Randomly shifted Gumbel convergence conjecture for the maximum of Ginzburg–Landau fields

Let $Q_N$ be the domain, let $\boldsymbol{P}^{Q_N,0}$ denote the law in the notation of Theorem 1.1, and let $\boldsymbol{g}$ and $m_N$ be the corresponding constants and centering term. Set

$$c^* = \frac{2}{\sqrt{\boldsymbol{g}}}.$$

Randomly shifted Gumbel convergence conjecture. There exists a random variable $Z>0$ such that

$$\lim_{N\to\infty}\boldsymbol{P}^{Q_N,0}\left(\max_{x\in Q_N}\phi(x)-\sqrt{\boldsymbol{g}}m_N\leq t\right)=\mathbb{E}\left(\exp\left(-Z\exp(-c^*t)\right)\right),\qquad \forall t\in\mathbb{R}.$$

Theorem 1.1 establishes the tightness of the centered maximum, and this conjecture predicts the stronger convergence in distribution to a randomly shifted Gumbel law. The existence and identification of the limiting random shift remain open.