Abstract
We introduce an elliptic extension of Dyson’s Brownian motion model, which is a temporally inhomogeneous diffusion process of noncolliding particles defined on a circle. Using elliptic determinant evaluations related to the reduced affine root system of type \(A\), we give determinantal martingale representation (DMR) for the process, when it is started at the configuration with equidistant spacing on the circle. DMR proves that the process is determinantal and the spatio-temporal correlation kernel is determined. By taking temporally homogeneous limits of the present elliptic determinantal process, trigonometric and hyperbolic versions of noncolliding diffusion processes are studied.
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1 Introduction
Eigenvalue distributions of random-matrix ensembles provide important examples of determinantal point processes, in which any correlation function is given by a determinant specified by a single continuous function called the correlation kernel [3, 15, 34, 39, 40]. Dyson’s Brownian motion model with parameter \(\beta =2\) [10, 41], which we simply call the Dyson model in this paper, and other noncolliding diffusion processes [8, 18, 21, 25, 29] are dynamical extensions of random-matrix ensembles. There any spatio-temporal correlation function is expressed by determinant [12, 35] and such processes are said to be determinantal [7, 26]. The noncolliding diffusion processes have attracted much attention in probability theory also by the fact that they are realized as \(h\)-transforms in the sense of Doob of absorbing particle systems in the Weyl chambers [18, 25, 29]. The relationship between the above mentioned integrability as spatio-temporal models and \(h\)-transform constructions as stochastic processes has been clarified by introducing a notion of determinantal martingales in [23, 24, 28]. The purpose of the present paper is to report elliptic extensions of these determinantal processes. Since the Dyson model can be regarded as a multivariate extension of the three-dimensional Bessel process, BES(3) [5, 27], first we discuss an elliptic extension of BES(3).
Let \(i=\sqrt{-1}, v, \tau \in \mathbb {C}\) and put
The Jacobi theta function \(\vartheta _1\) is defined as
(Note that the present function \(\vartheta _1(v; \tau )\) is represented as \(\vartheta _1(\pi v,q)\) in [44].) For \(\mathfrak {I}\tau >0\), \(\vartheta _1(v; \tau )\) is holomorphic for \(|v| < \infty \) and satisfies the partial differential equation
With parameters \(N \in \mathbb {N}\equiv \{1,2, \ldots \}, \alpha > 0\), and \(0 < t_{*} < \infty \), we introduce the following function of \((t, x) \in [0, t_{*}) \times \mathbb {R}\),
where \(\vartheta _1^{\prime }(v; \tau )=d \vartheta _1(v; \tau )/dv\). As a function of \(x \in \mathbb {R}\), it is odd,
and periodic with period \(\alpha \)
It has only simple poles at \(x= m \alpha , m \in \mathbb {Z}\), and simple zeroes at \(x=(m+1/2) \alpha , m~\in ~\mathbb {Z}\).
Let \(r > 0\). Suppose that \(\check{X}(t), t \in [0, t_{*})\) satisfies the stochastic differential equation (SDE)
started at \(\check{X}(0)=x \in (0, 2 \pi r)\), where \(B\) denotes the one-dimensional standard Brownian motion (BM). (From now on, BM means a one-dimensional standard Brownian motion unless specially mentioned). By periodicity (1.6) with period \(\alpha =2 \pi r\), \(\check{X}\) can be considered to describe a diffusion process of a particle moving around a circle with radius \(r>0\); \(\mathrm{S}^1(r)=\{x \in \mathbb {R}: x + 2 \pi r = x\}\). Note that this system is temporally inhomogeneous defined only in a time interval \([0, t_{*})\). Independently of \(t \in [0, t_{*})\) and \(N\), however, we have
It implies that the behavior of \(\check{X}\in (0, 2 \pi r)\) in the vicinity of 0 (and \(2 \pi r\)) is similar to that of BES(3) near 0. We define a process \(X \in [0, 2 \pi r)\) by
It gives a Markov process showing a position on the circumference \([0, 2 \pi r)\) of \(\mathrm{S}^1(r)\). We write the probability law of \(X(t), t \in [0, t_{*})\) started at \(x=X(0) \in (0, 2 \pi r)\) as \({\mathbb P}_x\).
The backward Kolmogorov equation for the SDE (1.7) is given as
Let \(q(t-s, y|x)\) be a solution of diffusion equation \(-\partial v(s,x)/\partial s=(1/2) \partial ^2 v(s,x)/\partial x^2\), \(0 \le s \le t < \infty , x \in (0, 2 \pi r)\) satisfying \(\lim _{s \uparrow t} v(s,x)=\delta _y(\{x\})\), \(y \in (0, 2 \pi r)\). Then
solves (1.10) under the condition \(\lim _{s \uparrow t} u(s,x)=\delta _y(\{x\})\), \(y \in (0, 2 \pi r)\), since \(\vartheta _1(v; \tau )\) satisfies (1.3). For \(0 < y < 2 \pi r\), \(\vartheta _1(y/2 \pi r; i(t_{*}-t)/2 \pi r^2) >0\) and it has simple zeroes at \(y=0\) and \(y= 2 \pi r\), \(t \in [0, t_{*})\). Then, if \(q(t-s, y|x)\) is chosen as the transition probability density (tpd) of the absorbing BM in the interval \([0, 2 \pi r]\) with absorbing walls at \(x=0\) and \(x=2 \pi r\), (1.11) is strictly positive and finite for any \(x, y \in (0, 2 \pi r)\), \(0 \le s \le t < t_{*}\), and thus \(p(t, y|s, x)\) gives the tpd for the process \(X(t), t \in [0, t_{*})\). Let \(W(t), t \ge 0\) be BM started at \(x=W(0) \in (0, 2 \pi r)\), where its probability law is denoted by \(\mathrm{P}_x\). Consider a filtration \(\{\mathcal{F}_W(t): t \ge 0\}\) generated by \(W(t), t \ge 0\), which satisfies the usual conditions, and introduce a stopping time \(T_W = \inf \{ t >0: W(t) \in \{0, 2 \pi r\} \}\). Then the above fact implies that, for \(t \in [0, t_{*})\),
Note that
Thus, in the limit \(t_{*} \rightarrow \infty \), (1.7) becomes a temporally homogeneous SDE,
and (1.12) becomes
If we take the further limit \(r \rightarrow \infty \) in (1.13), we have \(X(t)=\check{X}(t), t \ge 0\) and
which is the SDE for BES(3) on \(\mathbb {R}_+=\{x > 0 : x \in \mathbb {R}\}\), and (1.14) becomes
where \(T^{\prime }_W=\inf \{t >0 : W(t)=0\}\). The relation (1.16) states that BES(3) is the Doob h-transform of the absorbing BM in \([0, \infty )\) with an absorbing wall at the origin, where the harmonic function is given by \(h(x)=x, x \ge 0\). We regard (1.13) as a trigonometric extension, and (1.7) as an elliptic extension of (1.15), respectively. The equality (1.16) is generalized to (1.14) and (1.12), respectively. We can also discuss a scaling limit realizing \(q \rightarrow 1\), in which hyperbolic version of (1.7) is obtained (see Sect. 5).
