1998
DOI: 10.1007/s004400050146
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On local behaviour of the phase separation line in the 2D Ising model

Abstract: The aim of this note is to discuss some statistical properties of the phase separation line in the 2D low-temperature Ising model. We prove the functional central limit theorem for the probability distributions describinḡ uctuations of the phase boundary in the direction orthogonal to its orientation. The limiting Gaussian measure corresponds to a scaled Brownian bridge with direction dependent parameters. Up to the temperature factor, the variances of local increments of this limiting process are inversely pr… Show more

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Cited by 22 publications
(17 citation statements)
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“…Though our error term is likely not optimal-according to [15] the optimal error term may be of order log l-it is enough of an improvement over the corresponding results in [11] and [16] to enable us to establish an apparently near-optimal bound on the local roughness.…”
Section: Lower Bounds For Open Dual Circuit Probabilitiesmentioning
confidence: 88%
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“…Though our error term is likely not optimal-according to [15] the optimal error term may be of order log l-it is enough of an improvement over the corresponding results in [11] and [16] to enable us to establish an apparently near-optimal bound on the local roughness.…”
Section: Lower Bounds For Open Dual Circuit Probabilitiesmentioning
confidence: 88%
“…The main result of this paper is that in the random cluster model context, with probability approaching 1 as l → ∞, the average local roughness is O(l 1/3 (log l) 2/3 ). Results of Dobrushin and Hryniv [10] and Hryniv [15] (at very low temperatures) strongly suggest that the the fluctuations of the droplet boundary about the shrunken Wulff shape should be Gaussian, heuristically resembling roughly a rescaled Brownian bridge added radially to the Wulff shape. In particular, the long-wave fluctuations should be of order l 1/2 .…”
Section: Introductionmentioning
confidence: 97%
“…without the area term. From [29] we know that for L → ∞ and rescaling the horizontal (resp. vertical) space direction by L −(2/3−ǫ) (resp.…”
Section: Proof Of Theorems 2 Andmentioning
confidence: 99%
“…Dobrushin [61] proved that at low temperatures, interfaces perpendicular to the axis directions are localized (see also [83] for the extension to FK percolation). Refined estimates are available in dimensions two for the Ising model [62,63,89] and for Bernoulli percolation [5]. As far as we know, there is still no proof or disproof for the existence of this transition in dimensions d = 3.…”
Section: What Do We Know About the Wulff Crystal?mentioning
confidence: 99%
“…By using Pfister's approach and certain coarse-graining techniques from [112], Ioffe [90,91] extended the basic large deviation principle for the magnetization up to the critical temperature. Several other works investigated the phase separation phenomenon in dimension 2: [45,62,89,111,118,119,120]. Several other works investigated the phase separation phenomenon in dimension 2: [45,62,89,111,118,119,120].…”
Section: Bibliographical Commentsmentioning
confidence: 99%