Brownian Gibbs property for Airy line ensembles

Authors: Ivan Corwin Alan Hammond

Publish Date: 2013/03/21

Volume: 195, Issue: 2, Pages: 441-508

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Abstract

Consider a collection of N Brownian bridges B iNN to mathbbR B i −N=B i N=0 1≤i≤N conditioned not to intersect The edgescaling limit of this system is obtained by taking a weak limit as N→∞ of the collection of curves scaled so that the point 021/2 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 xtranslation invariant process sometimes called the multiline 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 nonintersecting 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 wellNote 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 ab⊆−NN then replace this erased curve with a new curve on ab according to the law of a Brownian bridge between the two existing endpoints aB k a and bB k b conditioned to intersect neither the curve above nor the one below We show that this property is preserved under the edgescaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs propertyAn 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 longstanding 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 multiline Airy process and readily yields several other interesting properties of this processThis 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 AH 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 32 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 IC recognizes support and travel funding from the NSF through grant DMS1056390 and the PIRE grant OISE0730136 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 AH was supported principally by EPSRC grant EP/I004378/1

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