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

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.