Let \(N \in \{2,3, \ldots \}\) and consider the following bounded region
which is called a scaled alcove of the affine Weyl group of type \(A_{N-1}\) (with scale \(2 \pi r\)) [19, 31]. With an additional condition \(x_1 \ge 0\), we also consider the space
Note that it is different from the scaled alcove of type \(C_N\) defined by \(\mathcal{A}^{C_N}_{2 \pi r}=\{{{\varvec{x}}}\in \mathbb {R}^N: 0 < x_1 < x_2 < \cdots < x_N < 2 \pi r\}\) [19, 31] which excludes \(x_1=0\) from \(\mathcal{A}_{[0, 2 \pi r)^N}\).
Now we introduce an \(N\)-particle extension of the above process, \(\check{{{\varvec{X}}}}^A(t)=(\check{X}^A_1(t), \ldots , \check{X}^A_N(t))\), \(t \in [0, t_{*})\). Assume that the initial configuration is chosen in the alcove
and an index \(\delta \in \pi r \mathbb {Z}\) is determined so that
Let
Then \(\check{{{\varvec{X}}}}^A(t), t \in [0, t_{*})\) is defined as a solution of the following set of SDEs on \(\mathbb {R}\),
\(1 \le j \le N, t \in [0, t_{*})\), where \(B_j, 1 \le j \le N\) are independent BMs on \(\mathbb {R}\). By (1.5), \(\sum _{1 \le j, k \le N, j \not = k}A_N^{2 \pi r}(t_{*}-s, x_j-x_k)=0\), and the summation of (1.17) over \(j=1,2, \ldots , N\) gives
where \(B\) is BM on \(\mathbb {R}\). We then define the process \({{\varvec{X}}}^A(t)=(X^A_1(t), \ldots , X^A_N(t)) \in [0, 2 \pi r)^N, t \in [0, t_*)\) by
It represents a Markov process showing the positions of \(N\) particles on the circumference \([0, 2 \pi r)\) of \(\mathrm{S}^1(r)\).
Let \(\mathfrak {M}([0, 2 \pi r))\) be the space of nonnegative integer-valued Radon measures on the interval \([0, 2 \pi r)\), which is a Polish space with the vague topology. Any element \(\xi \) of \(\mathfrak {M}([0, 2 \pi r))\) can be represented as \(\xi (\cdot ) = \sum _{j \ge 1}\delta _{x_j}(\cdot )\), in which the sequence of points in \([0, 2 \pi r)\), \({{\varvec{x}}}=(x_j)_{j \ge 1}\), satisfies \(\xi (K)=\sharp \{x_j : x_j \in K \} < \infty \) for any subset \(K \subset [0, 2 \pi r)\). Now we consider the process \({{\varvec{X}}}^A(t)\) as an \(\mathfrak {M}([0, 2 \pi r))\)-valued process and write it as
The probability law of \(\Xi ^A(t, \cdot ), t \in [0, t_{*})\) starting from a fixed configuration \(\xi \in \mathfrak {M}([0, 2 \pi r))\) is denoted by \({\mathbb P}^A_{\xi }\) and the process specified by the initial configuration is expressed by \(\big (\Xi ^A(t), t \in [0,t_{*}), {\mathbb P}^A_{\xi }\big )\). The expectations with respect to \({\mathbb P}^A_{\xi }\) is denoted by \({\mathbb E}^A_{\xi }\). We introduce a filtration \(\{\mathcal{F}_{\Xi ^A}(t) : t \in [0,t_{*}) \}\) generated by \(\Xi ^A(t), t \in [0, t_*)\), which satisfies the usual conditions. Let \(\mathrm{C}([0, 2 \pi r))\) be the set of all continuous real-valued functions on \([0, 2 \pi r)\). We set
which denotes a collection of configurations without any multiple points.
In the present paper, we study the case that the initial state \(\eta =\sum _{j=1}^N \delta _{v_j}\) is corresponding to the configuration with equidistant spacing on \(\mathrm{S}^1(r)\);
and we will prove that the process \((\Xi ^A(t), t \in [0, t_{*}), {\mathbb P}^A_{\eta })\) is determinantal. In this case, \(\sum _{j=1}^N v_j=\pi r(N-1)\) and the index \(\delta \in \pi r \mathbb {Z}\) is determined as
so that \(\overline{v}_{\delta }= \pi r \in (0, 2 \pi r)\). We will present that this determinantal process can be considered as an elliptic extension of the Dyson model. The key lemmas to construct the elliptic determinantal process are obtained from the elliptic determinant evaluations related to infinite families of irreducible reduced affine root systems studied in [13–15, 30, 37, 42, 43]. According to Macdonald’s classification of reduced affine root systems [33], the present process is related to the system of type \(A_{N-1}\). Further study concerning other types is in progress. Connection between the elliptic determinantal processes and probabilistic discrete models with elliptic weights [4, 6, 38] will be an interesting future problem. We note that the function \(A_N^{\alpha }(t_{*}-t, z)\) can be regarded as Villat’s kernel for an annulus \(\mathbb {A}_q=\{z \in \mathbb {C}: q < |z| < 1\}\) with \(0 < q=e^{-2 \pi ^2 N(t_*-t)/\alpha ^2} < 1\). It is the reason why it also appears in the study of stochastic Komatu-Loewner evolution in doubly connected domains [2, 45].
The paper is organized as follows. In Sect. 2 preliminaries of elliptic functions and their related functions are given. Useful determinantal identities are obtained from the elliptic determinant evaluations related to the affine root system of types \(A_{N-1}\) [13–15, 30, 37, 42, 43]. There the generalized \(h\)-transform and determinantal martingale for \(\big (\Xi ^A(t), t \in [0, t_{*}), {\mathbb P}^A_{\eta }\big )\) are introduced. The main results are given in Sect. 3. First the determinantal martingale representation (DMR) is given for \(\big (\Xi ^A(t), t \in [0, t_{*}), {\mathbb P}^A_{\eta }\big )\) (Theorem 3.1). As a result of Theorem 1.3 of [23], DMR proves that the process is determinantal and the correlation kernel is determined (Corollary 3.3). Explicit expression for the correlation kernel of \(\big (\Xi ^A(t), t \in [0, t_*), {\mathbb P}^A_{\eta }\big )\) is shown and the infinite-particle limit is discussed. In Sect. 4 the temporally homogeneous limit \(t_{*} \rightarrow \infty \) is studied both in the level of SDEs and in the level of determinantal process. We study the system of noncolliding Brownian motions on a circle, \(\widehat{\Xi }^A(t), t \in [0, \infty )\), obtained from \(\Xi ^A(t), t \in [0, t_*)\) by this reduction. It is different from the dynamical circular unitary ensemble (CUE) model studied in [20, 23, 36]. Finally in Sect. 5, the results are expressed by using Gosper’s \(q\)-sine function [17] as well as the \(q\)-gamma function [1]. Then \(q \rightarrow 1\) limit is discussed, in which temporally homogeneous processes expressed by hyperbolic functions are obtained.
