Erdős–Taylor conjecture on the supremum of planar random-walk local times

For $x\in\mathbb{Z}^2$ and $N\geq 1$, let $\ell_x^N$ be the total number of visits to $x$ by a planar simple random walk started at the origin before exiting $[-N,N]^2$. Define the random Borel measure $\mu_N$ on $\mathbb{R}^2\times\mathbb{R}$ by

$$\mu_N(A\times T)=\sum_{x\in\mathbb{Z}^2}\mathbf{1}_{\{x/N\in A\}}\mathbf{1}_{\{\sqrt{\ell_x^N}-2\pi^{-1/2}\log N+\pi^{-1/2}\log\log N\in T\}}$$

for Borel sets $A\subset\mathbb{R}^2$ and $T\subset\mathbb{R}$. Let $\mu$ be the critical Brownian multiplicative chaos in $[-1,1]^2$ with the origin as starting point. Erdős–Taylor conjecture. There exist constants $c_1,c_2>0$ such that $(\mu_N,N\geq1)$ converges in distribution, for the topology of vague convergence on $\mathbb{R}^2\times(\mathbb{R}\cup\{+\infty\})$, towards

$$\operatorname{PPP}(c_1\mu\otimes c_2e^{-c_2t}\,dt).$$

In particular, for every $t\in\mathbb{R}$,

$$\mathbb{P}\left(\sup_{x\in\mathbb{Z}^2}\sqrt{\ell_x^N}\leq\frac{2}{\sqrt{\pi}}\log N-\frac{1}{\sqrt{\pi}}\log\log N+t\right)\xrightarrow[N\to\infty]{}\mathbb{E}\left[\exp\left(-c_1\mu([-1,1]^2)e^{-c_2t}\right)\right].$$

The conjecture concerns the extremal process and maximum of planar random-walk local times, whose logarithmic centering is predicted by the theory of extremes of log-correlated fields. The leading-order term $2\pi^{-1/2}\log N$ was conjectured by Erdős and Taylor and has been proved; the full extremal-process and limiting-law statement remains unresolved.