Glassy-phase convergence to a stable random measure

For $\gamma^2>2d$, let

$$M_t^\gamma(dx)=e^{\gamma X_t(x)-\frac{\gamma^2}{2}\mathbb{E}[X_t(x)^2]}\\,dx,$$

set $\alpha=\frac{\sqrt{2d}}{\gamma}\in(0,1)$, and let $M'$ be the derivative-martingale measure. Let $N_\alpha$ be an independently scattered random measure whose conditional law given $M'$ is characterized by

$$\mathbb{E}\left[e^{-qN_\alpha(A)}\mid M'\right]=e^{-q^\alpha M'(A)}$$

for every $A\in\mathcal{B}(\mathbb{R}^d)$ and $q\;\geqslant\;0$. Glassy-phase convergence conjecture. There is a positive constant $c_\gamma$ depending on $\gamma$ such that

$$t^{\frac{3\gamma}{2\sqrt{2d}}}e^{t\left(\frac{\gamma}{\sqrt{2}}-\sqrt d\right)^2}M_t^\gamma(dx)\stackrel{\mathrm{law}}{\longrightarrow}c_\gamma N_\alpha(dx),\qquad t\to\infty.$$

The preceding proposition proves tightness and nontriviality of convergent subsequences, while this conjecture strengthens that statement to convergence and identifies the limit. The source notes that the analogous result is proved for branching random walks.