Truncated Linear Statistics Associated with the Eigenvalues of Random Matrices II. Partial Sums over Proper Time Delays for Chaotic Quantum Dots

Abstract

Invariant ensembles of random matrices are characterized by the distribution of their eigenvalues $\{\lambda _1,\ldots ,\lambda _N\}$. We study the distribution of truncated linear statistics of the form $\tilde{L}=\sum _{i=1}^p f(\lambda _i)$ with $p<N$. This problem has been considered by us in a previous paper when the p eigenvalues are further constrained to be the largest ones (or the smallest). In this second paper we consider the same problem without this restriction which leads to a rather different analysis. We introduce a new ensemble which is related, but not equivalent, to the “thinned ensembles” introduced by Bohigas and Pato. This question is motivated by the study of partial sums of proper time delays in chaotic quantum dots, which are characteristic times of the scattering process. Using the Coulomb gas technique, we derive the large deviation function for $\tilde{L}$. Large deviations of linear statistics $L=\sum _{i=1}^N f(\lambda _i)$ are usually dominated by the energy of the Coulomb gas, which scales as $\sim N^2$, implying that the relative fluctuations are of order 1 / N. For the truncated linear statistics considered here, there is a whole region (including the typical fluctuations region), where the energy of the Coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect. Because the entropy scales as $\sim N$, the relative fluctuations are of order $1/\sqrt{N}$. Our analysis relies on the mapping on a problem of p fictitious non-interacting fermions in N energy levels, which can exhibit both positive and negative effective (absolute) temperatures. We determine the large deviation function characterizing the distribution of the truncated linear statistics, and show that, for the case considered here ($f(\lambda )=1/\lambda$), the corresponding phase diagram is separated in three different phases.

Appendix 1: Distribution of the Injectance for \(\beta =2\)

In this section, we derive the distribution of the injectance of a quantum dot defined in Sect. 2.1:

$$\begin{aligned} \overline{\nu }_\alpha = \frac{1}{2\pi } \sum _{i\in \mathrm {contact}\,\alpha }\mathcal {Q}_{ii}. \end{aligned}$$

(7.1)

We will restrict our study to the situation where $\beta =2$ because the joint distribution of the $\mathcal {Q}_{ii}$’s is known only in this case. We will derive the distribution of $\overline{\nu }_\alpha$ using a Coulomb gas approach. The computation is very similar to the one described in [48] where the Wigner time delay (case where all contacts contribute) is studied. We start from the distribution of a subblock $\mathcal {Q}_{p}$ of size $p\times p$ on the diagonal of $\mathcal {Q}$ given in [46] for $\beta =2$:

$$\begin{aligned} P(\mathcal {Q}_{p}^{-1}) \propto (\det \mathcal {Q}_{p})^{-N} \mathrm {e}^{- \mathop {\mathrm {tr}}\nolimits \left\{ \mathcal {Q}_{p}^{-1} \right\} }. \end{aligned}$$

(7.2)

If we suppose that the contact $\alpha$ has $p$ open channels, we can rewrite the injectance as:

$$\begin{aligned} \overline{\nu }_\alpha = \frac{1}{2\pi } \mathop {\mathrm {tr}}\nolimits \left\{ \mathcal {Q}_{p} \right\} . \end{aligned}$$

(7.3)

Denoting $\lbrace \lambda _i \rbrace$ the eigenvalues of $\mathcal {Q}_{p}^{-1}$, we can write

$$\begin{aligned} \overline{\nu }_\alpha = \frac{1}{2\pi } \sum _{i=1}^{p} \frac{1}{\lambda _i}, \end{aligned}$$

(7.4)

where the joint probability density function of the $\lbrace \lambda _i \rbrace$ is given by

$$\begin{aligned} P(\lambda _1 ,\ldots , \lambda _{p}) \propto \prod _{i<j} \left| \lambda _i - \lambda _j \right| ^2 \prod _{i=1}^{p} \lambda _i^{N} \mathrm {e}^{-\lambda _i}. \end{aligned}$$

(7.5)

Rescale the eigenvalues as $\lambda _i = px_i$ and introduce the empirical density

$$\begin{aligned} \rho (x) = \frac{1}{p} \sum _{i=1}^{p} \delta (x - x_i). \end{aligned}$$

(7.6)

The distribution of

$$\begin{aligned} s = 2\pi \overline{\nu }_\alpha = \frac{1}{p} \sum _{i=1}^{p} \frac{1}{x_i} = \int \frac{\rho (x)}{x} \mathrm{d}x \end{aligned}$$

(7.7)

is given by

$$\begin{aligned} P_{N,\kappa }(s) \simeq \frac{ \displaystyle \int \mathcal {D}\rho \, \mathrm {e}^{-\kappa ^2 N^2 \mathscr {E}[\rho ]} \, \delta \left( \int \rho (x) \mathrm{d}x - 1 \right) \delta \left( \int \frac{\rho (x)}{x} \mathrm{d}x - s \right) }{ \displaystyle \int \mathcal {D}\rho \, \mathrm {e}^{- \kappa ^2 N^2 \mathscr {E}[\rho ]} \, \delta \left( \int \rho (x) \mathrm{d}x - 1 \right) }, \end{aligned}$$

