Let (X_n)_n≥ 0 be a centrally excited random walk on Z^d, d≥ 2, which otherwise moves as simple random walk, and let V_n=X_0,X_1,…,X_n. Kozma conjectured that, as n→∞, the visited set V_n approaches a deterministic Euclidean ball whose radius has order n^1/(d+1); equivalently, its spatial scale is Theta!(n^1/(d+1)).

Kozma’s centrally excited walk conjecture

Let $(X_n)_{n\ge 0}$ be a centrally excited random walk on $\mathbb{Z}^d$ , $d\ge 2$ , which otherwise moves as simple random walk, and let $V_n=\{X_0,X_1,\ldots,X_n\}$ . Kozma conjectured that, as $n\to\infty$ , the visited set $V_n$ approaches a deterministic Euclidean ball whose radius has order $n^{1/(d+1)}$ ; equivalently, its spatial scale is $\Theta\!\left(n^{1/(d+1)}\right)$ .

References

Primary source

arXiv

Additional references

Kozma’s centrally excited walk converges to a Euclidean ball — arXiv — Ahmed Bou-Rabee, Yuval Peres

Posted 13 hours ago · edited 13 hours ago

Progress summary

Refreshed 13 hours ago

Claimed progress

A recent unrefereed preprint claims a limit-shape theorem for a specific centrally excited random walk, but the broader conjecture remains unsettled.

Kozma’s conjecture, raised in 2007, predicts a deterministic asymptotic shape for the visited set and visit counts of a centrally excited walk, together with extensions to norm-induced drifts.

Recent shape theorem

Ahmed Bou-Rabee and Yuval Peres claim convergence for the origin-biased walk on $\mathbb{Z}^d$ , $d\ge 2$ : after time scaling by $R^{d+1}$ and spatial scaling by $R^{-1}$ , the visited set approaches an explicit $\ell^1$ ball, with a linearly decreasing limiting local-time profile. The retrieved mathematical text does not establish the Euclidean-ball formulation or the claimed general norm-induced extension; no independent assessment was found.

Current status (as of October 2026): A specialist preprint claims substantial progress for the origin-biased model, but the full Euclidean-shape and norm-induced-drift conjecture remains open and the claim is unverified.

Sources

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