Communications in Mathematical Physics Log-Gamma Polymer Free Energy Fluctuations via a Fredholm Determinant Identity
Communications in Mathematical Physics Log-Gamma Polymer Free Energy Fluctuations via a Fredholm Determinant Identity
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发表时间：
2013
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通讯作者：
A. Borodin;Ivan Corwin;Daniel Remenik
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作者：
A. Borodin;Ivan Corwin;Daniel Remenik
我们证明了在N1/3标度下，SeppäLäinen离散的对数伽马定向聚合物的自由能的极限分布为n→∞。我们提供的主要技术创新是一类n重轮廓积分和一类Fredholm型行列式之间的一般恒等式。将这一恒等式应用于Corwin等人证明的积分公式。热带组合学和惠特克函数。Http://arxiv.org/abs/1110.3489v3[Math.PR]，2012年)对于对数-伽马聚合物配分函数的拉普拉斯变换，我们得到了一个弗雷德霍尔姆行列式，该行列式适合于渐近分析(从而产生自由能极限定理)。在Borodin和Corwin(Macdonald过程)中，预计会有Fredholm决定因素。Http://arxiv.org/abs/1111.4408v3[Math.PR]，2012)通过麦克唐纳过程的形式主义，然而，它的严格证明到目前为止还缺乏，因为这种方法需要某些衰变估计的非平凡。
We prove that under n1/3 scaling, the limiting distribution as n → ∞ of the free energy of Seppäläinen’s log-Gamma discrete directed polymer is GUE TracyWidom. The main technical innovation we provide is a general identity between a class of n-fold contour integrals and a class of Fredholm determinants. Applying this identity to the integral formula proved in Corwin et al. (Tropical combinatorics and Whittaker functions. http://arxiv.org/abs/1110.3489v3 [math.PR], 2012) for the Laplace transform of the log-Gamma polymer partition function, we arrive at a Fredholm determinant which lends itself to asymptotic analysis (and thus yields the free energy limit theorem). The Fredholm determinant was anticipated in Borodin and Corwin (Macdonald processes. http://arxiv.org/abs/1111.4408v3 [math.PR], 2012) via the formalism of Macdonald processes yet its rigorous proof was so far lacking because of the nontriviality of certain decay estimates required by that approach.