Damron–Gravner–Junge–Lyu–Sivakoff parking-model questions

Consider the parking process on $\mathbb{Z}^d$ in which each site initially contains a car with probability $p$ and a parking spot with probability $1-p$ , independently, and cars perform independent discrete-time simple random walks until they park at the first available spot they encounter. For a fixed site $0$ , let $V_n(0)$ be the number of visits to $0$ during the first $n$ rounds, let $V(0)=\lim_{n\to\infty}V_n(0)$ , and let $\tau$ denote a car's parking time. Determine the sharp asymptotic behavior of $\mathbb{E}_{1/2}[V_n(0)]$ as $n\to\infty$ , the tail $\mathbb{P}_p(\tau>t)$ for $0<p<1/2$ , and the divergence of $\mathbb{E}_p[V(0)]$ as $p\uparrow 1/2$ . The claimed target orders are $\mathbb{E}_{1/2}[V_n(0)]\asymp n^{(4-d)/4}$ for $d\leq 3$ and $\mathbb{E}_{1/2}[V_n(0)]\asymp\log n$ for $d\geq 4$ ; stretched-exponential parking-time bounds with exponent $d/(d+2)$ below criticality; and $\mathbb{E}_p[V(0)]\asymp(1-2p)^{-3}$ , $(1-2p)^{-1}$ , $(1-2p)^{-1/3}$ , and $\log\!\bigl(1/(1-2p)\bigr)$ in dimensions $d=1$ , $d=2$ , $d=3$ , and $d\geq4$ , respectively.

References

Primary source

arXiv

Additional references

Sharpness and critical scaling of parking — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new unrefereed preprint claims substantial progress on several lattice parking-process questions, but it does not establish that every original question is settled.

The questions concern visit counts, parking times, and critical behavior in the lattice parking process. Earlier work established important phase-transition and one-dimensional bounds, while higher-dimensional sharpness remained incomplete.

Known results

Damron, Lyu, and Sivakoff (2020): stretched-exponential parking-time bounds for small $p$ , with an explicit dimension-dependent threshold; finiteness of $\mathbb{E}_p V$ for all $p<1/2$ in dimensions $d\geq 2$ remained open.
Przykucki, Roberts, and Scott (2019): at $p=1/2$ in one dimension, $\mathbb{E}[\min\{\tau,t\}]$ lies between logarithmically corrected $t^{3/4}$ bounds; for $p<1/2$ , $\mathbb{E}[\tau]=O((1/2-p)^{-6})$ .
Earlier work established finite root visits for $p<1/2$ and infinite visits for $p\geq 1/2$ .

September 2026 preprint

A new manuscript reports sharp asymptotic orders for site visits at criticality, stretched-exponential parking-time bounds below criticality, and dimension-dependent divergence as $p\uparrow 1/2$ , linking these estimates to divisible-sandpile methods. These are claims from an unrefereed preprint, and its abstract does not specify which original questions are fully sharp.

Current status (as of September 2026): Earlier phase-transition and partial bounds are established, while the new preprint claims further sharp progress; independent verification and the exact scope of the resolution remain open.

Sources

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