Identification of the Polaron measure I: Fixed coupling regime and the central limit theorem for large times

Abstract

We consider the Fröhlich model of the Polaron whose path integral formulation leads to the transformed path measure $${\hat{\mathbb{P}}}_{\alpha ,T}(\mathrm{d}\omega )=Z_{\alpha ,T}^{-1}\exp \left\{\frac{\alpha }{2}{\int }_{-T}^T{\int }_{-T}^T\frac{e^{-|t-s|}}{|\omega (t)-\omega (s)|}dsdt\right\}\mathbb{P}(\mathrm{d}\omega )$$ with respect to $\mathbb{P}$ which governs the law of the increments of the three dimensional Brownian motion on a finite interval $[-T,T]$, and $Z_{\alpha ,T}$ is the partition function or the normalizing constant and $\alpha >0$ is a constant. The Polaron measure reflects a self attractive interaction. According to a conjecture of Pekar that was proved in [DV83] $$g_0=\underset{\alpha \to \infty }{\lim }\frac{1}{{\alpha }^2}\left[\underset{T\to \infty }{\lim }\frac{\log Z_{\alpha ,T}}{2T}\right]$$ exists and has a variational formula. In this article we show that for any $\alpha >0$, the infinite-volume limit ${\hat{\mathbb{P}}}_{\alpha }={\lim }_{T\to \infty }{\hat{\mathbb{P}}}_{\alpha ,T}$ exists which is also identified explicitly. As a corollary, we deduce the central limit theorem (for any $\alpha >0$ and as $T\to \infty$) for the distribution of $\frac{\omega (T)-\omega (-T)}{\sqrt{2T}}$ both under the finite-volume Polaron measure ${\hat{\mathbb{P}}}_{\alpha ,T}$ and its infinite-volume counterpart ${\hat{\mathbb{P}}}_{\alpha }$, and obtain an expression for the limiting variance.

Theorem 4.5 (in the current version) was earlier stated (as Thm 4.8 in v3 and in the published version) for $\alpha \in (0,{\alpha }_0)\cup ({\alpha }_1,\infty )$, but its proof for $\alpha >{\alpha }_1$ there had a gap. In v4 (v5-v6 contain some simplifications) this gap has been fixed where Theorem 4.5 is shown for all coupling parameter $\alpha >0$. Consequently, the main results hold also for all $\alpha$

References in corpus (1)

• Identification of the Polaron measure in strong coupling and the Pekar variational formula

Cited by in corpus (4)

• Effective mass of the Polaron: a lower bound
• Identification of the Polaron measure in strong coupling and the Pekar variational formula
• Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron
• Asymptotic series for low-energy excitations of the Fröhlich Polaron at strong coupling