2 Preliminaries
2.1 Elliptic functions and their related functions
The Jacobi theta function \(\vartheta _1\) defined by (1.2) has the following infinite-product expressions,
with
It is easy to see from these expressions that, when \(\mathfrak {R}\tau =0, \mathfrak {I}\tau >0\) (that is, \(0 < q < 1\)), \(\vartheta _1(v; \tau )>0\) for \(v \in (0, 1)\) and it has simple zeroes at \(v=0\) and \(v=1\). It is odd with respect to \(v\)
and has quasi-periodicity
We define
as usual. (Note that the present functions \(\vartheta _{\mu }(v; \tau )\), \(\mu =1,2,3\) are denoted by \(\vartheta _{\mu }(\pi v,q)\), and \(\vartheta _0(v; \tau )\) by \(\vartheta _4(\pi v, q)\) in [44].) They solve the partial differential equation
We will use the following formulas; \(n \in \mathbb {N}\), \(q=e^{\pi i \tau }\),
where
and
The formula (2.6) is obtained from Eq. (2.3) of [37] (see (2.17) below) and (2.7) is obtained as its \(v \rightarrow 0\) limit, since \(\vartheta _1(v; \tau )\) has a simple root at \(v=0\). The formula (2.9) is found in ‘Miscellaneous Examples’ of Chapter XXI in [44].
Let \(\omega _1\) and \(\omega _3\) be fundamental periods and set
The Weierstrass \(\wp \) function and zeta function \(\zeta \) are defined as the following meromorphic functions
Let \(\omega _2=-(\omega _1+\omega _3)\), and put \(\zeta (\omega _{\nu })=\eta _{\nu }, \nu =1,2,3\). The relation \(\eta _1+\eta _2+\eta _3=0\) holds. By definition, \(\wp (z)\) is an elliptic function with fundamental periods \(\omega _1, \omega _2\) and \(\omega _3\),
and it is even, \(\wp (-z)=\wp (z)\). The function \(\zeta \) is odd, \(\zeta (-z)=-\zeta (z)\), and is quasi-periodic in the sense
By definition,
Moreover, the relation
holds (see Section 20.41 in [44]). From this, we obtain the following identity.
Lemma 2.1
For \(a, b, c \in \mathbb {C}\),
Proof
Put \(z=a-b, u=b-c\) in (2.11). Then by the fact that \(\zeta \) is odd, the equality is derived. \(\square \)
The following relation is established,
It has the following expansion with respect to \(q = q(\tau ) = e^{\pi i \tau }\),
The function \(A_N^{\alpha }(t_{*}-t, x)\) defined by (1.4) is then expressed as
for \(t \in [0, t_{*})\), \(x \in \mathbb {R}\), where
The formula (2.12) gives
From this expression, we can readily observe (1.5) and (1.6) and other properties of \(A_N^{\alpha }\). In particular, we can see (1.8).
2.2 Elliptic determinant identities
Let \(p\) be a fixed complex number such that \(0<|p|<1\). We use the standard notations
Following [37, 43], here we use ‘multiplicative notation’ for theta functions,
The function \(E(s;p)\) is holomorphic for \(s \not =0\). The zero set of \(E(s;p)\) is given by \(p^{\mathbb {Z}} \equiv \{p^j: j \in \mathbb {Z}\}\), and all zeroes are single. The inversion formula
the quasi-periodicity \(E(ps; p)=-E(s; p)/s\), and the Laurent expansion
are known. If \(g_n\) denotes a primitive \(n\)th root of unity, \(n \in \mathbb {N}\), the following equality holds [37],
With (1.1), the function \(E(s;p)\) is related to the Jacobi theta function \(\vartheta _1\) by
The equality (2.17) is then rewritten as (2.6).
For \(N \in \mathbb {N}\), \({{\varvec{s}}}=(s_1, \ldots , s_N) \in \mathbb {C}^N\), put
where the second equality is proved by the inversion formula (2.16). It is the Macdonald denominator for the reduced affine root systems of type \(A_{N-1}\) [37].
We start with the following two lemmas, which are readily obtained from the results given in [30, 37, 42, 43].
Lemma 2.2
For \({{\varvec{r}}}=(r_1, \ldots , r_N) \in \mathbb {C}^N\) and \(\kappa \in \mathbb {C}\) assume that
Then for \({{\varvec{s}}}\in \mathbb {C}^N\)
Proof of Lemma 2.2
Let \(s_1, \ldots , s_N\), \(a_1, \ldots , a_{N-1}\), \(b_1, \ldots , b_N\), \(c_2, \ldots , c_N\), and \(\kappa \) be indeterminates, which satisfy
The following equality holds [42] (see Corollary 4.5 and Remark 4.6 in [37], and see also Section 5.11 in [30]),
We put \(a_j=1/r_j, 1 \le j \le N-1\), \(b_j=\alpha /r_j, 1 \le j \le N\), \(c_j=1/r_j, 2 \le j \le N\) with \(\alpha \in \mathbb {C}\). Then the condition (2.21) gives
and (2.22) becomes
Under the assumption (2.19) we divide the both sides of (2.24) by \(E(\alpha ; p)^N \prod \nolimits _{1 \le k, \ell \le N, k \not = \ell } E(r_k/r_{\ell }; p)\) and obtain the equality
If we use (2.23), we obtain (2.20). \(\square \)
The following equality is given in the first line of Proposition 6.1 in [37].
Lemma 2.3
For \({{\varvec{s}}}\in \mathbb {C}^N\), \(\kappa \in \mathbb {C}\),
From now on, we assume \( \mathfrak {R}\tau = 0, \mathfrak {I}\tau > 0\), that is, \(0 < q=e^{\pi i \tau } < 1\). It is obvious from (1.2) that if \(v \in \mathbb {R}\), then \(\vartheta _1(v; \tau ) \in \mathbb {R}, |\vartheta _1(v; \tau )| < \infty \). We set \(p=q^2\) and \(s_j = e^{i x_j/r}\), \(r_j = e^{i u_j/r}\), \(\kappa =e^{i \delta /r}\) in Lemma 2.2, \(x_j, u_j \in \mathbb {R}, 1 \le j \le N\), \(\delta \in \pi r \mathbb {Z}\). Then, through (2.18), these lemmas are rewritten as follows.
Lemma 2.4
Assume that \({{\varvec{u}}}=(u_1, \ldots , u_N) \in \mathcal{A}_{[0, 2 \pi r)^N}\) and \(\overline{u}_{\delta } = \delta +\sum _{j=1}^N u_j \in (0, 2 \pi r)\). Let \(\overline{x}_{\delta }=\delta +\sum _{j=1}^N x_j\). Then
Similarly Lemma 2.3 gives the following.
Lemma 2.5
with
Let \(\eta (x)\) denotes Dedekind’s \(\eta \)-function [11, 33],
The following equalities were proved as Proposition 5.6.3 in [15] (see also [13, 14]).
