Limit Theorems for Orthogonal Polynomials Related to Circular Ensembles

Abstract

For a natural extension of the circular unitary ensemble of order n, we study as $n\rightarrow \infty$ the asymptotic behavior of the sequence of monic orthogonal polynomials $(\varPhi _{k,n}, k=0, \ldots , n)$ with respect to the spectral measure associated with a fixed vector, the last term being the characteristic polynomial. We show that, as $n\rightarrow \infty$, the sequence of processes $(\log \varPhi _{\lfloor nt\rfloor ,n}(1), t \in [0,1])$ converges to a deterministic limit, and we describe the fluctuations and the large deviations.

Appendix

1.1 Some Properties of $\ell = \log \varGamma$ and $\varPsi = \ell '$

From the Binet formula (Abramowitz and Stegun [1] or Erdélyi et al. [9], p. 21), we have for ${\mathfrak {Re }\,}\ x > 0$

$$\begin{aligned} \ell (x)= & {} \left( x-\frac{1}{2}\right) \log x -x +1 + \int _0^\infty f(s)[\mathrm{e}^{-sx} - \mathrm{e}^{-s}]\, \mathrm{d}s\nonumber \\= & {} \left( x-\frac{1}{2}\right) \log x -x + \frac{1}{2} \log (2\pi ) + \int _0^\infty f(s)\mathrm{e}^{-sx}\, \mathrm{d}s, \end{aligned}$$

(5.1)

where the function f is defined by

$$\begin{aligned} f(s) = \left[ \frac{1}{2}-\frac{1}{s}+ \frac{1}{\mathrm{e}^s -1}\right] \frac{1}{s} = 2\sum _{k=1}^\infty \frac{1}{s^2 + 4\pi ^2 k^2}, \end{aligned}$$

(5.2)

and satisfies for every $s \ge 0$:

$$\begin{aligned} 0 < f(s) \le f(0)=1/12,\quad 0<\left( sf(s)+ \frac{1}{2}\right) <1. \end{aligned}$$

(5.3)

By differentiation (recall that $\varPsi$ is the digamma function $\varGamma '/\varGamma$),

$$\begin{aligned} \log x - \varPsi (x) = \frac{1}{2x} +\int _0 ^\infty s f(s) \mathrm{e}^{-sx}\, \mathrm{d}s = \int _0 ^\infty \mathrm{e}^{-sx} \left( sf(s)+ \frac{1}{2}\right) \mathrm{d}s.\qquad \end{aligned}$$

(5.4)

Moreover, since $\varPsi (x+1) = \frac{1}{x} + \varPsi (x)$, we have a variation of (5.4):

$$\begin{aligned} \log x-\varPsi (x+1)= & {} - \frac{1}{2x} +\int _0 ^\infty s f(s) \mathrm{e}^{-sx}\, \mathrm{d}s\nonumber \\= & {} \int _0 ^\infty \mathrm{e}^{-sx} \left( sf(s)- \frac{1}{2}\right) \mathrm{d}s. \end{aligned}$$

(5.5)

As easy consequences, we have, for every $x > 0$,

$$\begin{aligned} 0< & {} x\left( \log x - \varPsi (x)\right) \le 1, \end{aligned}$$

(5.6)

$$\begin{aligned} 0< & {} \log x - \varPsi (x) - \frac{1}{2x} \le \frac{1}{12x^2}. \end{aligned}$$

(5.7)

Differentiating again, we see that for $q\ge 1$, ${\mathfrak {Re }\,}\ x > 0$,

$$\begin{aligned} \varPsi ^{(q)}(x) = (-1)^{q-1} (q-1)! x ^{-q} + (-1)^{q-1} \int _0^\infty \mathrm{e}^{-sx} s^q \left( sf(s)+ \frac{1}{2}\right) \,\mathrm{d}s \end{aligned}$$

and then

$$\begin{aligned} |\varPsi ^{(q)}(x) - (-1)^{q-1} (q-1)! x ^{-q}| \le ({\mathfrak {Re }\,}\ x)^{-q-1} q!. \end{aligned}$$

(5.8)

Another useful formula is

$$\begin{aligned} \varPsi (z+1) = -\gamma - \sum _{k=1}^\infty \left( \frac{1}{k+z}-\frac{1}{k}\right) , \end{aligned}$$

(5.9)

for $z \notin (-\infty , -1]$.