(7.8)

where we denoted $\kappa = p/N$ and introduced the energy (see Sect. 3)

$$\begin{aligned} \mathscr {E}[\rho ] = - \int \mathrm{d}x \int \mathrm{d}y \, \rho (x) \rho (y) \ln \left| x-y \right| + \int \mathrm{d}x \left( x - \frac{1}{\kappa } \ln x \right) \rho (x). \end{aligned}$$

(7.9)

The integrals in (7.8) are dominated by the minimum of the energy under the constraints imposed by the $\delta$-functions. Therefore, we introduce the functional

$$\begin{aligned} \mathscr {F}[\rho ,\mu _0,\mu _1] = \mathscr {E}[\rho ] + \mu _0 \left( \int \rho - 1 \right) + \mu _1 \left( \int \frac{\rho (x)}{x} \mathrm{d}x - s \right) , \end{aligned}$$

(7.10)

where $\mu _0$ and $\mu _1$ are Lagrange multipliers. Denote $\rho ^\star (x;\kappa ,s)$ the density which dominates the numerator of (7.8). It is given by $\left. \frac{\delta \mathscr {F}}{\delta \rho } \right| _{\rho ^\star } = 0$, which reads:

$$\begin{aligned} 2 \int \rho ^\star (y;\kappa ,s) \ln \left| x-y \right| \mathrm{d}y = x - \frac{1}{\kappa } \ln x + \mu _0 + \frac{\mu _1}{x} \quad \text {for} \quad x \in \mathrm {Supp}(\rho ^\star ). \end{aligned}$$

(7.11)

It is more convenient to derive this relation with respect to x:

(7.12)

The values $\mu _0^\star$ and $\mu _1^\star$ taken by the Lagrange multipliers are fixed by imposing the constraints

$$\begin{aligned} \int \rho ^\star (x;\kappa ,s) \mathrm{d}x = 1, \quad \int \frac{\rho ^\star (x;\kappa ,s)}{x} \mathrm{d}x = s. \end{aligned}$$

(7.13)

Similarly, denote $\rho _0^\star (x)$ the density which dominates the denominator. The distribution of s is then given by:

$$\begin{aligned} P_{N,\kappa }(s) \sim \exp \left\{ - \kappa ^2 N^2 \left( \mathscr {E}[\rho ^\star (x;\kappa ,s)]-\mathscr {E}[\rho _0^\star (x)] \right) \right\} . \end{aligned}$$

(7.14)

The thermodynamic identity (3.21) mentioned in the body of the paper allows a direct computation of the energy via the relation:

$$\begin{aligned} \frac{\mathrm {d}\mathscr {E}[\rho ^\star (x;\kappa ,s)]}{\mathrm {d}s} = - \mu _1^\star (\kappa ;s). \end{aligned}$$

(7.15)

Our aim is now to compute the density $\rho ^\star$. As in the body of the paper, depending on the values of the parameters $\kappa$ and s, we will find different types of densities $\rho ^\star$, which we interpret as different “phases”.

1.1 Phase I: Solution with One Compact Support

Let us assume that the solution $\rho ^\star$ has one compact support [a, b]. Then, Eq. (7.12) can be solved using Tricomi’s theorem [52]. We obtain:

$$\begin{aligned} \rho ^\star (x;\kappa ,s) = \frac{x+c}{2\pi x^2} \sqrt{(x-a)(b-x)}. \end{aligned}$$

(7.16)

It is convenient to parametrize this solution in terms of $u = \sqrt{a/b}$. Then, imposing $\rho ^\star (a;\kappa ,s)= \rho ^\star (b;\kappa ,s) = 0$, along with the constraints (7.13) yields:

$$\begin{aligned}&\displaystyle v = \sqrt{a b} = \frac{2u}{\kappa } \frac{(2\kappa +1)u^2 -2u + (2\kappa +1)}{(1-u^2)^2}, \end{aligned}$$

(7.17)

$$\begin{aligned}&\displaystyle \mu _1^\star =- \frac{4u^2}{\kappa ^2} \frac{((2\kappa +1)u^2 -2u + (2\kappa +1))(u^2 -2(2\kappa +1)u +1)}{(1-u^2)^4}, \end{aligned}$$

(7.18)

$$\begin{aligned}&\displaystyle c = \frac{\mu _1^\star }{v}, \end{aligned}$$

(7.19)

$$\begin{aligned}&\displaystyle = \sigma _\kappa (u) = (1-u)^2 \frac{-u^4 + 4 (3\kappa +1)u^3 + 2(4\kappa -3)u^2 + 4(3\kappa +1)u-1}{16u^2((2\kappa +1)u^2 -2u + (2\kappa +1))}. \qquad \end{aligned}$$

(7.20)

Given $\kappa$ and s, the last equation allows to compute u, from which all the other parameters are deduced. One can check that in the limit $\kappa \rightarrow 1$, one recovers the equations given in [48].