Lemma 2.6
Let \(\alpha \in \mathbb {C}\). For \(N\) odd
while for \(N\) even
2.3 Generalized \(h\)-transform
The backward Kolmogorov equation for (1.17) is given as
We write the tpd of the process \(\Xi ^A(t), t \in [0, t_*)\) as \(p_N^A(t, {{\varvec{y}}}|s, {{\varvec{x}}})=p_N^A(t, {{\varvec{y}}}|s, {{\varvec{x}}}; r, t_*), 0 \le s \le t < t_{*}\), provided that \({{\varvec{x}}}, {{\varvec{y}}}\in \mathcal{A}_{[0, 2 \pi r)^N}\) and \(\overline{x}_{\delta }, \overline{y}_{\delta } \in (0, 2 \pi r)\). Since the configuration of the process \(\Xi ^A(t), t \in [0, t_*)\) is unlabeled as (1.20), we solve (2.30) to obtain \(p_N^A\) under the ‘initial condition’
where \(\mathcal{S}_N\) denotes a collection of all permutations of \(N\) indices. It is a moderated version of the usual one \(\lim _{s \uparrow t} u^A(s, {{\varvec{x}}})= \prod _{j=1}^N \delta _{y_j}( \{x_j\})\) for processes with labeled configurations.
We consider the Brownian motion \({{\varvec{V}}}^r(\cdot )\) started at \({{\varvec{u}}}\in \mathcal{A}_{[0, 2 \pi r)^N}\) with an index \(\delta \in \pi r \mathbb {Z}\) chosen as \(\overline{u}_{\delta } \in (0, 2 \pi r)\), which is killed when it arrives at the boundary of \(\mathcal{A}^{A_{N-1}}_{2 \pi r}\) and when \(\overline{V}^{r}_{\delta } (\cdot )\in \{0, 2 \pi r\}\). Let \(q_N^{A}(t-s, {{\varvec{y}}}|{{\varvec{x}}})=q_N^{A}(t-s, {{\varvec{y}}}|{{\varvec{x}}}; r)\), \({{\varvec{x}}}, {{\varvec{y}}}\in \mathcal{A}_{[0, 2 \pi r)^N}\), \(0 < s < t < t_{*}\) be the tpd of \({{\varvec{V}}}^r(\cdot )\), which satisfies
Lemma 2.7
The tpd of the process \(\Xi ^A(t), t \in [0, t_*)\) is given by
where \(\overline{x}_{\delta }, \overline{y}_{\delta } \in (0, 2 \pi r)\) and
\(t \in [0, t_{*}), {{\varvec{x}}}\in \mathcal{A}_{[0, 2 \pi r)^N}\).
Proof
Set
and put it into (2.30) assuming that \(f^A\) is \(\mathrm{C}^1\) in \(t\) and \(\mathrm{C}^2\) in \({{\varvec{x}}}\). Then we have
since \(q_N^{A}(t-s,{{\varvec{y}}}|{{\varvec{x}}})\) satisfies the diffusion equation. We put
where \(g^A\) is a \(\mathrm{C}^1\) function of time \(s\) to be determined. By definition (1.4), we see
and
Then (2.35) gives the equation
where the sum in the last term denotes the summation over \(1 \le j, k, \ell \le N\) conditioned that \(j, k, \ell \) are all distinct. By the setting (2.36),
where \(\dot{\vartheta }_1(x; \tau )=d \vartheta _1(x; \tau )/d \tau \) and (2.2) was used. Since (1.3) is satisfied, the above is equal to
Therefore, (2.38) becomes
Now we rewrite RHS of (2.39) by using the functions \(\wp \) and \(\zeta \) through (2.13). First we see
For
Lemma 2.1 gives
On the other hand, differentiation of (2.13) with respect to \(x\) gives
where (2.10) was used. Combining it with (2.37) gives
Then
It is easy to prove that
by using the fact that \(\zeta \) is odd. Then Eq. (2.39) is reduced to be
Since \(\eta _1(t_{*}-t)\) is explicitly given as (2.14) with \(\alpha =2 \pi r\), this equation can be solved as
with a constant \(c^{\prime }\). The solution (2.34) has been determined of the form
For (2.32), the condition (2.31) is satisfied, if and only if
Since \(\vartheta _1(x/2\pi r; iN(t_{*}-t)/2 \pi r^2)>0\) if \(x \in (0, 2 \pi r)\), \(t \in [0, t_{*})\), and \(q_N^A(t-s, {{\varvec{y}}}|{{\varvec{x}}})\) is assumed to be the tpd of \({{\varvec{V}}}^r(\cdot )\), we can conclude that \(0< p_N^{A}(t, {{\varvec{y}}}|s, {{\varvec{x}}}) < \infty \) for any \({{\varvec{x}}}, {{\varvec{y}}}\in \mathcal{A}_{[0, 2 \pi r)^N}\), \(0 \le s \le t < t_{*}\), where \(\delta \) is chosen so that \(\overline{x}_{\delta }, \overline{y}_{\delta } \in (0, 2 \pi r)\). Then the proof is completed. \(\square \)
Let \(\check{{{\varvec{W}}}}(t)=(\check{W}_1(t), \ldots , \check{W}_N), t \ge 0\) be \(N\)-dimensional Brownian motion on \((\mathrm{S}^1(r))^N\) started at \({{\varvec{u}}}\in \mathcal{A}^{A_{N-1}}_{2 \pi r}\). The expectation with respect to this process is denoted by \(\check{\mathrm{E}}_{{{\varvec{u}}}}\). Consider a stopping time
Put \(\overline{W}_{\delta }=\delta +\sum _{j=1}^N \check{W}_j(t)\), where the index \(\delta \in \pi r \mathbb {Z}\) is determined so that \(\overline{u}_{\delta } \in (0, 2 \pi r)\). Then we also consider the following stopping time
For the process \(\Xi ^A(t), t \in [0, t_{*})\) is \(\mathfrak {M}([0, 2 \pi r))\)-valued, measurable functions are symmetric functions of \(N\) variables \(X^A_j, 1 \le j \le N\) at each time. By the definition (1.19) for \({{\varvec{X}}}^A\), they should be periodic with period \(2 \pi r\). Let \(T \in [0, t_{*})\). Then any \(\mathcal{F}_{\Xi ^A}(T)\)-measurable function \(F\) will be given as follows. With an arbitrary integer \(M \in \mathbb {N}\) and arbitrary sequence of times \(0 \le t_1 < \cdots < t_M \le T\),
where \(g_m({{\varvec{x}}}), 1 \le m \le M\) are symmetric functions and
for \(1 \le m \le M\).
The indicator function of \(\omega \) is denoted by \(\mathbf{1}(\omega )\); \(\mathbf{1}(\omega )=1\) if \(\omega \) is satisfied, and \(\mathbf{1}(\omega )=0\) otherwise. Lemma 2.7 implies the following equality.
