The one-big-excursion conjecture for conditioned return times

Let $G=(V,E)$ be a locally finite, connected, transitive, transient graph with spectral radius $\rho$, fix $o\in V$, and let $(X_n)_{n\geq 0}$ be simple random walk started at $o$. Let $f_n$ be the first-return probability, $F(z)=\sum_{n=1}^{\infty}f_nz^n$, and let $\mathsf{d}$ be the period. Conditioned on $\{X_n=X_0=o\}$, define the returning-time vectors $\mathsf{V}_n$ from the returns before and after the midpoint as in the source. Let $(\xi_i)_{i\geq 1}$ be i.i.d. with

$$\mathbb{P}[\xi_i=k]=\frac{f_k\rho^{-k}}{F(\rho^{-1})},$$

let $T_j$ be their partial sums, and let $L$ be independent with geometric parameter $1-F(\rho^{-1})$; take an independent copy for the hatted variables. The one-big-excursion conjecture. As $n\to\infty$ along $\mathsf{d}\mid n$, the conditional distribution of $\mathsf{V}_n$ converges to

$$\bigl((T_1,\ldots,T_L,0,0,\ldots),(\hat T_1,\ldots,\hat T_{\hat L},0,0,\ldots)\bigr).$$

This describes the conjectured limiting configuration of returns conditioned on a long return: with high probability, returns occur only in finite excursions near the two endpoints. The paper relates this conjecture to the first-return asymptotic conjecture and provides sufficient conditions for it, but does not prove it in full generality.