Consider p(v, t), the probability that a walk beginning from the origin ends at the point v on the d-dimensional integer

Definitions and mathematical statement

Let $(S_t)_{t\ge0}$ be simple symmetric nearest-neighbor random walk on $\mathbb Z^d$, with $S_0=0$ and independent increments uniformly distributed over $\{\pm e_1,\ldots,\pm e_d\}$. For a fixed site $v\in\mathbb Z^d$, write $p(v,t)=\Pr(S_t=v)$ and $m=\|v\|_1=\sum_i|v_i|$. The probability is zero when $t<m$ or $t\not\equiv m\pmod2$. Remove these forced parity zeros by putting \[ q_j=p(v,m+2j)\qquad(j\ge0). \] Is this sequence unimodal for every $d\ge1$ and every site $v$? Precisely, must there be an integer $J\ge0$ with \[ q_0\le q_1\le\cdots\le q_J \quad\text{and}\quad q_J\ge q_{J+1}\ge q_{J+2}\ge\cdots\,? \] Equal adjacent values are permitted, so a plateau at the maximum does not violate unimodality.

Dimension and target site remain fixed as time varies. The question concerns the probability of being at the site at a given time, including trajectories that visited it earlier; it is not a first-hitting-time distribution. Deleting the parity-forced zeros is essential, because the full sequence usually alternates between positive values and zero. No monotonicity in the location $v$, or in the dimension, is asserted.

Short English statement

After removing the forced parity zeros, is the probability that simple lattice random walk is at a fixed site unimodal in time?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3100082 (AMR-030-0082), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Cooper - Combinatorial Problems I Like (2020 snapshot), Probability, source-order bullet 82. Original source indexed by AMR-030-0082. https://people.math.sc.edu/cooper/combprob.html.