Radon-Nikodym derivative of inhomogeneous Brownian last passage percolation

TL;DR

This paper establishes quantitative Radon-Nikodym derivative estimates for the law of spatial increments in inhomogeneous Brownian LPP against Brownian motion, proving the RN derivative lies in $L^{\infty-}(\mu)$ on compacts and deriving $L^p$ bounds of order $O_p(e^{d_p m^2\log m})$. It develops a diffusion-interlacing framework via the Gelfand–Tsetlin cone and the Warren process to obtain explicit RN-derivative densities as ratios of determinant expressions, extending to a universal asymptotic growth in the number of curves. A key contribution is the comprehensive RN-derivative analysis for Brownian TASEP/BLPP (including the homogeneous case with optimal $L^p$ growth $\asymp O(e^{dm^2})$) and the extension to a toy KPZ fixed point model with random depth, providing tools for Brownian regularity in KPZ-related objects. The results have potential implications for rigorous control of KPZ universality objects and quantitative Brownian regularity in finitary initial-data settings.

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.

Figures (3)

• Figure 1: Visualisation of a possible path (blue) 'embedded' on the underlying environment (random ensemble $F:[x,y]\times \llbracket 1, 4\rrbracket\to \mathbb{R}$), here $(F_1, F_2, F_3, F_4)$ from top to bottom, and $m = 1, k = 4$. Here $\Delta_1 = F_4(t_1)-F_4(x)$, $\Delta_2 = F_3(t_2)-F_3(t_1)$, $\Delta_3 = F_2(t_3)-F_2(t_2)$, $\Delta_4 = F_1(y)-F_1(t_3)$ and $k = \sum_{i=1}^4 \Delta_i.$
• Figure 2: An illustration of the Pitman transform $WB$ of two Brownian motions $B_1, B_2$.
• Figure 3: Here, the contents of Lemma \ref{['lemma: Warren Markov density']} with $m=3$ are schematically depicted. It states that the law of the inhomogeneous Brownian LPP on the positive reals $(H_\ell)_{\ell =1}^3(\cdot)$ is a version of the regular conditional distribution of the Brownian melon $WB^3(\cdot + 1)$ conditioned on $WB^3(1) = (b_\ell)_{\ell =1}^3$. Thus, if one 'traces back' the inhomogeneous Brownian LPP by one unit to the left one recovers the Brownian $3$-melon $WB^3(\cdot + 1)$.

Theorems & Definitions (73)

• Theorem 1.1
• Definition 3.1: Random ensemble
• Definition 3.2: Path
• Remark
• Definition 3.3: Length
• Definition 3.4: Last passage value
• Remark
• Lemma 3.5: Metric composition law
• Lemma 3.6
• proof