1.2 The Density $g_r^{(\delta )}$ and Some Moments Related to It

Recall that for $r >0$ and $\delta$ such that $r + 2{\mathfrak {Re }\,}\ \delta +1>0$, the density $g_{r}^{(\delta )}$ on the unit disk ${\mathbb {D}}$ is given by

$$\begin{aligned} g_{r}^{(\delta )}(z) = c_{r, \delta } (1 -|z|^2)^{r -1} (1-z)^{{\bar{\delta }}} (1- \bar{z})^\delta \end{aligned}$$

where $c_{r, \delta }$ is the normalization constant. The following lemma is the key to compute $c_{r, \delta }$ and the moments of $g_r^{(\delta )}$.

Lemma 3

Let $s,t,\ell$ be complex numbers such that: ${\mathfrak {Re }\,}\ \ell$, ${\mathfrak {Re }\,}(s + \ell +1)$, ${\mathfrak {Re }\,}( t +\ell + 1 )$ and ${\mathfrak {Re }\,}( s + t +\ell + 1 )$ are strictly positive. Then, the following identity holds:

$$\begin{aligned} \int _{{\mathbb {D}}} (1 -|z|^2)^{\ell -1} (1-z)^s (1- \bar{z})^t {\mathrm {d}}^2z = \pi \varGamma \left[ \begin{matrix} \ell \ ,\ \ell +1+s+t\\ \ell +1+s\ , \ \ell +1+t \end{matrix}\right] , \end{aligned}$$

(5.10)

where for the sake of simplicity we use the polygamma symbol

$$\begin{aligned} \varGamma \left[ \begin{matrix} a, b, \ldots \\ c,d, \ldots \end{matrix}\right] := \frac{\varGamma (a)\varGamma (b)\cdots }{\varGamma (c)\varGamma (d)\cdots }. \end{aligned}$$

A proof of this result is given in [5]. A first consequence is that

$$\begin{aligned} c_{r, \delta }= \pi ^{-1} \varGamma \left[ \begin{matrix} r +1+\delta \ , \ r+1+{\bar{\delta }}\\ r\ ,\ r+1+\delta + {\bar{\delta }}\\ \end{matrix}\right] . \end{aligned}$$

(5.11)

A second consequence is that if $\gamma$ has the density $g_{r}^{(\delta )}$, then we have

$$\begin{aligned} {\mathbb {E}} (1- \gamma )^a (1- \bar{\gamma })^b = \varGamma \left[ \begin{matrix} r+1+\delta +{\bar{\delta }}+a+b,\,r+1+\bar{\delta },\,r+1+\delta \\ r+1+\delta +{\bar{\delta }},\,r+1+{\bar{\delta }}+ a,\,r+1+\delta +b \end{matrix}\right] \qquad \quad \end{aligned}$$

(5.12)

as soon as all the real parts of the arguments of the gamma functions are strictly positive.

Let us notice that for $r=0$, the RHS of (5.12) is the Mellin–Fourier transform of $1-\gamma$ when $\gamma \in {\mathbb {T}}$ is distributed according to $\lambda ^{(\delta )}$.

In this paper, we need the following computations, in order to deduce the moments of $\log (1-\gamma )$. The quantities involved below are all well-defined as soon as $s > s_0$, where $s_0$ is some strictly negative quantity depending on r and $\delta$, and in particular, for (s, t) in the neighborhood of (0, 0), one can write:

$$\begin{aligned} \varLambda (s,t):= & {} \log {\mathbb {E}} \exp \left( 2s {\mathfrak {Re }\,}\log (1-\gamma ) +2 t {\mathfrak {Im }\,}\log (1-\gamma )\right) \nonumber \\= & {} \log {\mathbb {E}} \exp \left( {\mathfrak {Re }\,}\ (2(s-{\mathrm {i}}t )\log (1-\gamma )\right) \nonumber \\= & {} \log {\mathbb {E}} (1-\gamma )^{s-{\mathrm {i}}t} (1-\bar{\gamma })^{s+{\mathrm {i}}t}\nonumber \\= & {} \ell \big (r +1+\delta +\overline{\delta }+ 2s\big )- \ell \big (r+1+ \delta +\overline{\delta }\big )\nonumber \\&-\ell \left( r +1+ \overline{\delta }+s-{\mathrm {i}}t\Big ) - \ell \Big (r +1+ \delta + s+{\mathrm {i}}t\right) \nonumber \\&+\,\ell \big (r +1 +\overline{\delta }\big )+ \ell \big (r +1 +\delta \big ). \end{aligned}$$

