## Abstract

The path measure corresponding to the *Fröhlich polaron* appearing in quantum statistical mechanics is defined as the tilted measure \begin{equation*}\mathrm{d}\widehat{\mathbb{P}}_{\varepsilon ,T}=\frac{1}{Z(\varepsilon ,T)}\exp \biggl\{\frac{1}{2}\int _{-T}^{T}\int _{-T}^{T}\frac{\varepsilon \mathrm{e}^{-\varepsilon \vert t-s\vert }}{\vert \omega (t)-\omega (s)\vert }\,\mathrm{d}s\,\mathrm{d}t\biggr\}\,\mathrm{d}\mathbb{P}.\end{equation*} Here, $\varepsilon >0$ is a constant known as the *Kac parameter* or the *inverse-coupling parameter*, and $\mathbb{P}$ is the distribution of the increments of the three-dimensional Brownian motion. In (*Comm. Pure Appl. Math.* **73** (2020) 350–383) it was shown that, when $\varepsilon >0$ is sufficiently small or sufficiently large, the (thermodynamic) limit $\lim_{T\to \infty }\widehat{\mathbb{P}}_{\varepsilon ,T}={\widehat{\mathbb{P}}}_{\varepsilon }$ exists as a process with stationary increments, and this limit was identified explicitly as a mixture of Gaussian processes. In the present article the *strong coupling limit* or the *vanishing Kac parameter* limit $\lim_{\varepsilon \to 0}\widehat{\mathbb{P}}_{\varepsilon }$ is investigated. It is shown that this limit exists and coincides with the *increments* of the so-called *Pekar process*, a stationary diffusion with generator $\frac{1}{2}\Delta +(\nabla \psi /\psi )\cdot \nabla $, where $\psi $ is the unique (up to spatial translations) maximizer of the *Pekar variational problem* \begin{equation*}g_{0}=\mathop{\mathrm{sup}}_{\Vert \psi \Vert _{2}=1}\biggl\{\int _{\mathbb{R}^{3}}\int _{\mathbb{R}^{3}}\psi ^{2}(x)\psi ^{2}(y)\vert x-y\vert ^{-1}\,\mathrm{d}x\,\mathrm{d}y-\frac{1}{2}\Vert \nabla \psi \Vert _{2}^{2}\biggr\}.\end{equation*} As the Pekar process was also earlier shown (*Ann. Probab.* **44** (2016) 3934–3964; *Ann. Inst. Henri Poincaré Probab. Stat.* **53** (2017) 2214–2228; *Comm. Pure Appl. Math.* **70** (2017) 1598–1629) to be the limiting object of the *mean-field polaron measures*, the present identification of the strong coupling limit is a rigorous justification of the mean-field approximation of the polaron problem (on the level of path measures) conjectured by Spohn in (*Ann. Physics* **175** (1987) 278–318). Replacing the Coulomb potential by continuous function vanishing at infinity and assuming uniqueness (modulo translations) of the relevant variational problem, our proof also shows that path measures coming from a Kac interaction of the above form with translation invariance in space converge to the increments of the corresponding mean-field model.

## Citation

Chiranjib Mukherjee. S. R. S. Varadhan. "Identification of the Polaron measure in strong coupling and the Pekar variational formula." Ann. Probab. 48 (5) 2119 - 2144, September 2020. https://doi.org/10.1214/19-AOP1392

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