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On the Structure of QuasiStationary Competing Particles Systems
, 2007
"... We study point processes on the real line whose configurations X are locally finite, have a maximum, and evolve through increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q = {qij}i,j∈N. A probability measure on th ..."
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Cited by 42 (5 self)
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We study point processes on the real line whose configurations X are locally finite, have a maximum, and evolve through increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q = {qij}i,j∈N. A probability measure on the pair (X, Q) is said to be quasistationary if the joint law of the gaps of X and of Q is invariant under the evolution. A known class of universally quasistationary processes is given by the Ruelle Probability Cascades (RPC), which are based on hierarchally nested PoissonDirichlet processes. It was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasistationary. The main result of this work is a proof of this conjecture for the case where qij assume only a finite number of values. The result is of relevance for meanfield spin glass models, where the evolution corresponds to the cavity dynamics, and where the hierarchal organization of the Gibbs measure was first proposed as an ansatz.
A PHASE TRANSITION BEHAVIOR FOR BROWNIAN MOTIONS INTERACTING THROUGH THEIR RANKS
"... Abstract. Consider a timevarying collection of n points on the positive real axis, modeled as exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their s ..."
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Cited by 31 (4 self)
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Abstract. Consider a timevarying collection of n points on the positive real axis, modeled as exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as n tends to infinity. Under a certain ‘continuity at the edge ’ condition on the drifts, we show that one of the following must happen: either (i) all points converge to 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a nontrivial PoissonDirichlet distribution. The proof employs, among other things, techniques from Talagrand’s analysis of the low temperature phase of Derrida’s Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the PoissonDirichlet law. 1.
Competing particle systems and the GhirlandaGuerra identities
, 2007
"... E l e c t r o n J o u r n a l ..."
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A COMBINATORIAL ANALYSIS OF INTERACTING DIFFUSIONS
, 2009
"... We consider a particular class of ndimensional homogeneous diffusions all of which have an identity diffusion matrix and a drift function that is piecewise constant and scale invariant. Abstract stochastic calculus immediately gives us general results about existence and uniqueness in law and invar ..."
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Cited by 6 (3 self)
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We consider a particular class of ndimensional homogeneous diffusions all of which have an identity diffusion matrix and a drift function that is piecewise constant and scale invariant. Abstract stochastic calculus immediately gives us general results about existence and uniqueness in law and invariant probability distributions when they exist. These invariant distributions are probability measures on the ndimensional space and can be extremely resistant to a more detailed understanding. To have a better analysis, we construct a polyhedra such that the inward normal at its surface is given by the drift function and show that the finer structures of the invariant probability measure is intertwined with the geometry of the polyhedra. We show that several natural interacting Brownian particle models can thus be analyzed by studying the combinatorial fan generated by the drift function, particularly when these are simplicial. This is the case when the polyhedra is a polytope that is invariant under a Coxeter group action, which leads to an explicit description of the invariant measures in terms of iid Exponential random variables. Another class of examples is furnished by interactions indexed by weighted graphs all of which generate simplicial polytopes with n! faces. We show that the proportion of volume contained in each component simplex corresponds to a probability distribution on the group of permutations, some of which have surprising connections with the classical urn models.
QUASISTATIONARY RANDOM OVERLAP STRUCTURES AND THE CONTINUOUS CASCADES
, 806
"... Abstract. A random overlap structure (ROSt) is a measure on pairs (X, Q) where X is a locally finite sequence in R with a maximum and Q a positive semidefinite matrix of overlaps intrinsic to the particles X. Such a measure is said to be quasistationary provided that the joint law of the gaps of X ..."
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Abstract. A random overlap structure (ROSt) is a measure on pairs (X, Q) where X is a locally finite sequence in R with a maximum and Q a positive semidefinite matrix of overlaps intrinsic to the particles X. Such a measure is said to be quasistationary provided that the joint law of the gaps of X and overlaps Q is stable under a stochastic evolution driven by a Gaussian sequence with covariance Q. Aizenman et al. show in [1] that quasistationary ROSts serve as an important computational tool in the study of the SherringtonKirkpatrick (SK) spinglass model from the perspective of cavity dynamics and the related ROSt variational principle for its free energy. In this framework, the Parisi solution is reflected in the ansatz that the overlap matrix exhibit a certain hierarchical structure. Aizenman et al. pose the question in [1] of whether the ansatz could be explained by showing that the only ROSts that are quasistationary in a robust sense are given by a special class of hierarchical ROSts known as both the Ruelle Probability Cascades as well the GREM. Arguin and Aizenman give an affirmative answer in [5] and [2] in the special case that the set of values SQ taken on by the entries of Q is finite. We prove that this result holds even when SQ  = ∞ provided that Q satisfies the technical condition that SQ has no limit points from below. This is relevant to the understanding of the ground states of the SK model, as they satisfy SQ  = ∞. 1.
POISSONDIRICHLET STATISTICS FOR THE EXTREMES OF THE TWODIMENSIONAL DISCRETE GAUSSIAN FREE FIELD
, 2013
"... Abstract. In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a logcorrelated Gaussian field converge to a PoissonDirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions. The method is based on sh ..."
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Abstract. In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a logcorrelated Gaussian field converge to a PoissonDirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions. The method is based on showing that the model admits a onestep replica symmetry breaking in spin glass terminology. This implies PoissonDirichlet statistics by general spin glass arguments. In this note, this approach is used to prove PoissonDirichlet statistics for the twodimensional discrete Gaussian free field, where boundary effects demand a more delicate analysis. 1.