Fluctuations of the Phase Boundary in the 2D Ising Ferromagnet

Dobrushin, R. L.; Hryniv, Ostap

doi:10.1007/s002200050209

Cited by 34 publications

(25 citation statements)

References 22 publications

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“…We call p a polymer using the terminology of a mathematical method called the cluster expansion, which we will use to prove our main theorem. A method of the cluster expansion is developed in the study of the statistical physics and is applied for instance to convergence theorems of the phase separation line of the two dimensional Ising model [4,12]. Since the cluster expansion is defined in an abstract setting [5], it can be applied to a financial model [7,14].…”

Section: Modelmentioning

confidence: 99%

Signs of Market Orders and Human Dynamics

Murai

2015

Proceedings of the International Conference on Social Modeling and Simulation, Plus Econophysics Colloquium 2014

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A time series of signs of market orders was found to exhibit long memory. There are several proposed explanations for the origin of this phenomenon. A cogent one is that investors tend to strategically split their large hidden orders into small pieces before execution to prevent the increase in the trading costs. Several mathematical models have been proposed under this explanation.In this paper, taking the bursty nature of the human activity patterns into account, we present a new mathematical model of order signs that have a long memory property. In addition, the power law exponent of distribution of a time interval between order executions is supposed to depend on the size of hidden order. More precisely, we introduce a discrete time stochastic process for polymer model, and show it's scaled process converges to a superposition of a Brownian motion and countably infinite number of fractional Brownian motions with Hurst exponents greater than one-half. IntroductionEmpirical studies [2,6,8,11] on high frequency financial data of stock markets that employ the continuous double auction method have revealed a time series of signs of market orders has long memory property. In contrast, a time series of stock returns is known to have short memory property. A time series of order signs is defined by changing transactions at the best ask price into C1 and transactions at the best bid price into 1. The auto-correlation function of the order signs decays as a power law of the lag and the exponent of the decay is less than 1, which is equivalent to a Hurst exponent of the time series is greater than one-half.In this paper, we propose a new mathematical model which takes account of origin of the long memory in order signs. As a first step, we define a discrete time stochastic process of cumulative order signs in accordance with some explanation

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Section: Modelmentioning

confidence: 99%

Signs of Market Orders and Human Dynamics

Murai

2015

Proceedings of the International Conference on Social Modeling and Simulation, Plus Econophysics Colloquium 2014

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“…Some interesting results in this area can be found in the literature (see, e.g., [8,1,9,5,6]), where, however, only vertical displacements of the phase separation line were investigated. Such a description is natural only if one considers horizontal or``almost horizontal'' interfaces.…”

Section: Introductionmentioning

confidence: 91%

“…x YI , P T u x I , could be still de®ned (for details, see [6]). In what follows we will refer to this distribution as to the ensemble of phase boundaries in the vertical strip x I (consistent with the boundary conditions r u ).…”

Section: Con®gurationsmentioning

confidence: 99%

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On local behaviour of the phase separation line in the 2D Ising model

Hryniv

1998

Probab Theory Relat Fields

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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 proportional to the stiness.

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“…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%

Cube-Root Boundary Fluctuations¶for Droplets in Random Cluster Models

Alexander¹

2001

Communications in Mathematical Physics

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Abstract. For a family of bond percolation models on Z 2 that includes the FortuinKasteleyn random cluster model, we consider properties of the "droplet" that results, in the percolating regime, from conditioning on the existence of an open dual circuit surrounding the origin and enclosing at least (or exactly) a given large area A. This droplet is a close surrogate for the one obtained by Dobrushin, Kotecký and Shlosman by conditioning the Ising model; it approximates an area-A Wulff shape. The local part of the deviation from the Wulff shape (the "local roughness") is the inward deviation of the droplet boundary from the boundary of its own convex hull; the remaining part of the deviation, that of the convex hull of the droplet from the Wulff shape, is inherently long-range. We show that the local roughness is described by at most the exponent 1/3 predicted by nonrigorous theory; this same prediction has been made for a wide class of interfaces in two dimensions. Specifically, the average of the local roughness over the droplet surface is shown to be O(l 1/3 (log l) 2/3 ) in probability, where l = √ A is the linear scale of the droplet. We also bound the maximum of the local roughness over the droplet surface and bound the long-range part of the deviation from a Wulff shape, and we establish the absense of "bottlenecks," which are a form of self-approach by the droplet boundary, down to scale log l. Finally, if we condition instead on the event that the total area of all large droplets inside a finite box exceeds A, we show that with probability near 1 for large A, only a single large droplet is present.

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Fluctuations of the Phase Boundary in the 2D Ising Ferromagnet

Cited by 34 publications

References 22 publications

Signs of Market Orders and Human Dynamics

Signs of Market Orders and Human Dynamics

On local behaviour of the phase separation line in the 2D Ising model

Cube-Root Boundary Fluctuations¶for Droplets in Random Cluster Models

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