Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times |
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Authors: | Chiranjib Mukherjee S R S Varadhan |
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Institution: | 1. Department of Mathematics, University of Münster, Einsteinstr. 62 Münster 48147, Germany;2. Courant Institute, 251 Mercer St., Room 1313, New York, NY 10012 USA |
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Abstract: | We consider the Fröhlich model of the polaron, whose path integral formulation leads to the transformed path measure with respect to ℙ that governs the law of the increments of the three-dimensional Brownian motion on a finite interval −T, T] , and Zα, T is the partition function or the normalizing constant and α > 0 is a constant, or the coupling parameter. The polaron measure reflects a self-attractive interaction. According to a conjecture of Pekar that was proved in 9], exists and has a variational formula. In this article we show that when α > 0 is either sufficiently small or sufficiently large, the limit exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for under and obtain an expression for the limiting variance. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, LLC. |
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