Sharp upper tail behavior of line ensembles via the tangent method

TL;DR

<3-5 sentence high-level summary>This work develops a probabilistic geometric framework for continuum line ensembles with Brownian Gibbs interactions to study sharp upper-tail events. By combining Brownian resampling, monotonicity, and correlation inequalities (FKG/BK), the authors obtain precise one-point and multi-point tail asymptotics, including the canonical exp(-4/3 θ^{3/2}) decay, and characterize limit shapes under conditioning, via tangent-method-inspired convex hull constructions. The results apply to the KPZ line ensemble and the parabolic Airy line ensemble, and extend to extremal stationary ensembles and general initial data in a non-integrable setting, highlighting universality beyond integrable formulas. The methods blend probabilistic Brownian-bridge techniques with a Cameron–Martin energy viewpoint to produce sharp, geometrically interpretable tail bounds and limit shapes.

Abstract

We develop a new probabilistic and geometric method to obtain several sharp results pertaining to the upper tail behavior of continuum Gibbs measures on infinite ensembles of random continuous curves, also known as line ensembles, satisfying some natural assumptions. The arguments make crucial use of Brownian resampling invariance properties and correlation inequalities admitted by such Gibbs measures. We obtain sharp one-point upper tail estimates showing that the probability of the value at zero being larger than $θ$ is $\exp(-\frac{4}{3}θ^{3/2}(1+o(1)))$. A key intermediate step is developing a precise understanding of the profile when conditioned on the value at zero equaling $θ$. Our method further allows one to obtain multi-point asymptotics which were out of reach of previous approaches. As an example, we prove sharp explicit two-point upper tail estimates. This framework is then used to establish the corresponding results for the KPZ equation, which are all new. Even for the zero-temperature case of the Airy$_2$ process, our arguments yield new proofs for one-point estimates previously known due to its connections to random matrix theory, as well as new two-point asymptotics. To showcase the reach of the method, we obtain the same results in a purely non-integrable setting under only assumptions of stationarity and extremality in the class of Gibbs measures. Our method bears resemblance to the tangent method introduced by Colomo-Sportiello and mathematically realized by Aggarwal in the context of the six-vertex model.

Figures (8)

• Figure 1: The three cases of Theorem \ref{['mt.two point tail']}: from left to right, $\mathsf{ConHull}_{a,b}$ (in blue) has two, infinitely many, and one extreme point inside ${[-\theta^{1/2},\theta^{1/2}]}$. (The distance of $\pm\theta^{1/2}$ from the center of the parabola at 0 have been made to differ in the three figures just to visually better emphasize the geometric features of the three cases.)
• Figure 2: The blue solid curve is $\mathsf{Tri}_\theta:[-\theta^{1/2}, \theta^{1/2}]\to\mathbb{R}$.
• Figure 3: The setup for the argument for the lower bound on the upper tail. The interval on which we resample is now $[-\frac{1}{2}\theta^{1/2}, \frac{1}{2}\theta^{1/2}]$; note that the boundary points are such that $\mathsf{Tri}_\theta$ equals zero at them, and so the line connecting $(x, \mathsf{Tri}_\theta(x))$ and $(-x, \mathsf{Tri}_\theta(-x))$ when $x=\frac{1}{2}\theta^{1/2}$ is tangent to $-x^2$. Thus the Brownian bridge defined between these points will avoid $-x^2$ with constant probability, and the FKG inequality will be essentially sharp in lower bounding $\mathbb{P}(\mathcal{L}_1(-\frac{1}{2}\theta^{1/2}),\mathcal{L}_1(\frac{1}{2}\theta^{1/2})\geq 0)$.
• Figure 4: On the left panel we consider the situation where the limit shape for the top curve is non-convex on some interval; the non-intersection condition then pushes the second curve down on the same interval. In the second panel we resample the top curve on the interval of non-convexity. The Brownian bridge would typically approximately follow the straight line between its endpoints if it was unconditioned; here it is conditioned to avoid the second curve, but, since the second curve is already lower, the avoidance conditioning is met by the bridge's typical behaviour. Thus as we see in the third panel, the resampling removes the non-convexity from the top curve, thus contradicting the initial non-convexity, since resampling preserves the distribution of the ensemble.
• Figure 5: Left panel: A Brownian bridge whose endpoints lie on a line of given slope is unlikely to intersect another line of different slope if the endpoints of the bridge are well-separated from the second line. Right panel: The lower green curve is what $\mathcal{L}_2$ should look like, but which we cannot yet prove; the darker green tent-shaped function above it is the upper bound we are able to prove on $\mathcal{L}_2$, lines of slope $\pm2$; and the blue curve is the upper bound we are able to prove on $\mathcal{L}_1$ using resampling. In more detail, the blue curve is a Brownian bridge from $(-x_\theta, \theta-x_\theta)$ to $(0,\theta)$ to $(x_\theta,\theta-x_\theta)$ (which are endpoints of lines of slope $\pm 1$, drawn in dashed blue) conditioned to stay above the dark green tent; this latter conditioning has probability lower bounded by a uniformly positive constant by the argument from the left panel. Thus, at $\pm \theta$, $\mathcal{L}_1$ is with high probability below $\theta/2$, as the Brownian bridge that bounds it has mean $0$ and standard deviation $O(\theta^{1/2})$ at those locations.

Theorems & Definitions (75)

• Remark 1.1
• Theorem 1: One-point density asymptotics
• Theorem 2: One-point upper tail bounds
• Theorem 3: Two-point upper tail bounds
• Theorem 4: Sharpness of the FKG inequality
• Theorem 1.2: Corollary 2.12 of aggarwal2023strong
• Theorem 5: Tail asymptotics for extremal stationary ensembles
• Theorem 6: One-point limit shape
• Theorem 7: Two-point limit shape
• Proposition 1.3