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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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