Mathematics – Probability
Scientific paper
2011-08-10
Mathematics
Probability
45 pages, 6 figures
Scientific paper
Consider a system of Brownian bridges B_i:[-N,N] -> R, B_i(-N)=B_i(N)=0, i=1,...,N, conditioned not to intersect. Its edge-scaling limit is obtained by taking a weak limit as N->infty of the collection of curves scaled so that the point (0,2N) is fixed and space is dilated, horizontally by a factor of N^{2/3} and vertically by N^{1/3}. If a parabola is added to each curve in 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 lines 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 finite collection of Brownian bridges above has the property of being invariant under the following action. Select an index k and erase B_k on a fixed time interval (a,b); then sample a new curve on (a,b) in place of the erased one according to the law of Brownian bridge between the two existing endpoints (a,B_k(a)) and (b,B_k(b)), conditioned not to intersect the curve above or below. We call this the Brownian Gibbs property and establish that it is inherited by the Airy line ensemble (from the above scaling limit). An immediate consequence of the Brownian Gibbs property is that each line of the Airy line ensemble is locally absolutely continuous with respect to Brownian motion. We thus obtain a 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.
Corwin Ivan
Hammond Alan
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