Non-intersecting path explanation for block Pfaffians and applications into skew-orthogonal polynomials

Enumerative combinatorics is always playing a fundamental role in mathematics and physics. Among these combinatorial objects, the theory of non-intersecting paths is an extremely useful theory with applications in counting tilings, plane partitions, and tableaux. In particular, the famous Lindström-Gessel-Vionnet (LGV) lemma states that the generating function of non-intersecting paths between vertex sets $\mathbf{u}\to\mathbf{v}$, where $\mathbf{u}$ and $\mathbf{v}$ have the same number of vertices, could be written as a determinant [24, 11]. This result has many applications such as non-intersecting random walks [17, 18], domino tiling models [8], dimer models [4], and so on. It turns out that the generating functions of non-intersecting paths are usually expressed in terms of structured determinants such as Hankel or Toeplitz determinants. Moreover, it has been shown in [26, 6] that the Hankel determinant of some special combinatorial numbers could be evaluated by using orthogonal polynomials, continued fraction and discrete integrable systems. Therefore, there is a unified relation between LGV lemma, integrable models, probability models, and orthogonal polynomials.

In [31], the author developed another algebraic tool—Pfaffian, to describe non-intersecting paths between a set of vertices to an interval. This combinatorial description for Pfaffian was also applied to count skew Young tableaux and plane partitions [27, 31]. A minor summation formula for Pfaffian [16] was given by using similar combinatorial technique, called the path switching involution in [21]. The minor summation formula is as an analogy of Cauchy-Binet formula for determinants, and it was applied to evaluate a certain Catalan-Hankel type Pfaffian

where $\{\mu_{i}\}_{i\in\mathbb{N}}$ are moments sequence related to little $q$-Jacobi polynomials [15]. In fact, Pfaffian expression (1.1) is an important quantity in random matrix theory, especially for the characterization of symplectic invariant ensemble [25]. By realizing that (1.1) is a normalization factor for certain skew-orthogonal polynomials, various Catalan-Hankel type Pfaffians were computed by using skew-orthogonal polynomials under Askey-Wilson scheme in [29].

Inspired by these results, we attempt to give a unified frame between non-intersecting paths, Pfaffian formulas, and skew-orthogonal polynomials. It was known from [31] that the generating function of non-intersecting paths from $\mathbf{u}\to I$ could be written as a Pfaffian, namely,

where $\mathscr{P}_{0}(\mathbf{u};I)$ stands for all non-intersecting paths from an even-numbered vertex set $\mathbf{u}=(u^{(1)},\cdots,u^{(2n)})$ to an interval $I$ and $Q_{I}(u^{(i)},u^{(j)})$ are some computable weights. Different from the Pfaffian formula considered in (1.1), we show that this generating function is related to another type of skew-orthogonal polynomials, which are related to orthogonal invariant ensemble in random matrix theory. Besides, if the vertex set $\mathbf{u}=(u^{(1)},\cdots,u^{(2n+1)})$, then its corresponding generating function could be expressed as an augmented Pfaffian

where $Q_{I}(u^{(j)})$ is a weight function related to $u^{(j)}$ only. Since skew-orthogonal polynomials are only related to the even-ordered Pfaffian formula (1.2), we demonstrate that this augmented Pfaffian should be related to partial-skew-orthogonal polynomials proposed in [5].

In recent years, matrix-valued orthogonal polynomials play an important role in probability and combinatoric models such as hexagon tilings [14] and periodic Aztec diamond models [8]. Despite of block determinants, Pfaffians of skew symmetric block matrices were also proved to be useful in practice [22]. It was shown in [31] that the generating function for $\mathbf{u}\to\mathbf{v}\oplus I$ could be expressed as a block Pfaffian, with one block being all zeros. Here $\mathbf{u}$ and $\mathbf{v}$ are two sets of vertices and $I$ is an interval of vertices. Our first result is to give a non-intersecting path interpretation for block Pfaffians

We show that this Pfaffian could be expressed as the generating function of non-intersecting paths $\mathscr{P}_{0}(J\oplus\mathbf{u};\mathbf{v}\oplus I)$, where $\mathbf{u}$ and $\mathbf{v}$ are two sets of vertices and $I$ and $J$ are two intervals of vertices. Besides, we show that this generating function is related to a 2-component skew-orthogonal polynomials, which could be used to describe a non-intersecting random walk with two different sources. Moreover, a concept of multiple partial-skew-orthogonal polynomials is proposed if the number of vertices in $\mathbf{u}$ and $\mathbf{v}$ are odd. Furthermore, we extend formula (1.5) to a more general case, where we consider the paths from $\{\mathbf{u}_{1},\cdots,\mathbf{u}_{n},I_{1},\cdots,I_{m}\}$ to $\{J_{1},\cdots,J_{n},\mathbf{v}_{1},\cdots,\mathbf{v}_{m}\}$. We show that this generating function could also be written as a block Pfaffian. We remark that in the reference [3], a non-intersecting path explanation for Pfaffians (1.5) was given, but it was shown by using Grassmann algebra in a cyclic digraph.

