Low- and high-temperature behavior of the interacting path process

Let $\Upsilon$ be the space of polygonal paths equipped with the push-forward a priori probability measure $p$, let $f$ be the potential

$$f(\gamma_1,\gamma_2,\ldots)=-\sum_{n=1}^{\infty}J(n)\frac{d_{\mathbb{H}}^{\alpha}(\gamma_1,\gamma_n)}{1+d_{\mathbb{H}}^{\alpha}(\gamma_1,\gamma_n)},$$

where $J(n)\geq 0$ tends to zero sufficiently fast, $0<\alpha<1$, and $d_{\mathbb{H}}$ is the Hausdorff distance. For each inverse temperature $\beta>0$, let $\mu_{\beta f}$ be the corresponding equilibrium measure. Low- and high-temperature behavior conjecture. At very low temperatures, $\beta\gg1$, a typical configuration should be an infinite collection of paths close to one another and close to the origin; at very high temperatures, $0<\beta\ll1$, a typical configuration for $\mu_{\beta}$ should be similar to an infinite collection of independent diffusive paths. This is a qualitative prediction for the equilibrium measures of the infinite interacting path process; the source does not provide a proof or a resolution of either asymptotic regime.