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83
The stochastic randomcluster process and the uniqueness of randomcluster measures
, 1995
"... The randomcluster model is a generalisation of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn (see [29]). Not only is the randomcluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated phy ..."
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Cited by 86 (13 self)
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The randomcluster model is a generalisation of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn (see [29]). Not only is the randomcluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated physical models. This paper serves two functions. First, we introduce and survey randomcluster measures from the probabilist’s point of view, giving clear statements of some of the many open problems. Secondly, we present new results for such measures, as follows. We discuss the relationship between weak limits of randomcluster measures and measures satisfying a suitable DLR condition. Using an argument based on the convexity of pressure, we prove the uniqueness of randomcluster measures for all but (at most) countably many values of the parameter p. Related results concerning phase transition in two or more dimensions are included, together with various stimulating conjectures. The uniqueness of the infinite cluster is employed in an intrinsic way, in part of these arguments. In the second part of this paper is constructed a Markov process whose levelsets are reversible Markov processes with randomcluster measures as unique equilibrium measures. This construction enables a coupling of randomcluster measures for all values of p. Furthermore it leads to a proof of the semicontinuity of the percolation probability, and provides a heuristic probabilistic justification for the widely held belief that there is a firstorder phase transition if and only if the clusterweighting factor q is sufficiently large.
Stability and Convergence of Moments for Multiclass Queueing Networks via Fluid Limit Models
 IEEE Transactions on Automatic Control
, 1995
"... The subject of this paper is open multiclass queueing networks, which are common models of communication networks, and complex manufacturing systems such as wafer fabrication facilities. We provide sufficient conditions for the existence of bounds on longrun average moments of the queue lengths at ..."
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Cited by 78 (31 self)
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The subject of this paper is open multiclass queueing networks, which are common models of communication networks, and complex manufacturing systems such as wafer fabrication facilities. We provide sufficient conditions for the existence of bounds on longrun average moments of the queue lengths at the various stations, and we bound the rate of convergence of the mean queue length to its steady state value. Our work provides a solid foundation for performance analysis either by analytical methods or by simulation. These results are applied to several examples including reentrant lines, generalized Jackson networks, and a general polling model as found in computer networks applications. Keywords: Multiclass queueing networks, ergodicity, general state space Markov processes, polling models, generalized Jackson networks, stability, performance analysis. 1 Introduction The subject of this paper is open multiclass queueing networks, which are models of complex systems such as wafer fabri...
Exponential and Uniform Ergodicity of Markov Processes
 Ann. Probab
, 1995
"... Geometric convergence of Markov chains in discrete time on a general state has been studied in detail in [15]. Here we develop a similar theory for 'irreducible continuous time processes, and consider the following types of criteria for geometric convergence: (a) the existence of exponentially boun ..."
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Cited by 45 (13 self)
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Geometric convergence of Markov chains in discrete time on a general state has been studied in detail in [15]. Here we develop a similar theory for 'irreducible continuous time processes, and consider the following types of criteria for geometric convergence: (a) the existence of exponentially bounded hitting times on one and then all suitably "small" sets; (b) the existence of "FosterLyapunov" or "drift" conditions for any one and then all skeleton and resolvent chains; (c) the existence of drift conditions on the extended generator e A of the process. We use the identity e AR fi = fi(R fi \Gamma I) connecting the extended generator and the resolvent kernels R fi , to show that, under a suitable aperiodicity assumption, exponential convergence is completely equivalent to any of (a)(c). These conditions yield criteria for exponential convergence of unbounded as well as bounded functions of the chain. They enable us to identify the dependence of the convergence on the initial state ...
A Lyapunov Bound for Solutions of Poisson's Equation
 Ann. Probab
, 1996
"... In this paper we consider /irreducible Markov processes evolving in discrete or continuous time, on a general state space. We develop a Lyapunov function criterion that permits one to obtain explicit bounds on the solution to Poisson's equation and, in particular, obtain conditions under which the ..."
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Cited by 43 (25 self)
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In this paper we consider /irreducible Markov processes evolving in discrete or continuous time, on a general state space. We develop a Lyapunov function criterion that permits one to obtain explicit bounds on the solution to Poisson's equation and, in particular, obtain conditions under which the solution is square integrable. These results are applied to obtain sufficient conditions that guarantee the validity of a functional central limit theorem for the Markov process. As a second consequence of the bounds obtained, a perturbation theory for Markov processes is developed which gives conditions under which both the solution to Poisson's equation and the invariant probability for the process are continuous functions of its transition kernel. The techniques are illustrated with applications to queueing theory and autoregressive processes. AMS subject classifications: 68M20, 60J10 Running head: Poisson's Equation Keywords: Markov chain, Markov process, Poisson's equation, Lyapunov f...