This paper is organized as follows. In Section 2, we recall the LGV lemma and show some applications. We connect the LGV lemma with several different orthogonal polynomials (including orthogonal polynomials on the real line and on the unit circle) and random walks (including discrete-time random walks and continuous-time random walks). In Section 3, we recall Stembridge’s results on the non-intersecting path explanation for Pfaffians. Their connections with skew-orthogonal polynomials and partial-skew-orthogonal polynomials are given. We show that this generating function is related to some random walk models as well. Thus we give a unified frame between non-intersecting paths, Pfaffian formulas and skew-orthogonal polynomials. We generalize Stembridge’s result in Section 4, where we consider paths from $J\oplus\mathbf{u}\to\mathbf{v}\oplus I$. Multiple skew-orthogonal polynomials are introduced by following this generating function, with some applications in random walks starting from different sources. Moreover, we introduce a new concept of multiple partial-skew-orthogonal polynomials from this generating function. In Section 5, non-intersecting paths between several vertex sets and intervals are considered. This is the most general case between vertex sets and intervals.

In this part, we give some brief reviews on the connections between Lindström-Gessel-Vionnet lemma and orthogonal polynomials. There have been numerous references about this topic, and please refer to [12, 21, 30, 28] and references therein.

Let $P$ be a path from $u$ to $v$, that is, $P\in\mathscr{P}(u;v)$, then the weight of $P$ denoted by $\omega(P)$, is defined as the product of the weights of all edges on this path. Moreover, the weight between two vertices $u$ and $v$ is defined as $h(u,v)=\sum\limits_{P\in{\mathscr{P}}(u;v)}{\omega(P)}$.

Let $\xi=(P_{1},P_{2},\cdots,P_{r})\in\mathscr{P}(\mathbf{u};\mathbf{v})$ be an $r$-path starting from $\mathbf{u}$ to $\mathbf{v}$, where $P_{i}$ is a path from $u_{i}$ to $v_{i}$. The weight of $\xi$ is defined to be $\omega(\xi)=\omega(P_{1},P_{2},\cdots,P_{n})=\prod_{i=1}^{r}\omega(P_{i})$. Moreover, if we count the weight for all paths from $\mathbf{u}$ to $\mathbf{v}$, then we have

In general, we refer this weight function as the generating function of all paths from $\mathbf{u}$ to $\mathbf{v}$, and denote it as GF$[\mathscr{P}(\mathbf{u};\mathbf{v})]$. For non-intersecting paths, we have GF$[\mathscr{P}_{0}(\mathbf{u};\mathbf{v})]=\sum_{\xi\in\mathscr{P}_{0}(\mathbf{u};\mathbf{v})}\omega(\xi)$.

Now, we could formally state the LGV lemma.

In literatures, there have been many applications of Lindstörm-Gessel-Vionnet lemma. For example, it has been applied to some tiling problem and random growth models [17, 8]. Originally, non-intersecting paths arose in matroid theory [24], which was later used to count tableaux and plane partitions [11]. In the references mentioned above, there is a specialized binomial determinant related to Toeplitz (respectively Hankel) determinant and orthogonal polynomials on the unit circle (respectively real line). Here we start with a non-intersecting random walk model and demonstrate how LGV lemma works in the random walk models and orthogonal polynomial theory.

Let’s consider $N$ independent simple random walks $X_{1}(t),\cdots,X_{N}(t)$ starting from positions $\mathbf{a}=(\alpha_{1},\cdots,\alpha_{N})$ at $t=0$, and ending at positions $\mathbf{b}=(\beta_{1},\cdots,\beta_{N})$ at $t=2T$ with conditions $\alpha_{1}<\cdots<\alpha_{N}$ and $\beta_{1}<\cdots<\beta_{N}$. There are two different models related to this setting. One is a continuous-time model called non-intersecting Brownian motion with configuration $\mathbb{R}\times\mathbb{R}_{\geq 0}$ [1, 9]. We first assume $X(t)=(X_{1}(t),\cdots,X_{N}(t))$ to be $N$ independent standard Brownian motions with transition probability

which indicates the probability from $x$ to $y$ within time $t$. Moreover, let’s assume that $D_{N}:=\{x_{1}<\cdots<x_{N}\}\subset\mathbb{R}^{N}$ is a admissible configuration. In this case, non-intersecting means that $X(t)\in D_{N}$ for all $t\in(0,2T)$. According to the Karlin-McGregor’s theorem [19], the conditional probability density function (PDF) for passing through $\mathbf{x}=(x_{1},\cdots,x_{N})$ at time $t$ and ending at $\beta$, given the starting point $\alpha$ and the total duration of $2T$, is expressed by

