HALF-SPACE MACDONALD PROCESSES
HALF-SPACE MACDONALD PROCESSES
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DOI：
10.1017/fmp.2020.3
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发表时间：
2018-02
期刊：
影响因子：
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通讯作者：
Guillaume Barraquand;A. Borodin;Ivan Corwin
中科院分区：
文献类型：
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作者：
Guillaume Barraquand;A. Borodin;Ivan Corwin
Macdonald过程是对整数分割序列的度量，这些整数分割序列使用Macdonald对称函数的柯西求和恒等式构建。这些测度是揭示许多概率系统的可积性的有用工具，包括Kardar-Parisi-Zhang（KPZ）方程及其普适类中的许多其他模型。在本文中，我们开发了这些模型的半空间变体和相应的半空间Macdonald过程背后的结构理论。这些过程是使用Littlewood求和恒等式而不是Cauchy恒等式建立的，并且它们的分析比全空间的对应过程要困难得多。我们计算了一般半空间Macdonald测度下观测量的矩和拉普拉斯变换。引入新的动态保持这类措施，我们将它们与各种随机过程，特别是在半象限的对数伽马聚合物（它们也与随机六顶点模型在半象限和半空间ASEP）。对于聚合物模型，我们提供了明确的积分公式的拉普拉斯变换的配分函数。非严格的鞍点渐近产生的有向聚合物自由能的Tracy-Widom（相关的高斯正交或辛系综）或高斯分布取决于边界上的权重的平均大小收敛。
Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar–Parisi–Zhang (KPZ) equation and a number of other models in its universality class. In this paper, we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts. We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Nonrigorous saddle-point asymptotics yield convergence of the directed polymer free energy to either the Tracy–Widom (associated to the Gaussian orthogonal or symplectic ensemble) or the Gaussian distribution depending on the average size of weights on the boundary.