the initial random interface converges to a deterministic macroscopic function, then at later times the height of the scaled interface converges to the viscosity solution of a HamiltonJacobi equation. The proof idea is to decompose the interface into the shapes that grow from individual seeds of the initial interface. This decomposition converges to a variational formula that defines viscosity solutions of the macrosopic equation. The technical side of the proof involves subadditive methods and large deviation bounds for related firstpassage percolation processes.