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Brownian gibbs property for airy line ensembles
"... 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 (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 ad ..."
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Cited by 30 (8 self)
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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 (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 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 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 Bk 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, Bk(a) ) and ( b, Bk(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 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 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 process. 1.
MULTIDIMENSIONAL YAMADAWATANABE THEOREM AND ITS APPLICATIONS
"... Abstract. Multidimensional and matrix versions of the YamadaWatanabe theorem are proved. They are applied to particle systems of squared Bessel processes and to matrix analogues of squared Bessel processes: Wishart and Jacobi matrix processes. ..."
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Abstract. Multidimensional and matrix versions of the YamadaWatanabe theorem are proved. They are applied to particle systems of squared Bessel processes and to matrix analogues of squared Bessel processes: Wishart and Jacobi matrix processes.
Markov property of determinantal processes with extended sine, Airy, and Bessel kernels
, 2011
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RiemannHilbert approach to gap probabilities for the Bessel process. arXiv:1306.5663
, 2013
"... We consider the gap probability for the Bessel process in the singletime and multitime case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels a ́ la ItsIzerginKorepinSlavnov and thus related to suitable R ..."
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Cited by 2 (1 self)
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We consider the gap probability for the Bessel process in the singletime and multitime case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels a ́ la ItsIzerginKorepinSlavnov and thus related to suitable RiemannHilbert problems. In the singletime case, we construct a Lax pair formalism and we derive a Painleve ́ III equation related to the Fredholm determinant. 1