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The Thermodynamic Bethe Ansatz equation is an integral equation that was derived by Yang and Yang to study some interacting systems. In the simplest case, it is $$\epsilon(\beta)=R\cosh\beta-\int\frac{d\beta'}{\pi\cosh(\beta-\beta')}\log(1+\exp(-\epsilon(\beta'))).$$

A reference for this is Al.B. Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models, Nucl Phys B342 (1990) 695-720.

I am interested in the asymptotics of the solutions as $R\rightarrow 0$. Zamolodchikov gives a heuristic argument that for small $R\cosh\beta$ we can neglect the first term. Therefore, the $\beta\rightarrow\beta+const$ invariance is restored and the solution $\epsilon(\beta)$ becomes independent of $\beta$ for small $R\cosh\beta$. Can one deduce a more precise asymptotic behavior? In particular, I would like to write down the small $R$ corrections.

One can easily show that this integral operator acting on $\exp(-\epsilon)$ maps the ball of radius $(e^R-1)^{-1}$ in $C^0(\mathbf{R})$ to itself. Furthermore, it is a contraction for large $R$ (see e.g. http://arxiv.org/abs/0807.4723, appendix C).

Was this equation studied anywhere in the mathematical literature?

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Some pointers to the literature that may help: The small-$R$ asymptotics of the Thermodynamic Bethe Ansatz equation can be expressed in terms of a Painleve III function with independent variable $R$ [1], and the small-$R$ asymptotics has been studied in that connection [2]. The $R\rightarrow 0$ limit (called the "massless" or "ultraviolet" limit in the physics literature) has the form of an Airy function [3].

[1] C.A. Tracy and H. Widom, Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz, Commun.Math.Phys. 179 (1996) 667-680 [arXiv:solv-int/9509003].

[2] P. Fendley and H. Saleur, $N=2$ Supersymmetry, Painleve III and Exact Scaling Functions in 2D Polymers, Nucl.Phys.B 388 (1992) 609-626 [arXiv:hep-th/9204094].

[3] P. Fendley, Airy functions in the thermodynamic Bethe ansatz, Lett.Math.Phys. 49 (1999) 229-233 [arXiv:hep-th/9906114].

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The first and third papers deal with a slightly different version they call the $N=2$ TBA equation. Do their results generalize to other TBA-type equations (in particular, the one I mentioned)? –  Pavel Safronov Oct 30 '11 at 22:08
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