Results 1  10
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18
The RandomCluster Model
, 2008
"... 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 elec ..."
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Cited by 69 (21 self)
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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
Critical Region for Droplet Formation in the TwoDimensional Ising Model
, 2002
"... We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size L inverse temperature # > # c and overall magnetization conditioned to take the value m vL , w ..."
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Cited by 18 (8 self)
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We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size L inverse temperature # > # c and overall magnetization conditioned to take the value m vL , where # 1 c is the critical temperature, m (#) is the spontaneous magnetization and vL is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when v L L 2 tends to a definite limit. Specifically, we identify a dimensionless parameter #, proportional to this limit, a nontrivial critical value # c and a function ## such that the following holds: For # < # c , there are no droplets beyond log L scale, while for # > # c , there is a single, Wulffshaped droplet containing a fraction ## # c = 2/3 of the magnetization deficit and there are no other droplets beyond the scale of log L. Moreover, ## and # are related via a universal equation that apparently is independent of the details of the system.
SURFACE ORDER LARGE DEVIATIONS FOR 2D FKPERCOLATION AND POTTS MODELS
, 2003
"... By adapting the renormalization techniques of Pisztora, [32], we establish surface order large deviations estimates for FKpercolation on Z 2 with parameter q ≥ 1 and for the corresponding Potts models. Our results are valid up to the exponential decay threshold of dual connectivities which is wide ..."
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Cited by 9 (3 self)
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By adapting the renormalization techniques of Pisztora, [32], we establish surface order large deviations estimates for FKpercolation on Z 2 with parameter q ≥ 1 and for the corresponding Potts models. Our results are valid up to the exponential decay threshold of dual connectivities which is widely believed to agree with the critical point.
The spectral gap of the 2D stochastic Ising model with mixed boundary conditions
 J. Statist. Phys
"... Abstract. We establish upper bounds for the spectral gap of the stochastic Ising model at low temperature in an N ×N box, with boundary conditions which are “plus ” except for small regions at the corners which are either free or “minus. ” The spectral gap decreases exponentially in the size of the ..."
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Cited by 6 (1 self)
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Abstract. We establish upper bounds for the spectral gap of the stochastic Ising model at low temperature in an N ×N box, with boundary conditions which are “plus ” except for small regions at the corners which are either free or “minus. ” The spectral gap decreases exponentially in the size of the corner regions, when these regions are of size at least of order log N. This means that removing as few as O(log N) plus spins from the corners produces a spectral gap far smaller than the order N −2 gap believed to hold under the allplus boundary condition. Our results are valid at all subcritical temperatures.
Scaling limit and cuberoot fluctuations in sos surfaces above a wall
, 2013
"... Consider the classical (2 + 1)dimensional SolidOnSolid model above a hard wall on an L × L box of Z2. The model describes a crystal surface by assigning a nonnegative integer height ηx to each site x in the box and 0 heights to its boundary. The probability of a surface configuration η is prop ..."
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Cited by 5 (4 self)
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Consider the classical (2 + 1)dimensional SolidOnSolid model above a hard wall on an L × L box of Z2. The model describes a crystal surface by assigning a nonnegative integer height ηx to each site x in the box and 0 heights to its boundary. The probability of a surface configuration η is proportional to exp(−βH(η)), where β is the inversetemperature and H(η) sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability the height of almost all sites is concentrated on two levels, H(L) = ⌊(1/4β) logL ⌋ and H(L) − 1. Moreover, for most values of L the height is concentrated on the single value H(L). Next, we study the ensemble of level lines corresponding to the heights (H(L),H(L)−1,...). We prove that w.h.p. there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes Lk, the rescaled macroscopic level line at height H(Lk) − n has a limiting shape if the fractional parts of (1/4β) logLk converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square Q and its boundary has a flat component on the boundary of Q. Finally, the highest macroscopic level line has L 1/3+o(1) k fluctuations along the flat part of the boundary of its limiting shape.
On the 2d Ising Wulff crystal near criticality
, 2006
"... We study the behavior of the twodimensional Ising model in a finite box at temperatures that are below, but very close to, the critical temperature. In a regime where the temperature approaches the critical point and, simultaneously, the size of the box grows fast enough, we establish a large dev ..."
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Cited by 3 (0 self)
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We study the behavior of the twodimensional Ising model in a finite box at temperatures that are below, but very close to, the critical temperature. In a regime where the temperature approaches the critical point and, simultaneously, the size of the box grows fast enough, we establish a large deviation principle that proves the appearance of a round Wulff crystal.
Lower bounds for boundary roughness for droplets in Bernoulli percolation. Probab. Theory Related Fields
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Phase Coexistence for the Kac Ising Models
"... We derive the Wulff construction for Kac Ising models with long but finite range interaction in dimensions d > 2. Some open problems concerning the phase coexistence for more general models are also discussed. ..."
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Cited by 2 (1 self)
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We derive the Wulff construction for Kac Ising models with long but finite range interaction in dimensions d > 2. Some open problems concerning the phase coexistence for more general models are also discussed.
Separatedoccurrence inequalities for dependent percolation and Ising models
"... Abstract. Separatedoccurrence inequalities are variants for dependent lattice models of the van den BergKesten inequality for independent models. They take the form P(A◦rB) ≤ (1 + ce −ǫr)P(A)P(B), where A ◦r B is the event that A and B occur at separation r in a configuration ω, that is, there exi ..."
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Cited by 2 (0 self)
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Abstract. Separatedoccurrence inequalities are variants for dependent lattice models of the van den BergKesten inequality for independent models. They take the form P(A◦rB) ≤ (1 + ce −ǫr)P(A)P(B), where A ◦r B is the event that A and B occur at separation r in a configuration ω, that is, there exist two random sets of bonds or sites separated by at least distance r, one set responsible for the occurrence of the event A in ω, the other for the occurrence of B. We establish such inequalities for subcritical FK models, and for Ising models which are at supercritical temperature or have an external field, with A and B increasing or decreasing events.
The 2dIsing model near criticality: A FKpercolation analysis, ArXiv
, 2006
"... Abstract. We study the 2dIsing model defined on finite boxes at temperatures that are below but very close from the critical point. When the temperature approaches the critical point and the size of the box grows fast enough, we establish large deviations estimates on FKpercolation events that con ..."
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Cited by 2 (2 self)
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Abstract. We study the 2dIsing model defined on finite boxes at temperatures that are below but very close from the critical point. When the temperature approaches the critical point and the size of the box grows fast enough, we establish large deviations estimates on FKpercolation events that concern the phenomenon of phase coexistence. 1.