Log-concavity of the step TASEP distribution at rounded midpoints

Let $X(t)=(X_1(t),\dots,X_n(t))$ be the positions at time $t$ of $n$ particles of the continuous-time totally asymmetric simple exclusion process started from the step configuration $X_i(0)=-i$, and let $p_t(x)=\mathbb P(X(t)=x)$ for $x\in\mathbb Z^n$. We prove that \[ p_t(a)\,p_t(b)\le p_t\bigl(\lfloor \tfrac{a+b}2 \rfloor\bigr)\,p_t\bigl(\lceil \tfrac{a+b}2 \rceil\bigr) \] for all $n$, all $t\ge0$ and all $a,b\in\mathbb Z^n$, with rounding taken coordinatewise. In particular $p_t(a)p_t(b)\le p_t(\tfrac{a+b}2)^2$ whenever $a+b\in2\mathbb Z^n$, which answers a question from the open problem list of the 2025 AIM workshop on KPZ universality. The proof realizes $X(t)$ as a coordinate subvector of the Borodin–Ferrari interlacing array, whose fixed-time law has the explicit weight $e^{-nt}t^{|\lambda|}\prod_{i<j}(l_i-l_j)/\prod_i l_i!$ depending only on the top row. This weight satisfies the rounded-midpoint inequality factor by factor, and the discrete four-functions theorem of Halikias, Klartag and Slomka carries the inequality from the array to any coordinate marginal. The same argument shows that the position of each single particle has a log-concave law.