Proposition 2.8
Suppose \(\xi =\sum _{j=1}^N \delta _{u_j} \in \mathfrak {M}_0([0, 2 \pi r))\). Let \(T \in [0, t_{*})\). For any \(\mathcal{F}_{\Xi ^A}(T)\)-measurable observable \(F\),
See Remark 2 at the end of Sect. 3.1.
2.4 Markov process \({{\varvec{W}}}^r\)
We write the tpd of BM on \(\mathbb {R}\) as
By wrapping it on \(\mathrm{S}^1(r)\), we define
\(x, y \in [0, 2 \pi r)\), \(t \ge 0\). Using the Jacobi theta functions (2.5), it is written as
We find that by Jacobi’s imaginary transformations [44],
the above is further rewritten as
Lemma 2.6 given by Forrester [15] implies the following.
Proposition 2.9
For \(N \in \{2,3, \ldots \}\), \({{\varvec{v}}}\) is given by (1.21). Then for \({{\varvec{y}}}\in \mathcal{A}_{[0, 2 \pi r)^N}\), \(t>0\),
Proof
For \(N\) odd, we put \(\alpha =1/N+\tau /2\) in (2.28), and for \(N\) even, \(\alpha =1/N+(1+\tau )/2\) in (2.29). Let \(\tau =it/2 \pi r^2\). Then we have
where quasi-periodicity of \(\vartheta _{\mu }, \mu =0,1,2,3\) has been used. By expression (2.43) with (2.5), it is obvious that (2.45) satisfies the diffusion equation. This expression (2.45) guarantees the positivity and finiteness of \(\det _{1 \le j, k \le N}[ p^r_{A_{N-1}}(t, y_j| v_k)]\) for \({{\varvec{y}}}\in \mathcal{A}_{[0, 2 \pi r)^N}\) and \(\overline{y}_{-\pi r (N-2)} \in (0, 2 \pi r)\). Equation (2.45) also shows that it vanishes when \(y_j=y_k\) for any \(j \not = k\) and when \(\overline{y}_{-\pi r (N-2)} \in \{0, 2\pi r\}\). By the expression (2.42) and the argument given by Liechty and Wang [32] (see also [16]), we can prove that (2.45) satisfies the moderated initial configuration
Then the proof is completed. \(\square \)
Remark 1
The Karlin-McGregor-type [22] determinantal formula (2.44) for \(q_N^A(t, {{\varvec{y}}}|{{\varvec{x}}})\) given in Proposition 2.9 is crucial for DMR which will be proved in Theorem 3.1. In the present paper, we obtained it for the special initial configuration (1.21) due to the explicit evaluation (2.45) given by Lemma 2.6 of Forrester [15]. If we obtain the Karlin-McGregor-type determinantal formula for \(q_N^A\) for other initial configuration, we can prove DMR for the process and it will immediately conclude that the process is determinantal by Theorem 1.3 of [23].
Given \(N \in \mathbb {N}\), let \(W^r(t), t \ge 0\) be a Markov process in \([0, 2 \pi r)\) such that its transition density is given by \(p^r_{A_{N-1}}(t, y|x), t \ge 0, x, y \in [0, 2 \pi r)\), defined by (2.42). Then we introduce an \(N\) independent copies of \(W^r(t), t \ge 0\), denoted by \(W^r_j(t), t \ge 0, 1 \le j \le N\) and let \({{\varvec{W}}}^r(t)=(W^r_1(t), \ldots , W^r_N(t))\), \(t \ge 0\). The probability space of the process is denoted by \((\Omega _{W^r}, \mathcal{F}_{W^r}, \mathrm{P}_{{{\varvec{v}}}}^r)\), and the expectation is written as \(\mathrm{E}_{{{\varvec{v}}}}^r\), where the initial configuration is given by \({{\varvec{v}}}\) with (1.21). A filtration \(\{\mathcal{F}_{W^r}(t) : t \ge 0\}\) is generated by \({{\varvec{W}}}^r(t), t \ge 0\), which satisfies the usual conditions.
2.5 Martingales and complex Brownian motions
Let \(0 < t_{*} < \infty \) and \(\xi =\sum _{j=1}^N \delta _{u_j} \in \mathfrak {M}_0([0, 2 \pi r))\). For \(1 \le k \le N\), define
and
Since \(\Phi ^A_{\xi , u_k}(z), 1 \le k \le N\) are holomorphic for \(|z| < \infty \), (2.47) is written as
where \(\widetilde{W}\) denotes a BM on \(\mathbb {R}\) started at 0, which is independent of \({{\varvec{W}}}^r\), and \(\widetilde{\mathrm{E}}\) does the expectation for \(\widetilde{W}\). Then the following is proved.
Lemma 2.10
Assume \(v_j, 1 \le j \le N\) are given by (1.21) and \(\eta =\sum _{j=1}^N \delta _{v_j}\). Then
-
(i)
\(\mathcal{M}^A_{\eta , v_k}(t, W^r(t)), 1 \le k \le N, t \in [0, t_*)\) are continuous-time martingales; \(\mathrm{E}^r[ \mathcal{M}^A_{\eta , v_k}(t, W^r(t)) | \mathcal{F}_{W^r}(s)] =\mathcal{M}^A_{\eta , v_k}(s, W^r(s))\) a.s. for any two bounded stopping times with \(0 \le s \le t < t_*\).
-
(ii)
For any \(t \in [0, t_*), \mathcal{M}^A_{\eta , v_k}(t, x),1 \le k \le N\), are linearly independent functions of \(x \in [0, 2 \pi r)\),
-
(iii)
\(\mathcal{M}^A_{\eta , v_k}(0,v_j)=\delta _{jk}, \quad 1 \le j, k \le N\).
