Atypical stars on a directed landscape geodesic

Abstract

In random geometry, a recurring theme is that any two geodesics emanating from a typical point part ways at a strictly positive distance from the above point, and we call such points as $1$-stars. However, the measure zero set of atypical stars, the points where such coalescence fails, is typically uncountable and the corresponding Hausdorff dimensions of these sets have been heavily investigated for a variety of models including the directed landscape, Liouville quantum gravity and the Brownian map. In this paper, we consider the directed landscape -- the scaling limit of last passage percolation as constructed in the work Dauvergne-Ortmann-Vir\'ag '18 -- and look into the Hausdorff dimension of the set of atypical stars lying on a geodesic. We show that the above dimension is almost surely equal to $1/3$. This is in contrast to Ganguly-Zhang '22, where it was shown that set of atypical stars on the line $\{x=0\}$ has dimension $2/3$. This reduction of the dimension from $2/3$ to $1/3$ yields a quantitative manifestation of the smoothing of the environment around a geodesic with regard to exceptional behaviour.