(5.13)

To compute moments, we need differentiation. First we have:

$$\begin{aligned} \frac{\partial }{\partial s}\varLambda (s, t)= & {} 2 \varPsi \left( r +1+\delta +\overline{\delta }+2s\right) \nonumber \\&-\varPsi \left( r +1+ \delta + s+ {\mathrm {i}}t\right) -\varPsi \left( r +1+ \overline{\delta }+s- {\mathrm {i}}t\right) \nonumber \\ \frac{\partial }{\partial t}\varLambda (s, t)= & {} {\mathrm {i}}\varPsi \Big (r +1+{\bar{\delta }}+ s -{\mathrm {i}}t\Big ) - {\mathrm {i}}\varPsi \Big (r +1+\delta +s + {\mathrm {i}}t\Big ). \end{aligned}$$

(5.14)

The first moment is then:

$$\begin{aligned} {\mathbb {E}}\,{\mathfrak {Re }\,}\log (1-\gamma )= & {} \varPsi (r+1 + \delta + {\bar{\delta }}) - \frac{1}{2}\varPsi (r+1+\delta ) -\frac{1}{2}\varPsi (r+1+{\bar{\delta }})\\ {\mathbb {E}}\,{\mathfrak {Im }\,}\log (1-\gamma )= & {} \frac{1}{2{\mathrm {i}}}\varPsi (r+1+\delta ) -\frac{1}{2{\mathrm {i}}}\varPsi (r+1+{\bar{\delta }}) \end{aligned}$$

or

$$\begin{aligned} {\mathbb {E}} \log (1-\gamma ) = \varPsi (r+1 + \delta + {\bar{\delta }}) - \varPsi (r+1+{\bar{\delta }}). \end{aligned}$$

(5.15)

Differentiating again (5.14), we get

$$\begin{aligned} \frac{\partial ^2}{\partial s^2} \varLambda (s, t)= & {} 4 \varPsi ' \big (r +1+\delta +\overline{\delta }+2s\big )\nonumber \\&- \varPsi '\Big (r +1+ \delta + s + {\mathrm {i}}t\Big )-\varPsi '\Big (r +1+ \overline{\delta }+s- {\mathrm {i}}t\Big )\nonumber \\ \frac{\partial ^2}{\partial t^2}\varLambda (s, t)= & {} \varPsi ' \Big (r +1+{\bar{\delta }}+s-{\mathrm {i}}t\Big ) +\varPsi ' \Big (r +1+\delta +s +{\mathrm {i}}t\Big )\nonumber \\ \frac{\partial ^2}{\partial s\partial t} \varLambda (s, t)= & {} -{\mathrm {i}}\varPsi '\Big (r+1+\delta +s + {\mathrm {i}}t\Big ) +{\mathrm {i}}\varPsi '\Big (r+1+{\bar{\delta }}+s-{\mathrm {i}}t\Big )\qquad \end{aligned}$$

(5.16)

and the second moments are

$$\begin{aligned}&\hbox {Var}\,{\mathfrak {Re }\,}\log (1-\gamma ) = \varPsi ' \big (r +1+\delta +\overline{\delta }\big ) -\frac{1}{4}\varPsi '\big (r +1+ \delta \big )-\frac{1}{4}\varPsi '\big (r +1+ \overline{\delta }\big )\nonumber \\&\hbox {Var}\,{\mathfrak {Im }\,}\log (1-\gamma ) = \frac{1}{4}\varPsi ' \big (r +1+\delta \big ) + \frac{1}{4}\varPsi ' \big (r +1+\overline{\delta }\big )\nonumber \\&\hbox {Cov}\,({\mathfrak {Re }\,}\log (1-\gamma ), {\mathfrak {Im }\,}\log (1-\gamma )) = \frac{1}{4{\mathrm {i}}}\varPsi '\big (r+1+\delta \big ) - \frac{1}{4{\mathrm {i}}}\varPsi '\big (r+1+{\bar{\delta }}\big ).\nonumber \\ \end{aligned}$$