Proof
(i) For the quasi-periodicity (2.3) of \(\vartheta _1\), the expression (2.48) with (2.46) implies that, for \(\ell \in \mathbb {Z}\),
Then, for \(0 \le s \le t < t_*, 1 \le k \le N\), (2.42) gives
By (2.49), it is equal to
By Definition (1.2) of \(\vartheta _1\), we will obtain the following expansions; for \(1 \le k \le N\),
where the coefficients \(b^{\ell }_{n_{\ell }}, 0 \le \ell \le N, \ell \not = k\) are functions of \(\{u_{\ell }\}_{\ell =1}^N\), \(t_*, N\) and \(r\). Then, if we introduce an \(N\)-component index \({{\varvec{n}}}=(n_0, n_1, \ldots , n_{k-1}, n_{k+1}, \ldots , n_N)\) for each \(1 \le k \le N\), and put \(B^k_{{{\varvec{n}}}}=\prod _{0 \le \ell \le N, \ell \not = k} b^{\ell }_{n_{\ell }}\), (2.48) with \(\xi =\eta , u_k=v_k, x=w\) is expanded as
where
For any \(\alpha \in \mathbb {C}\), it is easy to confirm that
Then (2.50) gives
and hence (i) is concluded. As a matter of course, \(\eta \in \mathfrak {M}_0([0, 2 \pi r))\), and then the zeroes of \(\Phi ^A_{\eta , v_j}(z)\) are distinct from those of \(\Phi ^A_{\eta , v_k}(z)\), if \(j \not = k\). Then (ii) is proved. By (2.48), \(\mathcal{M}^A_{\eta , v_k}(0, x)=\lim _{t \downarrow 0} \widetilde{\mathrm{E}}\left[ \Phi ^A_{\eta , v_k}(x+i \widetilde{W}(t))\right] =\Phi ^A_{\eta , v_k}(x), 1 \le k \le N\). Since \(\Phi ^A_{\eta , v_k}(v_j)=\delta _{jk}, 1 \le j, k \le N\) by Definition (2.46), (iii) is also satisfied. \(\square \)
Let \(\widetilde{W}_j(\cdot ), 1 \le j \le N\) be independent \(N\) copies of \(\widetilde{W}(\cdot )\). For \(\widetilde{{{\varvec{W}}}}(t)=(\widetilde{W}_1(t), \ldots , \widetilde{W}_N(t)), t \ge 0\), the probability space is denoted by \((\widetilde{\Omega }_W, \widetilde{\mathcal{F}}_W, \widetilde{\mathrm{P}})\) with expectation \(\widetilde{\mathrm{E}}\). We put
which are independent complex Brownian motions on \(\mathbb {C}(r) \equiv [0, 2 \pi r) \times i \mathbb {R}\). The probability space for \({{\varvec{Z}}}^r(t)=(Z^r_1(t), \ldots , Z^r_N(t))\), \(t \ge 0\) is given by the direct product of the two spaces, \((\Omega _{W^r}, \mathcal{F}_{W^r}, \mathrm{P}_{{{\varvec{u}}}}^r)\) for \({{\varvec{W}}}^r(\cdot )\) and \((\widetilde{\Omega }_W, \widetilde{\mathcal{F}}_W, \widetilde{\mathrm{P}})\) for \(\widetilde{{{\varvec{W}}}}(\cdot )\), which is denoted by \((\Omega ^r, \mathcal{F}^r, \mathbf{P}_{{{\varvec{u}}}}^r)\) with expectation \(\mathbf{E}_{{{\varvec{u}}}}^r\).
Proposition 2.11
Let \(\overline{Z}^r_{\delta }(t)=\delta +\sum _{j=1}^N Z^r_j(t)\) and \(\overline{W}^r_{\delta }=\delta +\sum _{j=1}^N W^r(t), t \ge 0\). Then the following equality holds,
Proof
By (2.25) in Lemma 2.5
where the multilinearity of determinant and independence of \(Z^r_j(t)\)’s have been used. Using the Laurent expansion (1.2), we have
Here
Then (2.52) is equal to
Put this into (2.51), we have
where (2.25) of Lemma 2.5 was used again. By (2.26),
If we set \(\tau =iNt_{*}/2 \pi r^2\), the equality is obtained. \(\square \)
For \({{\varvec{W}}}^r(t), t \ge 0\), define
which we call the determinantal martingale [23]. By Lemma 2.10, it is a continuous-time martingale. Then the following equality is established.
Lemma 2.12
Assume that \(\xi =\sum _{j=1}^N u_j \in \mathfrak {M}_0([0, 2 \pi r))\) and \(\overline{u}_{\delta } \in (0, 2 \pi r)\). Then
Proof
By multilinearity of determinant and independence of \(\widetilde{W}_j(\cdot ), 1 \le j \le N\), (2.53) with (2.46) and (2.48) gives
By Lemma 2.4, it is equal to
Then we apply Proposition 2.11. By definition (2.33) of \(h_N^{A}\), the equality is obtained.
\(\square \)
3 Main results
3.1 Determinantal martingale representation
By Lemmas 2.12 we obtain the following representation. We call it the determinantal martingale representation (DMR) for the process \((\Xi ^A(t), t \in [0, t_{*}), {\mathbb P}^A_{\eta })\).
Theorem 3.1
Suppose that \(N \in \mathbb {N}\), \(\eta =\sum _{j=1}^{N} \delta _{v_j}\) with (1.21) and (1.22). Let \(T \in [0, t_{*})\). For any \(\mathcal{F}_{\Xi ^A}(T)\)-measurable observable \(F\),
Note that the second representation of (3.1) is an elliptic extension of the complex Brownian motion representation reported in [28] for the Dyson model (i.e. the noncolliding BM).
Proof
It is sufficient to consider the case that \(F\) is given as (2.40). Moreover, by Markov property, it is enough to prove the case \(M=1\); \(0 \le t_1 \le T < \infty \). Here we prove the equalities
where \(g_1\) is a symmetric function having periodicity (2.41). By Proposition 2.8,
The definition of \(h_N^A\) given by (2.33) and the initial condition \(\eta \) give
By the determinantal formula (2.44) of \(q_N^A\) given in Proposition 2.9, the above is written as
where \({{\varvec{W}}}^r(t_1)=(W^r_1(t_1), \ldots , W^r_N(t_1))\) and the transition density of each \(W^r_j\) is given by (2.42), \(1 \le j \le N\). Here we used the notation \(\sigma ({{\varvec{x}}})=(x_{\sigma (1)}, \ldots , x_{\sigma (N)})\) for \(\sigma \in \mathcal{S}_N\). Since
(3.4) is equal to
Then by Lemma 2.12, we obtain the first line of (3.2). By definitions of \(\mathbf{E}_{{{\varvec{v}}}}^r\) and \(\mathcal{D}^A_{\eta }\) given by (2.53) with (2.48), the second line of (3.2) is also obtained. \(\square \)
Remark 2
The function \(h^A_N(t_*-t, {{\varvec{x}}}), t \in [0, t_*)\) is not a harmonic function of \({{\varvec{x}}}\), but Lemma 2.10 proves that \(\mathcal{D}^A_{\eta }(t, {{\varvec{W}}}^r(t))\) given by (2.53) is a continuous-time martingale, where \({{\varvec{W}}}^r(t)\) is a Markov process defined by using Brownian motion in Sect. 2.4. Then Itô’s formula implies
where \(\Delta =\sum _{j=1}^N \partial ^2/\partial x_j^2\). In this sense, DMR is a time-dependent extension of \(h\)-transform [23].
3.2 Determinantal process
For any integer \(M \in \mathbb {N}\), a sequence of times \({{\varvec{t}}}=(t_1,\ldots ,t_M)\) with \(0 \le t_1 < \cdots < t_M < t_{*}\), and a sequence of functions \({{\varvec{f}}}=(f_{t_1},\ldots ,f_{t_M}) \in \mathrm{C}([0, 2 \pi r))^M\), the moment generating function of multitime distribution of \(\left( \Xi ^A(t), t \in [0,t_{*}), {\mathbb P}^A_{\xi }\right) \) is defined by
It is expanded with respect to ‘test functions’ \(\chi _{t_m}(\cdot )=e^{f_{t_m}(\cdot )}-1, 1 \le m \le M\) as
and it defines the spatio-temporal correlation functions \(\rho _{\xi }(\cdot )\) for the process \((\Xi ^A(t), t \in [0,t_{*}), {\mathbb P}^A_{\xi })\).
