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arXiv · 2609.01562

Points of maximal traffic on a grid with obstruction

Juan Gil·Zhenni Liang·Ayodeji Odetola·Michael Weiner
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Abstract

For $n\in\mathbb{N}$, we consider the set of lattice paths from $(0,0)$ to $(n,n)$ using only unit north and east steps. Given a point $B$ to be avoided, we ask: at which point $A$ on the grid with corners $(0,0)$ and $(n,n)$, different from the endpoints, does the largest number of $B$-avoiding lattice paths pass through? We show that for $n\ge 9$, regardless of the location of $B$, the maximum is attained at one of ten specific points clustered near the two endpoints of the grid. This stability, however, conceals an interesting anomaly. When the obstruction $B$ lies on the antidiagonal $x+y=n$, the points of maximal traffic migrate from the near-corner points $(1,1)$ and $(n-1,n-1)$ to boundary points in the set of possible maximizers. The migration occurs for every $8\le n\le 375$, and intermittently up to $n=495$. We conjecture that the anomaly disappears for $n\ge 496$.

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Juan Gil, Zhenni Liang, Ayodeji Odetola, Michael Weiner. 2026-09-01. Points of maximal traffic on a grid with obstruction. https://arxiv.org/abs/2609.01562

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