A Short Note on the Scaling Function Constant Problem in the Two-Dimensional Ising Model

Thomas Bothner

doi:10.1007/s10955-017-1947-z

A Short Note on the Scaling Function Constant Problem in the Two-Dimensional Ising Model

Thomas Bothner

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We provide a simple derivation of the constant factor in the short-distance asymptotics of the tau-function associated with the 2-point function of the two-dimensional Ising model. This factor was first computed by Tracy (Commun Math Phys 142:297–311, 1991) via an exponential series expansion of the correlation function. Further simplifications in the analysis are due to Tracy and Widom (Commun Math Phys 190:697–721, 1998) using Fredholm determinant representations of the correlation function and Wiener–Hopf approximation
results for the underlying resolvent operator. Our method relies on an action integral representation of the tau-function and asymptotic results for the underlying Painlevé-III transcendent from McCoy et al. (J Math Phys 18:1058–1092, 1977).

Original language	English
Pages (from-to)	672-683
Number of pages	12
Journal	Journal of Statistical Physics
Volume	170
Issue number	4
Early online date	19 Dec 2017
DOIs	https://doi.org/10.1007/s10955-017-1947-z
Publication status	Published - Feb 2018

Keywords

two-dimensional Ising model
2-Point function
short distance expansion
action integral

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10.1007/s10955-017-1947-z

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abstract = "We provide a simple derivation of the constant factor in the short-distance asymptotics of the tau-function associated with the 2-point function of the two-dimensional Ising model. This factor was first computed by Tracy (Commun Math Phys 142:297–311, 1991) via an exponential series expansion of the correlation function. Further simplifications in the analysis are due to Tracy and Widom (Commun Math Phys 190:697–721, 1998) using Fredholm determinant representations of the correlation function and Wiener–Hopf approximationresults for the underlying resolvent operator. Our method relies on an action integral representation of the tau-function and asymptotic results for the underlying Painlev{\'e}-III transcendent from McCoy et al. (J Math Phys 18:1058–1092, 1977).",

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N2 - We provide a simple derivation of the constant factor in the short-distance asymptotics of the tau-function associated with the 2-point function of the two-dimensional Ising model. This factor was first computed by Tracy (Commun Math Phys 142:297–311, 1991) via an exponential series expansion of the correlation function. Further simplifications in the analysis are due to Tracy and Widom (Commun Math Phys 190:697–721, 1998) using Fredholm determinant representations of the correlation function and Wiener–Hopf approximationresults for the underlying resolvent operator. Our method relies on an action integral representation of the tau-function and asymptotic results for the underlying Painlevé-III transcendent from McCoy et al. (J Math Phys 18:1058–1092, 1977).

AB - We provide a simple derivation of the constant factor in the short-distance asymptotics of the tau-function associated with the 2-point function of the two-dimensional Ising model. This factor was first computed by Tracy (Commun Math Phys 142:297–311, 1991) via an exponential series expansion of the correlation function. Further simplifications in the analysis are due to Tracy and Widom (Commun Math Phys 190:697–721, 1998) using Fredholm determinant representations of the correlation function and Wiener–Hopf approximationresults for the underlying resolvent operator. Our method relies on an action integral representation of the tau-function and asymptotic results for the underlying Painlevé-III transcendent from McCoy et al. (J Math Phys 18:1058–1092, 1977).

KW - two-dimensional Ising model

KW - 2-Point function

KW - short distance expansion

KW - action integral

U2 - 10.1007/s10955-017-1947-z

DO - 10.1007/s10955-017-1947-z

M3 - Article (Academic Journal)

VL - 170

SP - 672

EP - 683

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 4

ER -

A Short Note on the Scaling Function Constant Problem in the Two-Dimensional Ising Model

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