Results 1  10
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21
Correlation function of Schur process with application to local geometry of a random 3dimensional Young Diagram
, 2001
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The RandomCluster Model
, 2006
"... Abstract. The class of randomcluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, an ..."
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Cited by 43 (18 self)
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Abstract. The class of randomcluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the randomcluster representation. This systematic summary of randomcluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinitevolume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for twodimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the clusterweighting factor q, and the problem of proving that the critical randomcluster model in two
Universality for mathematical and physical systems
, 2006
"... Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all ..."
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Cited by 21 (0 self)
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Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner’s model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting. 1.
Step fluctuations for a faceted crystal
"... A statistical mechanics model for a faceted crystal is the 3D Ising model at zero temperature. It is assumed that in one octant all sites are occupied by atoms, the remaining ones being empty. Allowed atom configurations are such that they can be obtained from the filled octant through successive re ..."
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Cited by 16 (0 self)
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A statistical mechanics model for a faceted crystal is the 3D Ising model at zero temperature. It is assumed that in one octant all sites are occupied by atoms, the remaining ones being empty. Allowed atom configurations are such that they can be obtained from the filled octant through successive removals of atoms with breaking of precisely three bonds. If V denotes the number of atoms removed, then the grand canonical Boltzmann weight is q V, 0 < q < 1. As shown by Cerf and Kenyon, in the limit q → 1 a deterministic shape is attained, which has the three facets (100), (010), (001), and a rounded piece interpolating between them. We analyse the step statistics as q → 1. In the rounded piece it is given by a determinantal process based on the discrete sinekernel. Exactly at the facet edge, the steps have more space to meander. Their statistics is again determinantal, but this time based on the Airykernel. In particular, the border step is well approximated by the Airy process, which has been obtained previously in the context of growth models. Our results are based on the asymptotic analysis for spacetime inhomogeneous transfer matrices. 1
RANDOM SAMPLING OF PLANE PARTITIONS
"... abstract. This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is O(n(ln n) ..."
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Cited by 4 (2 self)
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abstract. This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is O(n(ln n) 3) in approximatesize sampling and O(n4/3) in exactsize sampling (under a realarithmetic computation model). To our knowledge, these are the first polynomialtime samplers for plane partitions according to the size (there exist polynomialtime samplers of another type, which draw plane partitions that fit inside a fixed bounding box). The same principles yield efficient samplers for (a × b)boxed plane partitions (plane partitions with two dimensions bounded), and for skew plane partitions. The random samplers allow us to perform simulations and observe limit shapes and frozen boundaries, which have been analysed recently by Cerf and Kenyon for plane partitions, and by Okounkov and Reshetikhin for skew plane partitions.
Dbranes as defects in the CalabiYau crystal
"... We define the notion of Amodel Lagrangian Dbranes as introducing defects in the CalabiYau crystal. The crystal melting in the presence of these defects reproduces all genus string amplitudes as well as leads to additional nonperturbative terms. Contents 1 ..."
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Cited by 3 (0 self)
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We define the notion of Amodel Lagrangian Dbranes as introducing defects in the CalabiYau crystal. The crystal melting in the presence of these defects reproduces all genus string amplitudes as well as leads to additional nonperturbative terms. Contents 1
Zero” temperature stochastic 3D Ising model and Dimer covering fluctuation: a first step towards mean curvature motion
 Comm. Pure Appl. Math
"... Abstract. We consider the Glauber dynamics for the Ising model with “+ ” boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of “− ” spins disa ..."
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Cited by 3 (3 self)
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Abstract. We consider the Glauber dynamics for the Ising model with “+ ” boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of “− ” spins disappears within a time τ+ which is at most L 2 (log L) c and at least L 2 /(c log L), for some c> 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large timescales, the evolution of the interface between “+ ” and “− ” domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior τ+ ≃ const × L 2, conjectured on heuristic grounds [13, 7]. In dimension d = 2, τ+ can be shown to be of order L 2 without logarithmic corrections: the upper bound was proven in [8] and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [3].
Critical resonance in the . . .
, 2004
"... We study the phase transition in the honeycomb dimer model (equivalently, monotone nonintersecting lattice path model). At the critical point the system has a strong longrange dependence; in particular, periodic boundary conditions give rise to a “resonance” phenomenon, where the partition functio ..."
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Cited by 1 (0 self)
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We study the phase transition in the honeycomb dimer model (equivalently, monotone nonintersecting lattice path model). At the critical point the system has a strong longrange dependence; in particular, periodic boundary conditions give rise to a “resonance” phenomenon, where the partition function and other properties of the system depend sensitively on the shape of the domain.