$(q,t)$ -高斯过程
The $(q,t)$-Gaussian Process
摘要：
我们引入经典玻色子，费米和玻尔兹曼-福克空间的双参数变形，这是[BS 91]的$q$ -福克空间的改进。从实可分Hilbert空间$H$出发，构造了$(q,t)$-Fock空间及相应的生成算子和湮灭算子$\{a_{q,t}(h)^\ast\}_{h\in H}$和$\{a_{q,t}(h)\}_{h\in H}$，满足$h,g\in H$的$(q,t)$ -对易关系$a_{q,t}(f)a_{q,t}(g)^\ast-q \,a_{q,t}(g)^\ast a_{q,t}(f)= <f,g>_{_H}\, t^{N},$，其中$N$表示数算子.将$(q,t)$-Fock空间上的有界线性算子解释为非交换随机变量，高斯随机变量的模拟由变形场算子$s_{q,t}(h):=a_{q,t}(h)+a_{q,t}(h)^\ast$给出，对于$h\in H$。由此产生的改进是特别自然的，因为$s_{q,t}(h)$的矩是由成对分区中的交叉\emph{and nestings}的联合统计编码的。此外，与归一化的$(q,t)$ -高斯$s_{q,t}$相关联的正交多项式序列是$(q,t)$ -厄米特正交多项式的正交多项式序列，是$q$ -厄米特序列的变形，由具有$H_0(z;q,t)=1$、$H_1(z;q,t)=z$和$[n]_{q,t}=\sum_{i=1}^n q^{i-1}t^{n-i}$的递归$zH_n(z;q,t)=H_{n+1}(z;q,t)+[n]_{q,t}H_{n-1}(z;q,t),$给出. $q=0<t$专业化产生了一个新的单参数变形的完整的玻尔兹曼-福克空间的自由概率。与相应的变形半圆算子相关联的概率测度被证明以各种形式编码，通过Rogers-Ramanujan连分数，Rogers-Ramanujan恒等式，$t$ -Airy函数，Carlitz-Riordan的$t$ -Catalan数，以及约化Wigner过程的一阶统计。
英文摘要：
We introduce a two-parameter deformation of the classical Bosonic, Fermionic, and Boltzmann Fock spaces that is a refinement of the $q$-Fock space of [BS91]. Starting with a real, separable Hilbert space $H$, we construct the $(q,t)$-Fock space and the corresponding creation and annihilation operators, $\{a_{q,t}(h)^\ast\}_{h\in H}$ and $\{a_{q,t}(h)\}_{h\in H}$, satifying the $(q,t)$-commutation relation $a_{q,t}(f)a_{q,t}(g)^\ast-q \,a_{q,t}(g)^\ast a_{q,t}(f)= <f,g>_{_H}\, t^{N},$ for $h,g\in H$, with $N$ denoting the number operator. Interpreting the bounded linear operators on the $(q,t)$-Fock space as non-commutative random variables, the analogue of the Gaussian random variable is given by the deformed field operator $s_{q,t}(h):=a_{q,t}(h)+a_{q,t}(h)^\ast$, for $h\in H$. The resulting refinement is particularly natural, as the moments of $s_{q,t}(h)$ are encoded by the joint statistics of crossings \emph{and nestings} in pair partitions. Furthermore, the orthogonal polynomial sequence associated with the normalized $(q,t)$-Gaussian $s_{q,t}$ is that of the $(q,t)$-Hermite orthogonal polynomials, a deformation of the $q$-Hermite sequence that is given by the recurrence $zH_n(z;q,t)=H_{n+1}(z;q,t)+[n]_{q,t}H_{n-1}(z;q,t),$ with $H_0(z;q,t)=1$, $H_1(z;q,t)=z$, and $[n]_{q,t}=\sum_{i=1}^n q^{i-1}t^{n-i}$. The $q=0<t$ specialization yields a new single-parameter deformation of the full Boltzmann Fock space of free probability. The probability measure associated with the corresponding deformed semicircular operator turns out to be encoded, in various forms, via the Rogers-Ramanujan continued fraction, the Rogers-Ramanujan identities, the $t$-Airy function, the $t$-Catalan numbers of Carlitz-Riordan, and the first-order statistics of the reduced Wigner process.
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