Brownian bridge limit of path measures in the upper tail of KPZ models

TL;DR

This work proves that path measures in KPZ models conditioned on upper-tail events converge to Brownian bridges, both in the zero-temperature directed landscape and in the positive-temperature continuum directed random polymer. The authors develop a geometric-probabilistic framework centered on tent-profile shapes, coalescence, and shift-invariance, bypassing the need for exact solvable-formulas in the polymer case. A key outcome is the Brownian-bridge limit for finite-dimensional marginals and a quenched localization phenomenon around a random backbone in the polymer setting. The results illuminate the structure and fluctuations of geodesics and polymers under rare upper-tail conditioning, with potential implications for multi-point statistics and KPZ-related tail phenomena.

Abstract

For models in the KPZ universality class, such as the zero temperature model of planar last passage-percolation (LPP) and the positive temperature model of directed polymers, its upper tail behavior has been a topic of recent interest, with particular focus on the associated path measures (i.e., geodesics or polymers). For Exponential LPP, diffusive fluctuation had been established in Basu-Ganguly. In the directed landscape, the continuum limit of LPP, the limiting Gaussianity at one point, as well as of related finite-dimensional distributions of the KPZ fixed point, were established, using exact formulas in Liu and Wang-Liu. It was further conjectured in these works that the limit of the corresponding geodesic should be a Brownian bridge. We prove it in both zero and positive temperatures; for the latter, neither the one-point limit nor the scale of fluctuations was previously known. Instead of relying on formulas (which are still missing in the positive temperature literature), our arguments are geometric and probabilistic, using the results on the shape of the weight and free energy profiles under the upper tail from Ganguly-Hegde as a starting point. Another key ingredient involves novel coalescence estimates, developed using the recently discovered shift-invariance Borodin-Gorin-Wheeler in these models. Finally, our proof also yields insight into the structure of the polymer measure under the upper tail conditioning, establishing a quenched localization exponent around a random backbone.

Figures (6)

• Figure 1: An illustration of the profile $\mathcal{L}(0,0;\cdot, 1)$ conditional on it equaling $L$ at $0$, and the parabola $-x^2$ that it fluctuates around when there is no conditioning.
• Figure 2: A depiction of the parabolic Airy line ensemble.
• Figure 3: Left: An illustration of the coalescence phenomenon in time $[\frac{1}{3}, \frac{2}{3}]$, under upper tail. The geodesic from $(0,0)$ to $(0,1)$ is shown in brown except for the portion common to all the paths, which is in dark green. Right: a depiction of how coalescence implies equality in the quadrangle inequality. It is clear that for fixed $x< x'$ and $y<y'$, under coalescence, the union of the geodesic from $x$ to $y$ and the geodesic from $x'$ to $y'$ equals (with multiplicity) the union of the geodesic from $x$ to $y'$ and the geodesic from $x'$ to $y$.
• Figure 4: An illustration of the shift-invariance: the joint distributions of the three passage times/partition functions in the left and right panels are the same, under the condition that both endpoints of each path are shifted by the same amount, and the endpoints and shifts are such that the paths are all forced by planarity to intersect both before and after the shift.
• Figure 5: An illustration of transforming the weights of varying one side (left panel) into varying both sides (middle panel) using shift-invariance, then to two tents (right panel) by coalescence.

Theorems & Definitions (126)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Proposition 1.5
• Lemma 2.1: Lemma 10.2, DOV
• Lemma 2.2: DZ
• Lemma 2.3: DZ
• Lemma 2.4: GZ22
• proof : Proof of \ref{['eq:mult-quad']}