(5.17)

1.3 Complex Logarithm and Characteristic Polynomial

Let $E_k$ be the set of the complex $k \times k$ matrices with no eigenvalue on the interval $[1, \infty )$. For $V \in E_k$, let us define

$$\begin{aligned} \log {\text {det}} (I_k - V) := \sum _{j=1}^k \log (1 - \lambda _j), \end{aligned}$$

where the $\lambda _j$’s are the roots, counted with multiplicity, of the polynomial $z \mapsto \det (z I_k -V)$, and where in the right-hand side, one considers the principal branch of the logarithm. This definition is meaningful, since by assumption, $1 - \lambda _j \notin \mathbb {R}_-$ for all $j \in \{1, \ldots , k\}$. By the continuity of the set of roots of a polynomial with respect to its coefficients, the set $E_k$ is open and the function $V \mapsto \log {\text {det}} (I_k - V)$ defined just above is continuous on $E_k$. In fact, since $E_k$ is connected (this is easily checked by tridiagonalizing the matrices), this is the unique way to define the logarithm of ${\text {det}} (I_k - V)$ as a continuous function of $V \in E_k$ if we assume that it should take the value zero at $V =0$.

Now, with the notation of the beginning of the paper, the matrix $G_k(U_n)$ is a submatrix of the unitary matrix $U_n$, and all its eigenvalues have modulus bounded by 1. If we assume $\gamma _0, \ldots , \gamma _{n-1} \ne 1$ (which holds almost surely under $\hbox {CJ}_{\beta , \delta }^{(n)}$), then by (1.6), $\varPhi _{k,n}(1) \ne 0$, and one easily deduces that $G_k(U_n) \in E_k$, which allows to define $\log \varPhi _{k,n}(1)$ without ambiguity. Now, the map from $\mathbb {D}^{n-1} \times (\mathbb {U} \backslash \{1\})$ to $\mathbb {R}$, given by

$$\begin{aligned} (\gamma _1, \ldots , \gamma _{n-1}) \mapsto \sum _{j=0}^{k-1} \log (1 -\gamma _j) \end{aligned}$$

is continuous if we take the principal branch of the logarithm, and since $U_n$ depends continuously on $(\gamma _1, \ldots , \gamma _{n-1}) \in \mathbb {D}^{n-1} \times (\mathbb {U} \backslash \{1\})$, as it can be checked in [5], the map

$$\begin{aligned} (\gamma _1, \ldots , \gamma _{n-1}) \mapsto \log \varPhi _{k,n}(1) \end{aligned}$$

is also continuous. These two maps have the same exponential, and one can check that they are both real if the $\gamma _j$’s are all real. Hence, they are equal, which fully justifies the equation

$$\begin{aligned} \log \varPhi _{k,n} (1) = \sum _{j=0}^{k-1} \log (1-\gamma _j). \end{aligned}$$

(5.18)

1.4 Abel–Plana Summation Formula

Theorem 9

Let $m < n$ be integers and let g be a holomorphic function on the strip $\{t \in \mathbb {C}, n \le {\mathfrak {Re }\,}\ t \le m\}$ (i.e., g is continuous on this strip and holomorphic in its interior). We assume that $g(t) = o\left( \exp (2\pi |{\mathfrak {Im }\,}\ t|)\right)$ as ${\mathfrak {Im }\,}\ t \rightarrow \pm \infty$, uniformly with respect to ${\mathfrak {Re }\,}\ t \in [n,m]$. Then,

$$\begin{aligned} \sum _{j= m+1}^n g(j)= & {} \int _m^n g(t)\mathrm{d}t + \frac{g(n) - g(m)}{2}\nonumber \\&+\,\, {\mathrm {i}}\int _0^\infty \frac{g(m+{\mathrm {i}}y) - g(n+ {\mathrm {i}}y) - g(m -{\mathrm {i}}y) + g(n-{\mathrm {i}}y)}{\mathrm{e}^{2\pi y}-1} \mathrm{d}y.\nonumber \\ \end{aligned}$$

(5.19)

For a proof see [20, p. 290].