Seneta–Heyde scaling conjecture for the critical local-time measure on a regular tree

Let $\mathbb L_n$ be the set of vertices at level $n$ of the rooted regular tree with branching factor $b$, let $L_t(x)$ denote the local time at vertex $x$ up to time $t$ for simple random walk started at the root $\varrho$, and let $Z(t)$ be the limiting critical measure defined in the paper. Seneta–Heyde scaling conjecture. There is $c>0$ such that for all $t>0$, under $P^\varrho$,

$$n b^{-2n} \sum_{x \in \mathbb L_n} \operatorname{e}^{2\sqrt{\log b}\,\sqrt{L_t(x)}} \underset{t\to\infty}{\overset{\operatorname{law}}{\longrightarrow}} c Z(t).$$

This conjectures the critical normalization for the exponential local-time measure, with the extra factor $n$ accounting for the critical Seneta–Heyde scaling. The surrounding discussion relates it to the critical Gaussian multiplicative chaos and derivative-martingale constructions for the Gaussian free field on the tree; the statement is presented as an expectation rather than as an established result.