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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”.

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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[1] Abramowitz, M. and Stegun, I. (1970). Handbook of Mathematical Functions , 9th ed. Dover, New York. · Zbl 0171.38503
[2] Bolthausen, E. (1993). On the construction of a three-dimensional polymer measure. Probab. Theory Related Fields 97 81–101. · Zbl 0794.60104
[3] Coddington, E. A. and Levinson, N. (1955). Theory of Ordinary Differential Equations . McGraw-Hill, New York. · Zbl 0064.33002
[4] Dembo, A. and Zeitouni, O. (1996). Large Deviations Techniques and Applications . Jones and Bartlett, Boston. · Zbl 0793.60030
[5] Erdélyi, A. (1956). Asymptotic Expansions . Dover, Toronto. · Zbl 0070.29002
[6] Griffel, D. H. (1981). Applied Functional Analysis . Wiley, New York. · Zbl 0461.46001
[7] van der Hofstad, R. (1998). On the constants in the central limit theorem for the one-dimensional Edwards model. J. Statist. Phys. 90 1295–1306. · Zbl 0930.60089
[8] van der Hofstad, R. and den Hollander, F. (1995). Scaling for a random polymer. Comm. Math. Phys. 169 397–440. · Zbl 0821.60078
[9] van der Hofstad, R. and König, W. (2001). A survey of one-dimensional random polymers. J. Statist. Phys. 103 915–944. · Zbl 1126.82313
[10] van der Hofstad, R., den Hollander, F. and König, W. (1997). Central limit theorem for the Edwards model. Ann. Probab. 25 573–597. · Zbl 0873.60009
[11] van der Hofstad, R., den Hollander, F. and König, W. (2002). Weak interaction limits for one-dimensional random polymers. Probab. Theory Related Fields 125 483–521. · Zbl 1037.60023
[12] den Hollander, F. (2000). Large Deviations . Amer. Math. Soc., Providence, RI. · Zbl 0949.60001
[13] Lawler, G., Schramm, O. and Werner, W. (2002). On the scaling limit of planar self-avoiding walk. · Zbl 1069.60089
[14] Madras, N. and Slade, G. (1993). The Self-Avoiding Walk . Birkhäuser, Boston. · Zbl 0780.60103
[15] March, P. and Sznitman, A.-S. (1987). Some connections between excursion theory and the discrete Schrödinger equation with random potentials. Probab. Theory Related Fields 75 11–53. · Zbl 0592.60046
[16] Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion , 2nd ed. Springer, Berlin. · Zbl 0804.60001
[17] Vanderzande, C. (1998). Lattice Models of Polymers . Cambridge Univ. Press. · Zbl 0922.76004
[18] Varadhan, S. R. S. (1969). Appendix to “Euclidean quantum field theory,” by K. Symanzik. In Local Quantum Field Theory (R. Jost, ed.). Academic Press, New York.
[19] Westwater, J. (1981). On the Edwards model for long polymer chains. II. The self-consistent potential. Comm. Math. Phys. 79 53–73. · Zbl 0477.60100
[20] Westwater, J. (1984). On Edwards’ model for polymer chains. In Trends and Developments in the Eighties (S. Albeverio and Ph. Blanchard, eds.) 384–404. World Scientific, Singapore. · Zbl 0583.60066
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