Brownian Gibbs property for Airy line ensembles.(English)Zbl 1459.82117
Summary: Consider a collection of \(N\) Brownian bridges \(B_{i}:[-N,N] \to \mathbb{R}, B _{i }(-N) = B _{i}(N)=0, 1\leq i\leq N\), conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a weak limit as \(N\rightarrow \infty \) of the collection of curves scaled so that the point \((0,2^{1/2} N)\) 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 \(1\leq k\leq N\) and erase \(B _{k }\) on a fixed time interval \((a,b)\subseteq ( - N,N)\); 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 ofM. Prähofer andH. Spohn [J. Stat. Phys. 108, No. 5–6, 1071–1106 (2002;Zbl 1025.82010)] 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 ofK. Johansson [Commun. Math. Phys. 242, No. 1–2, 277–329 (2003;Zbl 1031.60084)] 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.
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 \(1\leq k\leq N\) and erase \(B _{k }\) on a fixed time interval \((a,b)\subseteq ( - N,N)\); 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 ofM. Prähofer andH. Spohn [J. Stat. Phys. 108, No. 5–6, 1071–1106 (2002;Zbl 1025.82010)] 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 ofK. Johansson [Commun. Math. Phys. 242, No. 1–2, 277–329 (2003;Zbl 1031.60084)] 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.
MSC:
| 82B43 | Percolation |
| 82B41 | Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics |
| 60J65 | Brownian motion |
| 60B20 | Random matrices (probabilistic aspects) |
| 15B52 | Random matrices (algebraic aspects) |
Cite
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