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arXiv · 2509.19414

Radon-Nikodym derivative of inhomogeneous Brownian last passage percolation

Pantelis Tassopoulos·Sourav Sarkar
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Abstract

We show that the Radon-Nikodym derivative of the law of the spatial increments (with endpoints away from the origin) of inhomogeneous Brownian last passage percolation (LPP) with non-decreasing initial data against the Wiener measure $μ$ on compacts is in $L^{\infty-}(μ)$; and for any fixed $p>1$, the $L^p$ norm is at most of the order $O_p(\mathrm{e}^{d_pm^2\log m})$ for some $p$-dependent constant $d_p>0$. Furthermore, when the initial data is homogeneous, we establish optimal growth on $L^p$ norms ($\asymp O(\exp(dm^2))$) of the Radon-Nikodym derivative of the Brownian LPP (i.e. top line of an $m$-level Dyson Brownian motion) away from the origin, as the number of curves $m$ tends to infinity, for all $p>1$ sufficiently large. As an application of our framework, we show that the Radon-Nikodym derivative of certain toy models for the KPZ fixed point lies in $L^{\infty-}(μ)$, inspired by its variational characterisation in terms of the directed landscape.

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Pantelis Tassopoulos, Sourav Sarkar. 2025-11-03. Radon-Nikodym derivative of inhomogeneous Brownian last passage percolation. https://arxiv.org/abs/2509.19414

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