Trimmed-range large deviations for simple random walk

For $n\in\mathbb N$ and $A\in\mathbb N$, define the set of sites visited between one and $A$ times and its associated local time by

$$R_{n,A}^- = \{x\in\mathbb Z^d:1\leq\ell_n(x)\leq A\},\qquad \gamma_{n,A}^- = \sum_{x\in R_{n,A}^-}\ell_n(x).$$

Let $\lambda_d(A)$ and $J(A,s)$ denote the quantities in the conjecture. Trimmed-range conjecture. For every $A\in\mathbb N$ and $s\in[0,1]$ there exists $J(A,s)$ such that, for $\theta>0$ and $n\geq n_0(A,s,\theta)$,

$$P\bigl(|R_{n,A}^-|\geq s\theta n,\ \gamma_{n,A}\leq\theta n\bigr)\leq e^{-J(A,s)\theta n},$$

with

$$d=2:\quad J(A,s)>0\text{ for }s>0,$$

and, for $d\geq 3$,

$$J(A,s)=0\text{ for }0\leq s\leq1/\lambda_d(A),\qquad J(A,s)>0\text{ for }1/\lambda_d(A)<s\leq1.$$

Moreover,

$$d=2:\quad\lim_{s\downarrow0}-s\log J(A(s),s)=\frac1{\lambda_2},\quad A(s)\gg s^{-10},$$

and, for $d\geq3$, $\lambda_d(A)<\lambda_d$ with $\lim_{A\to\infty}\lambda_d(A)=\lambda_d$. This conjecture concerns upward large deviations of the range after trimming sites with excessive local time. The paper states that it would supply the technical estimate needed for the sharp small-bias critical-curve asymptotics; it remains open.