From duality to determinants for q-TASEP and ASEP
From duality to determinants for q-TASEP and ASEP
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DOI：
10.1214/13-aop868
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发表时间：
2012-07
影响因子：
2.3
通讯作者：
A. Borodin;Ivan Corwin;T. Sasamoto
中科院分区：
文献类型：
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作者：
A. Borodin;Ivan Corwin;T. Sasamoto
我们证明了两个相互作用的粒子系统的对偶关系：$q$-变形的完全不对称简单不相容过程（$q$-TASEP）和非对称简单不相容过程（ASEP）。对偶泛函的期望分别对应于粒子位置或集成电流的某些联合力矩。对偶性意味着它们解ode的系统。这些系统是可积的，对于特定的阶跃和半平稳初始数据，我们使用嵌套的轮廓积分分析来提供系统解的显式公式，从而也提供了矩。我们形成了这些矩的拉普拉斯变换生成函数，并通过剩余演算计算了两种不同类型的Fredholm生成函数的行列式公式。对于ASEP，第一种类型的公式是新的，很容易用于渐近分析（这是必要的，以反驳GUE Tracy—Widom的ASEP分布波动），而第二种类型的公式与Tracy和Widom的ASEP公式密切相关[Comm. Math]。[j] .中国科学：物理学报，2009(6):815—844。[j].中国科学：物理学报，2009(5):619—634。对于$q$-TASEP，这两个公式与通过Borodin和Corwin的麦克唐纳过程计算的公式一致。理论相关领域（2014）158 225—400]。$q$-TASEP和ASEP对连续介质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解都有极限跃迁。因此，$q$-TASEP和ASEP是这些连续体对象的可积离散化；与其对偶性相关的ode系统是变形离散量子δ玻色气体；我们从对偶函数的期望过渡到描述生成函数的过程是物理学中复制技巧的严格版本。
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.