This result could be also recognized as an application of LGV lemma, where $\operatorname{det}\left(p\left(\alpha_{i},x_{j};t\right)\right)_{i,j=1}^{N}$ could be viewed as the generating function from $\mathbf{a}$ to $\mathbf{x}$, and $\operatorname{det}\left(p\left(x_{j},\beta_{i};2T-t\right)\right)_{i,j=1}^{N}$ could be viewed as the generating function from $\mathbf{x}$ to $\mathbf{b}$. To calculate the conditional probability of event $\mathcal{N}_{0}$, we normalize it by dividing a normalization factor for reaching $\mathbf{b}$ from $\mathbf{a}$ within the total time of $2T$. In fact, this normalization factor is given by $\prod_{j=1}^{N}p(\alpha_{j}-\beta_{j};2T)$, where $p(x;T)$ is a normal distribution with mean zero and variance $T$. Therefore, the conditional probability of the event $\mathcal{N}_{0}$ could expressed by

In fact, the PDF (2.2) induces a bi-orthogonal system. We demonstrate this result in the following proposition.

This is a general bi-orthogonal system [2]. If we consider confluent starting and ending point, then we could define multiple orthogonal polynomials and multiple orthogonal polynomials of mixed type [7]. To be precise, let’s consider a confluent case where $N$ non-intersecting Brownian motions start from $k$ different points $\alpha_{j}$ with multiplicity $a_{j}$ for $j=1,\cdots,k$, and end at $l$ different points $\beta_{j}$ with multiplicity $b_{j}$ for $j=1,\cdots,l$, such that $\sum_{j=1}^{k}a_{j}=\sum_{j=1}^{l}b_{j}=N$. Under this circumstance, we have the following proposition for multiple orthogonal polynomials of mixed type.

Another application is to consider a discrete-time random walk, defined in the configuration space $\mathbb{Z}\times\mathbb{Z}_{\geq 0}$. At each discrete time step, the simple random walk increase or decrease by one step with equal probability. If we denote $\alpha_{j}=2x_{j}$ and $\beta_{j}=2y_{j}$ for all $j=1,2,\cdots,N$, and assume that all paths don’t intersect, then the number of all non-intersecting paths is given by a binomial determinant [20, 10]

Such a model is sometimes referred to as vicious random walkers which means that the walkers can not be in the same position at each discrete step. It was shown in [10] that this binomial determinant is related to symmetric Hahn polynomials. However, the following proposition demonstrates that such a binomial determinant could also be related to a Toeplitz determinant and orthogonal polynomials on the unit circle (OPUC).

Similar to the Proposition 2.6, if we define

then one could obtain the following orthogonal relation

where $\bar{z}$ is the complex conjugate of $z$ and $H_{n}=\det(m_{k-j})_{k,j=0}^{n-1}\det(m_{k-j})_{k,j=0}^{n}.$ Regarding with general $\{x_{j},y_{j}\}$ and slat Toeplitz determinants, one could refer to [12, 13] for more examples.

In [31], Stembridge generalized the result of Lindström-Gessel-Viennot, by considering non-intersecting paths from a set of vertices to an ordered subset $I\subset V$, whose number of vertices is indefinite. In this section, let’s assume that $\mathbf{u}=\left(u^{(1)},u^{(2)},\ldots,u^{(r)}\right)$ is a finite sequence of vertices in $V$.

Especially, if $\mathbf{u}=\{u\}$, which is a single vertex, then

If $\mathbf{u}=(u^{(1)},u^{(2)})$, then according to LGV theorem, we have

where sgn is a sign function defined by

It was found by Stembridge that $\operatorname{GF}[\mathscr{P}_{0}(\mathbf{u};I)]$ could be written as a Pfaffian, whose elements are given by (3.1) and (3.2). To this end, let’s introduce the concept of 1-factor, and the graphic definition of Pfaffians.

For example, for a set $\mathbf{u}=\left({u^{(1)}},{u^{(2)}},{u^{(3)}},{u^{(4)}}\right)$, the perfect matching $\left\{\left\{{u^{(1)},u^{(2)}}\right\},\left\{{u^{(3)},u^{(4)}}\right\}\right\}$ is a 1-factor in $\mathscr{F}(\mathbf{u})$.