Drift transforms and Green function estimates for discontinuous processes
 JOURNAL OF FUNCTIONAL ANALYSIS
, 2003
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Efficient Markovian couplings: examples and counterexamples
, 1999
"... In this paper we study the notion of an efficient coupling of Markov processes. Informally, an efficient coupling is one which couples at the maximum possible exponential rate, as given by the spectral gap. This notion is of interest not only for its own sake, but also of growing importance arising ..."
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Cited by 33 (18 self)
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In this paper we study the notion of an efficient coupling of Markov processes. Informally, an efficient coupling is one which couples at the maximum possible exponential rate, as given by the spectral gap. This notion is of interest not only for its own sake, but also of growing importance arising from the recent advent of methods of "perfect simulation": it helps to establish the "price of perfection" for such methods. In general one can always achieve efficient coupling if the coupling is allowed to "cheat" (if each component's behaviour is affected by future behaviour of the other component), but the situation is more interesting if the coupling is required to be coadapted. We present an informal heuristic for the existence of an efficient coupling, and justify the heuristic by proving rigorous results and examples in the contexts of finite reversible Markov chains and of reflecting Brownian motion in planar domains. Keywords: DIFFUSION, CHENOPTIMAL COUPLING, COADAPTED COUPLING,...
Fast Equilibrium Selection by Rational Players Living in a Changing World
 Econometrica
, 1996
"... We study a coordination game with randomly changing payoffs and small frictions in changing actions. Using only backwards induction, we find that players must coordinate on the risk dominant equilibrium. More precisely, a continuum of fully rational players are randomly matched to play a symmetric 2 ..."
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Cited by 29 (7 self)
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We study a coordination game with randomly changing payoffs and small frictions in changing actions. Using only backwards induction, we find that players must coordinate on the risk dominant equilibrium. More precisely, a continuum of fully rational players are randomly matched to play a symmetric 2 \Theta 2 game. The payoff matrix changes over time according to some Brownian motion. Players observe these payoffs and the population distribution of actions as they evolve. The game has frictions: opportunities to change strategies arrive from independent random processes, so that the players are locked into their actions for some time. We solve the game using only backwards induction. As the frictions disappear, each player ignores what the others are doing and switches at her first opportunity to the risk dominant equilibrium. History dependence emerges in some cases when frictions remain positive. As an application we show how frictions and aggregate cost shocks can lead to the selecti...
Harnack inequality for some classes of Markov processes
"... In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. ..."
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Cited by 25 (13 self)
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In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps.
Generalized Resolvents and Harris Recurrence of Markov Processes
, 1992
"... In this paper we consider a #irreducible continuous parameter Markov process # whose state space is a general topological space. The recurrence and Harris recurrence structure of # is developed in terms of generalized forms of resolvent chains, where we allow statemodulated resolvents and embedd ..."
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Cited by 24 (15 self)
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In this paper we consider a #irreducible continuous parameter Markov process # whose state space is a general topological space. The recurrence and Harris recurrence structure of # is developed in terms of generalized forms of resolvent chains, where we allow statemodulated resolvents and embedded chains with arbitrary sampling distributions. We show that the recurrence behavior of such generalized resolvents classifies the behavior of the continuous time process; from this we prove that hitting times on the small sets of a generalized resolvent chain provide criteria for, successively, (i) Harris recurrence of # (ii) the existence of an invariant probability measure # (or positive Harris recurrence of #) and (iii) the finiteness of #(f) for arbitrary f.
Gaugeability and Conditional Gaugeability
 TRANS. AMER. MATH. SOC
, 2001
"... New Kato classes are introduced for general transient Borel right processes, under which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Greentight measures in the classical Brownian motion case. However the main focus of this paper is on establishing ..."
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Cited by 21 (5 self)
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New Kato classes are introduced for general transient Borel right processes, under which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Greentight measures in the classical Brownian motion case. However the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrödinger operators are equivalent for transient Borel right processes with strong duals. for transient Borel standard processes having strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrödinger operators are also established.