Given an integral kernel \( \mathbb {K}(s,x;t,y), (s,x), (t,y) \in [0, t_{*}) \times [0, 2 \pi r), \) the Fredholm determinant is defined as
We put the following definition [7, 26].
Definition 3.2
For a given initial configuration \(\xi \), if any moment generating function (3.5) is expressed by a Fredholm determinant, we say the process \(\big (\Xi ^A(t), t \in [0, t_{*}), {\mathbb P}^A_{\xi }\big )\) is determinantal. In this case, all spatio-temporal correlation functions are given by determinants as
\(0 \le t_1 < \cdots < t_M < t_{*}\), \(1 \le m \le M\), \(1 \le N_m \le N\), \({{\varvec{x}}}^{(m)}_{N_m} \in [0, 2 \pi r)^{N_m}, 1 \le m \le M \in \mathbb {N}\). Here the integral kernel \(\mathbb {K}_{\xi }: ([0, t_{*}) \times [0, 2 \pi r))^2 \mapsto \mathbb {R}\) is called the (spatio-temporal) correlation kernel.
By Theorem 1.3 in [23], DMR given by Theorem 3.1 leads to the following result.
Corollary 3.3
For \(\eta =\sum _{j=1}^N \delta _{v_j}\) with (1.21) and (1.22), the process \((\Xi ^A(t), t \in [0, t_{*}), {\mathbb P}^A_{\eta })\) is determinantal with the correlation kernel
\((s,x), (t,y) \in [0, t_{*}) \times [0, 2 \pi r)\).
3.3 Explicit expression of \(\mathbb {K}^A_{\eta }\) and infinite-particle limit
For \(\eta =\sum _{j=1}^N \delta _{v_j}\) with (1.21) and (1.22), the entire functions (2.46) become
\(1 \le k \le N\), with \(\tau =\tau (t_*)=i N t_*/2 \pi r^2\), where we have used (2.3). Using the formulas (2.6) and (2.7), it is written as
\(1 \le k \le N\). If we apply the formula (2.9) and the Laurent expansion (1.2) of \(\vartheta _1\), we have
since \(\cot (\pi /2)=0\) and \(\sin (\theta +m \pi )=(-1)^m \sin \theta , m \in \mathbb {Z}\). For \(M \in \mathbb {N}\), let
It is easy to confirm the equality
Then we see
Here we have used the fact that, for \(M \in \mathbb {N}\), \(2 \sigma _M(m)+1=2 \sigma _{M-1}(m+1)\) if \(M\) is odd, and \(2 \sigma _M(m)+1=2 \sigma _{M-1}(m)\) if \(M\) is even, and that \(\sigma _{(2n-1)N-1}(m)=\sigma _{N-1}(m)\) for \(n, N \in \mathbb {N}\).
The functions
which give continuous-time martingales if we put \(x=W^r_j(\cdot ), 1 \le j \le N\) (Lemma 2.10 (i)), are calculated by performing Gaussian integrals for each term. By setting \(\tau =i N t_*/2 \pi r^2\), the result is expressed as
Then by Corollary 3.3, the correlation kernel is determined as
\((s,x), (t,y) \in [0, \infty ) \times [0, 2 \pi r)\), where
with (2.43). We note that, by using (3.10), (2.43) is written as
We find that, by using the identity \(\sum _{k=1}^N e^{-2i (k-1) \alpha \pi /N}=N \sum _{k \in \mathbb {Z}} \mathbf{1}(\alpha =k N)\), the above is expressed as follows,
where \(\vartheta _2\) and \(\vartheta _3\) are defined by (2.5).
We consider the infinite-particle limit with fixed particle-density;
We can see that the first term in (3.13) vanishes in this limit because of the overall factor \(1/r\), and obtain
The correlation kernel in this limit (3.14) is given by
\((s,x), (t,y) \in [0, t_*) \times \mathbb {R}\). The convergence of correlation kernel (3.15) in the limit (3.14) implies well-definedness of elliptic determinantal process with an infinite number of particles. Further study will be reported elsewhere (see Sect. 4.3 below).
4 Reduced processes in temporally homogeneous limit
4.1 Reduction in SDEs
Since we have
by (2.15), the limit \(t_{*} \rightarrow \infty \) of (1.17) gives a temporally homogeneous system of the SDEs,
\(1 \le j \le N\), and (1.18) becomes
The system of SDEs without the third term in RHS of (4.2) has been studied as a dynamical extension of the circular unitary ensemble (CUE) of random matrix theory [15, 20, 34]. It is interesting to see that a one-parameter extension of that system is discussed as a driving system for a multiple Schramm-Loewner evolution by Cardy [9]. Moreover, for
and \(\delta =\pi r n\) with a fixed \(n \in \mathbb {Z}\) determined by the initial configuration, the \(r \rightarrow \infty \) limit of the system (4.2) is given by
where \(X^A_j=\check{X}^A_j, 1 \le j \le N\) in \(r \rightarrow \infty \). It is the system of SDEs of the Dyson model (i.e., the noncolliding Brownian motion on \(\mathbb {R}\)).
4.2 Reduction in correlation kernel of determinantal process
Next we study the reduction as determinantal processes for \((\Xi ^A(t), t \in [0, t_*), {\mathbb P}^A_{\eta })\). For the configuration \(\eta =\sum _{j=1}^N \delta _{v_j}\) with (1.21) and (1.22), an explicit expression of correlation kernel \(\mathbb {K}^A_{\eta }\) was given in Sect. 3.3. In (3.13) we can take the temporally homogeneous limit \(t_* \rightarrow \infty \) as follows. We see from (2.8) and (2.5) that \(\vartheta _1^{\prime }(0; \tau ) \sim 2 \pi e^{\pi i \tau /4}\), \(\vartheta _2(v; \tau ) \sim 2 e^{\pi i \tau /4} \cos (\pi v)\), and \(\vartheta _3(v; \tau ) \rightarrow 1\) as \(\mathfrak {I}\tau \rightarrow + \infty \). Then, we have
where
The correlation kernel is given by
\((s,x), (t,y) \in [0, \infty ) \times [0, 2 \pi r)\). The determinantal process defined by the correlation kernel (4.6) is denoted by \((\widehat{\Xi }^A(t), t \in [0, \infty ), \widehat{{\mathbb P}}^A_{\eta })\). From the explicit expression (4.5), we can show that \(\left( \widehat{\Xi }^A(t), t \in [0, \infty \right) , \widehat{{\mathbb P}}^A_{\eta })\) exhibits a typical nonequilibrium phenomenon; relaxation to equilibrium.
Let \((\widehat{\Xi }^A(t), t \in [0, \infty ), \widehat{{\mathbb P}}^A_\mathrm{eq})\) be the equilibrium determinantal process, whose correlation kernel is homogeneous both in space \([0, 2 \pi r)\) and time \([0, \infty )\) and given by
\((s, x), (t,y) \in [0, \infty ) \times [0, 2 \pi r)\). Note that the spatial dependence of (4.7) is on a distance of two points \(|y-x|\).
