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Large deviations for the one-dimensional Edwards model. (English) Zbl 1052.60019
The authors present the study of a \(T\)-polymer measure \(Q_{T}^{\beta}\) with strength of self-repellence \(\beta\), given by the Edwards model \[ \frac{dQ_{T}^{\beta}}{dP}(\cdot)=\frac{1}{Z_{T}^{\beta}}e^{-\beta H_{T}(\cdot)}. \] Here \(Z_{T}^{\beta}=E(e^{-\beta H_{T}})\), and \(H_{T}(\{B_{t}\}_{t\in[0,T]}) =\int_{R}dx L(T,x)^{2}\) is the Brownian intersection local time up to time \(T\). The authors [Ann. Probab. 25, No. 2, 573–597 (1997; Zbl 0873.60009)] proved a central limit theorem for this model. In the present paper they prove a large deviations principle. The rate function is given in terms of the principal eigenvalues of a one-parameter family of Strum-Liouville operators. The proof is based on the description of the joint distribution of the local times at the endpoint at a fixed time \(T\) in terms of a combination of Ray-Knight theorems. The main idea is that, conditional on \(B_{T}\), \(L(T,B_{T})\) and \(L(T,0)\), “middle part” of the local times is a two-dimensional squared Bessel process, while the two “boundary parts” are two zero-dimensional squared Bessel processes. One also uses a certain Girsanov transformation of the “middle part”.

MSC:
60F05 Central limit and other weak theorems
60F10 Large deviations
60J55 Local time and additive functionals
82D60 Statistical mechanics of polymers
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