### Abstract

We consider the Fröhlich model of the polaron, whose path integral formulation leads to the transformed path measure (Formula presented.) with respect to ℙ that governs the law of the increments of the three-dimensional Brownian motion on a finite interval [−T, T], and Z_{α, 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], (Formula presented.) exists and has a variational formula. In this article we show that when α > 0 is either sufficiently small or sufficiently large, the limit (Formula presented.) exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for (Formula presented.) under (Formula presented.) and obtain an expression for the limiting variance.

Original language | English (US) |
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Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21858

**Identification of the Polaron Measure I : Fixed Coupling Regime and the Central Limit Theorem for Large Times.** / Mukherjee, Chiranjib; Varadhan, Srinivasa.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Identification of the Polaron Measure I

T2 - Fixed Coupling Regime and the Central Limit Theorem for Large Times

AU - Mukherjee, Chiranjib

AU - Varadhan, Srinivasa

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We consider the Fröhlich model of the polaron, whose path integral formulation leads to the transformed path measure (Formula presented.) 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], (Formula presented.) exists and has a variational formula. In this article we show that when α > 0 is either sufficiently small or sufficiently large, the limit (Formula presented.) exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for (Formula presented.) under (Formula presented.) and obtain an expression for the limiting variance.

AB - We consider the Fröhlich model of the polaron, whose path integral formulation leads to the transformed path measure (Formula presented.) 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], (Formula presented.) exists and has a variational formula. In this article we show that when α > 0 is either sufficiently small or sufficiently large, the limit (Formula presented.) exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for (Formula presented.) under (Formula presented.) and obtain an expression for the limiting variance.

UR - http://www.scopus.com/inward/record.url?scp=85070717004&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85070717004&partnerID=8YFLogxK

U2 - 10.1002/cpa.21858

DO - 10.1002/cpa.21858

M3 - Article

AN - SCOPUS:85070717004

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -