Painleve Functions of the Third Kind
Nov, 1976138 pages
Published in:
- J.Math.Phys. 18 (1977) 1058
DOI:
Report number:
- ITP-SB-76-21
Citations per year
Abstract: (AIP)
We explicitly construct the one‐parameter family of solutions, η (ϑ;ν,λ), that remain bounded as ϑ→∞ along the positive real ϑ axis for the Painlevé equation of third kind w w′′= (w′)2−ϑ−1 w w′+2νϑ−1(w 3−w) +w 4−1, where, as ϑ→∞, η ∼ 1−λΓ (ν+1/2)2−2νϑ−ν−1/2 e −2ϑ. We further construct a representation for ψ (t;ν,λ) =−ln[η (t/2;ν,λ)], where ψ (t;ν,λ) satisfies the differential equation ψ′′+t −1ψ′= (1/2)sinh(2ψ)+2νt −1 sinh(ψ). The small‐ϑ behavior of η (ϑ;ν,λ) is described for ‖λ‖<π−1 by η (ϑ;ν,λ) ∼ 2σ Bϑσ. The parameters σ and B are given as explicit functions of λ and ν. Finally an identity involving the Painlevé transcendent η (ϑ;ν,λ) is proved. These results for the special case ν=0 and λ=π−1 make rigorous the analysis of the scaling limit of the spin–spin correlation function of the two‐dimensional Ising model.- STATISTICAL MECHANICS: ISING
- CORRELATION: SPIN
- SPIN: CORRELATION
- SCALING
- MATHEMATICS
- NUMERICAL CALCULATIONS
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