Proposition 4.1
The process \((\widehat{\Xi }^A(t+T), t \in [0, \infty ), \widehat{{\mathbb P}}^A_{\eta })\) converges to \((\widehat{\Xi }^A(t), t \in [0, \infty ), \widehat{{\mathbb P}}^A_\mathrm{eq})\) as \(T \rightarrow \infty \) weakly in the sense of finite-dimensional distributions.
Note that if we ignore the terms containing \(\mathcal{K}^A\), the kernel (4.7) is equal to the correlation kernel of the equilibrium determinantal process of the noncolliding BM on a circle \(\mathrm{S}^1(r)\) with \(N-1\) particles, \(\mathbb {K}_\mathrm{eq}^\mathrm{CUE}(t-s, y-x; N-1,r)\) [23, 36]. It is a reversible process with respect to the eigenvalue-angle distribution of unitary random matrices with size \(N-1\) in the CUE (if we set \(r=1\)). With contribution from the term \(\mathcal{K}^A\), we have the particle density
which is uniform on \([0, 2 \pi r)\) as it should be.
Proof of Proposition 4.1
Since \(e^{-k^2 N^2 T/2r^2} \rightarrow 0, k \not = 0\) and \(\vartheta _3(v; i N^2 T/2 \pi r^2) \rightarrow 1\) as \(T \rightarrow \infty \), (4.5) gives
which is \(\widehat{\mathcal{G}}^A_\mathrm{eq}(t-s, y-x; N, r)\). If we set \(t=s\), the second term becomes
Combining the above results with the expression (3.12) of \(p^r_{A_{N-1}}\), we obtain (4.7). The convergence of correlation kernel \(\widehat{\mathbb {K}}^A_{\eta } \rightarrow \widehat{\mathbb {K}}^A_\mathrm{eq}\) in the long-term limit guarantees the convergence of moment generating function of multitime distribution \(\Psi ^A_{\eta , {{\varvec{t}}}}[{{\varvec{f}}}]\) given by the Fredholm determinant (3.6) of \(\widehat{\mathbb {K}}^A_{\eta }\). It implies the convergence \(\widehat{{\mathbb P}}^A_{\eta } \rightarrow \widehat{{\mathbb P}}^A_\mathrm{eq}\) in the long-term limit in the sense of finite-dimensional distributions. Then the proof is completed. \(\square \)
4.3 Reduction in infinite-particle system
The temporally homogeneous limit \(t_* \rightarrow \infty \) of (3.16) with (3.15) gives
\((s, x), (t, y) \in [0, \infty ) \times \mathbb {R}\). If we set \(\rho =1\), this correlation kernel is exactly the same as Eq. (1.5) in [26]. There we gave a set \(\mathfrak {X}\) of infinite-particle configurations, started at which the Dyson model is well defined as a determinantal process. The function (4.8) was derived as the correlation kernel of the Dyson model started at \(\xi ^{\mathbb {Z}} \equiv \sum _{j \in \mathbb {Z}} \delta _{j} \in \mathfrak {X}\). As shown in [26], for \((s,x), (t, y) \in [0, \infty ) \times \mathbb {R}\),
The limit kernel is the extended sine kernel with density \(\rho \) [15]. The convergence implies the infinite-dimensional relaxation phenomenon of the Dyson model from \(\xi ^{\mathbb {Z}}\) to equilibrium [26]. This observation is consistent with Proposition 4.1, since \(\lim _{N \rightarrow \infty , r \rightarrow \infty , \rho =\mathrm{const.}} \mathcal{K}^A(x; N, r)=0\).
5 Expressions by Gosper’s \(q\)-sine function and hyperbolic limit
Gosper defined his \(q\)-sine function as [17]
By the product form (2.1) of \(\vartheta _1(v;\tau )\), we find the equality
with (1.1). Then the function \(A_N^{\alpha }(t_{*}-t,x)\) defined by (1.4) is expressed as
with
The entire functions (2.46), with which the determinantal martingales and correlation functions are expressed, are rewritten as
\(1 \le k \le N, z \in \mathbb {C}\), with \(q=e^{-N(t_*-t)/2r^2}\). We remark that, the \(q\)-extension of the gamma function is defined as [1]
and the \(q\)-analogue of Euler’s reflection formula
holds. Then (5.1) and (5.3) are also expressed using the \(q\)-gamma function \(\Gamma _{q^2}\).
In the previous section, we studied the temporally homogeneous limit \(t_{*} \rightarrow \infty \) with fixed \(\alpha =2 \pi r\). By (5.2), it corresponded to the limit \(q \rightarrow 0\). In contrast to it, here we consider the following scaling limit,
By (5.2), it gives the limit \(q \rightarrow 1\). Since \( \lim _{q \uparrow 1} \sin _q(\pi z)=\sin (\pi z), \) and \(\sin (\pi i z)=i \sinh (\pi z)\), we obtain the limits
In this scaling limit the SDE (1.7) discussed in Sect. 1 becomes a temporally homogeneous process on \(\mathbb {R}_+\)
Similarly (5.5) and (5.6) enable us to discuss the determinantal process of a hyperbolic version of the Dyson model such that
\(1 \le j \le N\), \(t \ge 0\), summation of which over \(j=1,2, \ldots , N\) gives
Note that \(\lim _{q \uparrow 1} \Gamma _q(z)=\Gamma (z)\) and in this limit (5.4) with \(z \rightarrow iz\) gives \(\sinh (\pi z)=-\pi i/\{\Gamma (iz) \Gamma (1-iz)\}\), and thus the above limit is expressed also by using the gamma functions. If we take the further limit \(a \rightarrow \infty \) with \(\delta /a=\mathrm{constant} >0\) in (5.7), we obtain the Dyson model (4.4). In the argument in the present paper, analyticity of functions \(\Phi ^A_{\xi , u_k}(z), 1 \le k \le N\) plays an essential role to determine correlation kernels of determinantal processes. Since (5.3) is an entire function of \(z\), the calculations and results for the trigonometric version of the Dyson model given in Sect. 4.2 will be readily mapped to those for the hyperbolic versions with (5.7) and (5.8) by analytic continuation of \(z\) and suitable change of parameters.
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Acknowledgments
The present author would like to thank Saburo Kakei and Tomoyuki Shirai for useful comments on the manuscript. He also thank Masatoshi Fukushima for valuable comments on Villat’s kernel and the Komatu-Loewner evolution. This work is supported in part by the Grant-in-Aid for Scientific Research (C) (No.26400405) of Japan Society for the Promotion of Science.
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Katori, M. Elliptic determinantal process of type A. Probab. Theory Relat. Fields 162, 637–677 (2015). https://doi.org/10.1007/s00440-014-0581-9
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DOI: https://doi.org/10.1007/s00440-014-0581-9
Keywords
- Noncolliding diffusion process
- Dyson’s Brownian motion model
- Elliptic determinant evaluations
- Determinantal process
- Determinantal martingale
- Alcove of affine Weyl group