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Brownian Gibbs property for Airy line ensembles

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Abstract

Consider a collection of N Brownian bridges \(B_{i}:[-N,N] \to \mathbb{R} \), 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 weak limit as N→∞ of the collection of curves scaled so that the point (0,21/2 N) is fixed and space is squeezed, horizontally by a factor of N 2/3 and vertically by N 1/3. If a parabola is added to each of the curves of this scaling limit, 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 the curves are almost surely 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)⊆(−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 of the Brownian Gibbs property is a confirmation of the prediction of M. Prähofer and H. Spohn that each line of the Airy line ensemble is locally absolutely continuous with respect to Brownian motion. We also obtain a proof of the long-standing conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point. This establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights. Our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi-line Airy process, and readily yields several other interesting properties of this process.

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Notes

  1. One way of seeing this is as follows: For a fixed δ observe that as the starting and ending points go to zero, the distributions of the height of the N lines at ±(N−δ) converge to a non-trivial limit which can be explicitly calculated via the Karlin-McGregor formula [47]. The resulting ensemble on the interval [−N+δ,N−δ] with this non-trivial entrance and exit law is continuous and non-intersecting and has the Brownian Gibbs property. As δ goes to zero this procedure yields a consistent family of measures which one identifies as the desired line ensemble with starting and ending height all identically zero.

  2. The rth correlation function is given essentially by the probability of finding points in small neighborhoods of (s i ,x i ) for s i ∈A, \(x_{i}\in \mathbb{R} \) and i=1,…,r.

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Acknowledgements

This project was initiated at the 2010 Clay Mathematics Institute Summer School in Buzios, Brazil. The authors also thank the Mathematical Science Research Institute, the Fields Institute and the Mathematisches Forschungsinstitut Oberwolfach for their hospitality and support, as much of this work was completed during stays at these institutes. We thank Jinho Baik, Jeremy Quastel and Herbert Spohn for their input and interest. We also thank our referee for a thorough reading of this work and many useful comments. A.H. would like to thank Scott Sheffield for drawing attention to a talk in 2006 in which Andrei Okounkov proposed problems closely related to the discussion in Sect. 3.2 and for interesting ensuing conversations, and Neil O’Connell and Jon Warren for useful early discussions regarding approaches to proving the results in this article. I.C. recognizes support and travel funding from the NSF through grant DMS-1056390 and the PIRE grant OISE-07-30136 as well as Microsoft Research New England’s support through the Schramm Memorial Fellowship and the Clay Mathematics Institute’s support through a Clay Research Fellowship. A.H. was supported principally by EPSRC grant EP/I004378/1.

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  1. Microsoft Research New England: Theory group, 1 Memorial Drive, Cambridge, MA, 02142, USA

    Ivan Corwin

  2. Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG, UK

    Alan Hammond

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Corwin, I., Hammond, A. Brownian Gibbs property for Airy line ensembles. Invent. math. 195, 441–508 (2014). https://doi.org/10.1007/s00222-013-0462-3

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