Brownian Gibbs property for Airy line ensembles (1108.2291v2)

Abstract: Consider N Brownian bridges B_i:[-N,N] -> R, B_i(-N) = B_i(N) = 0, 1 <= i <= N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a limit as N -> infinity of these curves scaled around (0,2^{1/2} N) horizontally by a factor of N^{2/3} and vertically by N^{1/3}. If a parabola is added to each limit curve, an x-translation invariant process sometimes called the multi-line Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which curves are a.s. everywhere continuous and non-intersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with "wanderers" and "outliers". We formulate our results to treat these relatives as well. Note that the law of the finite collection of Brownian bridges above has the property -- called the Brownian Gibbs property -- of being invariant under the following action. Select an index 1 <= k <= N and erase B_k on a fixed time interval (a,b) subset of (-N,N); then replace this erased curve with a new curve on (a,b) according to the law of a Brownian bridge between the two existing endpoints (a,B_k(a)) and (b,B_k(b)), conditioned to intersect neither the curve above nor the one below. We show that this property is preserved under the edge-scaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs property. An immediate consequence is a proof of M. Prahofer and H. Spohn's prediction that the lines of the Airy line ensemble are locally absolutely continuous with respect to Brownian motion. We also prove the conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point, thus establishing the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights.

• The paper establishes the Brownian Gibbs property for Airy line ensembles through the scaling limits of non-intersecting Brownian bridges.
• It employs a robust probabilistic coupling method to prove convergence in distribution and tightness for multi-line Airy processes along curved trajectories.
• The findings enhance theoretical foundations in random matrix theory and growth processes, offering new avenues for research in statistical physics.

Analysis of the Brownian Gibbs Property for Airy Line Ensembles

The paper by Corwin and Hammond investigates the intricate properties of Airy line ensembles and establishes the Brownian Gibbs property for these mathematical objects. This work is pivotal in understanding systems composed of collections of non-intersecting Brownian bridges and their scaling limits.

The Airy line ensemble is shown to embody the edge-scaling limit of a collection of these Brownian bridges as the number of bridges, $N$, approaches infinity. The remarkable aspect of this scaling involves the addition of a parabola to achieve an x-translation invariant process known as the multi-line Airy process. Notably, the authors prove the existence of a version of the Airy process, revealing its applicability to growth processes and random matrix theory, fields where related processes with “wanderers” and “outliers” arise.

Central to this exploration is the Brownian Gibbs property, a concept that ensures invariance under specific conditions, which is shown to be preserved even under the complicated scaling limits considered. A key implication of this property is the continuity of line measures under the condition that they do not intersect each other, confirmed by Prahofer and Spohn’s predictions. This directly impacts the asymptotic analysis of last passage percolation models.

The authors employ a probabilistic approach to complement exactly solvable systems often used in the paper of multi-line Airy processes. They illustrate that these probabilistic methods yield insightful results about the process, supplementing the analytical methods traditionally employed.

This work substantially contributes to the theoretical framework surrounding Airy line ensembles by addressing two main challenges. The first is establishing a continuous, non-intersecting version of the multi-line Airy process supported on curved trajectories. The second is validating the convergence in distribution of finite systems of non-intersecting Brownian bridges to these limiting ensembles.

Mathematically, the authors leverage a robust coupling argument to demonstrate that the Brownian Gibbs property is upheld in limit ensembles. Furthermore, they show that tightness for this family of curves is achievable, solidifying the theoretical underpinning for Airy line ensembles.

The implications of these findings extend beyond theoretical exposition; they propose concrete advancements in models of statistical physics. By identifying scaling limits associated with these non-intersecting line ensembles, the authors connect their findings to the Dyson sine process and other well-known stochastic processes.

The paper leaves open numerous research avenues for future exploration. The universality of the Brownian Gibbs property in different stochastic environments, particularly Airy-like line ensembles derived from perturbed non-intersecting systems, presents immense potential for further paper. Additionally, examining the extremal nature and uniqueness of these Gibbs measures remain key areas of interest, promising to enrich the field's understanding significantly.

Conclusively, Corwin and Hammond’s paper lays a comprehensive mathematical foundation for the paper and application of Airy line ensembles while opening up new pathways for exploration in scaling limits, probabilistic techniques, and statistical mechanics.