Brownian Gibbs property for Airy line ensembles
Brownian Gibbs property for Airy line ensembles
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DOI：
10.1007/s00222-013-0462-3
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发表时间：
2011-08
影响因子：
3.1
通讯作者：
Ivan Corwin;A. Hammond
中科院分区：
文献类型：
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作者：
Ivan Corwin;A. Hammond
考虑N个布朗桥的集合，Bi（−N）=Bi（N）=0，1≤i≤N，条件是不相交。这个系统的边标度极限是通过取一个弱极限作为标度曲线集合的N →∞来得到的，使得点（0，21/2N）是固定的，空间在水平方向上被压缩了N2/3倍，在垂直方向上被压缩了N1/3倍。如果在这个标度极限的每一条曲线上加上一条抛物线，就得到了一个平移不变过程，有时称为多线艾里过程。我们证明了这个过程的一个版本（我们称之为艾里线系综）的存在，其中的曲线几乎肯定是处处连续和不相交。这个过程自然出现在增长过程和随机矩阵集合的研究中，就像“流浪者”和“离群值”的相关过程一样。我们用公式表示我们的结果来处理这些关系。注意，上面的布朗桥的有限集合的定律具有在下面的作用下不变的性质-称为布朗吉布斯性质。选择一个指数1≤k≤ N，在一个固定的时间间隔（a，B）（−N，N）内擦除B;然后根据两个现有端点（a，Bk（a））和（B，Bk（B））之间的布朗桥定律，用（a，B）上的一条新曲线替换擦除的曲线，条件是既不与上面的曲线相交，也不与下面的曲线相交。我们证明了这一性质在边标度极限下是保持的，从而建立了Airy线系综具有Brownian Gibbs性质，Brownian Gibbs性质的直接结果是M. Prähofer和H. Spohn证明了Airy线系综的每条线相对于布朗运动是局部绝对连续的。我们还得到了长期存在的猜想K.约翰森的艾里线系综的顶线减去抛物线达到其最大值在一个独特的点的证明。建立了带几何权重的末道渗流横向涨落的渐近规律。我们的概率方法补充的角度来看，正是可解的系统，这是经常在研究多线艾里过程，并很容易产生其他几个有趣的性质，这个过程。
Consider a collection ofNBrownian bridges,Bi(−N)=Bi(N)=0, 1≤i≤N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a weak limit asN→∞ of the collection of curves scaled so that the point (0,21/2N) is fixed and space is squeezed, horizontally by a factor ofN2/3and vertically byN1/3. If a parabola is added to each of the curves of this scaling limit, anx-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≤Nand eraseBkon 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 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.