从对偶性到q-TASEP和ASEP的行列式
From duality to determinants for q-TASEP and ASEP
摘要：
证明了两个相互作用粒子系统：$q$ -变形的完全非对称简单排斥过程（$q$ -TASEP）和非对称简单排它过程（ASEP）的对偶关系.对偶泛函的期望分别对应于粒子位置或积分电流的某些联合矩。对偶性意味着它们能解常微分方程组。这些系统是可积的，并为特定的步骤和半固定的初始数据，我们使用嵌套的围道积分animals提供明确的公式系统的解决方案，因此也时刻。我们构造了这些矩的类拉普拉斯变换生成函数，并通过留数演算计算了这类生成函数的两种不同类型的弗雷德霍姆行列式公式。对于ASEP，第一种类型的公式是新的，并且容易使其自身适合于渐近分析（如重新证明ASEP的GUE Tracy-Widom分布波动所必需的），而第二种类型的公式被认为与Tracy和Widom的ASEP公式密切相关[Comm.Math.Phys.279（2008）815- 844，J. Stat. 132（2008）291- 300，Comm.Math.Phys.290（2009）129- 154，J. Stat. 140（2010）619- 634]。对于$q$ -TASEP，两个公式与通过Borodin和Corwin的Macdonald过程计算的公式一致[Probab.理论相关领域（2014）158 225--400]。$q$ -TASEP和ASEP都有极限跃迁到连续定向聚合物的自由能、随机热方程解的对数或Kardar-Parisi-Zhang方程的Hopf-Cole解。因此，$q$ -TASEP和ASEP是这些连续体对象的可积离散化;与它们的对偶性相关的常微分方程系统是变形的离散量子δ玻色气体;我们从它们的对偶泛函的期望到表征生成函数的过程是物理学中复制技巧的严格版本。
英文摘要：
We prove duality relations for two interacting particle systems: the $q$-deformed totally asymmetric simple exclusion process ($q$-TASEP) and the asymmetric simple exclusion process (ASEP). Expectations of the duality functionals correspond to certain joint moments of particle locations or integrated currents, respectively. Duality implies that they solve systems of ODEs. These systems are integrable and for particular step and half-stationary initial data we use a nested contour integral ansatz to provide explicit formulas for the systems' solutions, and hence also the moments. We form Laplace transform-like generating functions of these moments and via residue calculus we compute two different types of Fredholm determinant formulas for such generating functions. For ASEP, the first type of formula is new and readily lends itself to asymptotic analysis (as necessary to reprove GUE Tracy--Widom distribution fluctuations for ASEP), while the second type of formula is recognizable as closely related to Tracy and Widom's ASEP formula [Comm. Math. Phys. 279 (2008) 815--844, J. Stat. Phys. 132 (2008) 291--300, Comm. Math. Phys. 290 (2009) 129--154, J. Stat. Phys. 140 (2010) 619--634]. For $q$-TASEP, both formulas coincide with those computed via Borodin and Corwin's Macdonald processes [Probab. Theory Related Fields (2014) 158 225--400]. Both $q$-TASEP and ASEP have limit transitions to the free energy of the continuum directed polymer, the logarithm of the solution of the stochastic heat equation or the Hopf--Cole solution to the Kardar--Parisi--Zhang equation. Thus, $q$-TASEP and ASEP are integrable discretizations of these continuum objects; the systems of ODEs associated to their dualities are deformed discrete quantum delta Bose gases; and the procedure through which we pass from expectations of their duality functionals to characterizing generating functions is a rigorous version of the replica trick in physics.
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