The optimal density $\rho _0^\star$ for the denominator can be deduced from $\rho ^\star$ by releasing the constraint (setting $\mu _1^\star = 0$).

1.1.1 Domain of Validity

This solution exists as long as $\rho ^\star$ is positive, which corresponds to $x+c \ge 0$. This gives the condition

$$\begin{aligned} \kappa \ge \frac{1 - 3 u + 3 u^2 - u^3}{2 u (3 + u^2)}, \end{aligned}$$

(7.21)

which can be rewritten as $s < s_{I}(\kappa )$. This corresponds to the lower domain delimited by the upper solid line on Fig. 10 (left).

1.1.2 Typical Fluctuations

The typical fluctuations are controlled by the minimum of the energy, which is given by $\mu _1^\star =0$. The density is then

$$\begin{aligned} \rho _0^\star (x) = \frac{\sqrt{(x-a_0)(b_0-x)}}{2\pi x}, \end{aligned}$$

(7.22)

where

$$\begin{aligned} a_0 = \frac{2 \kappa +1 - 2\sqrt{\kappa (\kappa +1)}}{\kappa }, \quad b_0 = \frac{2 \kappa +1 + 2\sqrt{\kappa (\kappa +1)}}{\kappa }. \end{aligned}$$

(7.23)

The corresponding value of s is $\kappa$, and expanding Eqs. (7.18, 7.20) for s close to $\kappa$ yields:

$$\begin{aligned} \mu _1^\star = - \frac{s-\kappa }{\kappa ^3(\kappa +1)} + O((s-\kappa )^2). \end{aligned}$$

(7.24)

The energy is obtained by simple integration, via Eq. (7.15):

$$\begin{aligned} \mathscr {E}[\rho ^\star (x;\kappa ,s)]-\mathscr {E}[\rho _0^\star (x)] = \frac{(s-\kappa )^2}{2\kappa ^3(\kappa +1)} + \mathcal {O}((s-\kappa )^3). \end{aligned}$$

(7.25)

Thus, the distribution of s near $\kappa$ is given by:

$$\begin{aligned} P_{N,\kappa }(s) \sim \exp \left\{ - N^2 \frac{(s-\kappa )^2}{2\kappa (\kappa +1)} \right\} . \end{aligned}$$

(7.26)

We recover the leading term of the variance given in the introduction, Eq. (2.10):

$$\begin{aligned} \mathrm {Var}(s) \simeq \frac{\kappa (1+\kappa )}{N^2}. \end{aligned}$$

(7.27)

1.1.3 Limiting Behaviour $s \rightarrow 0$

The limit $s \rightarrow 0$ corresponds to $\mu _1^\star \rightarrow +\infty$. Expanding Eqs. (7.17, 7.18, 7.19, 7.20) in this limit gives:

$$\begin{aligned} \mu _1^\star = \frac{1}{s^2} - \frac{\kappa +2}{2\kappa } \frac{1}{s} + \mathcal {O}(1). \end{aligned}$$

(7.28)

Using again Eq. (7.15), we obtain:

$$\begin{aligned} \mathscr {E}[\rho ^\star (x;\kappa ,s)] = \frac{1}{s} + \frac{\kappa +2}{2\kappa } \ln s + \mathcal {O}(1), \end{aligned}$$

(7.29)

thus:

$$\begin{aligned} P_{N,\kappa }(s) \sim s^{-N^2\kappa (\kappa +2)/2 } e^{- \kappa ^2 N^2/s} \quad \text {for}\quad s \rightarrow 0. \end{aligned}$$

(7.30)

1.2 Phase II: Solution with an Isolated Eigenvalue

As for the Wigner time delay [48] ($\kappa =1$) and in Section 5, we look for a solution with an isolated eigenvalue:

$$\begin{aligned} \rho ^\star (x;\kappa ,s) = \frac{1}{p} \delta (x-x_1) + \tilde{\rho }(x), \end{aligned}$$

(7.31)

with now $\int \tilde{\rho } = 1-1/p$. The minimization of $\mathscr {F}$ with respect to $\tilde{\rho }$ reads:

(7.32)

And minimization with respect to $x_1$ gives

$$\begin{aligned} 2 \int \frac{\tilde{\rho }(y)}{x_1 - y} = 1 - \frac{1}{\kappa x_1} - \frac{\mu _1}{x_1^2}, \quad \text {for}\; x \in \mathrm {Supp}(\tilde{\rho }). \end{aligned}$$

(7.33)

In addition, the constraint becomes:

$$\begin{aligned} s = \frac{1}{px_1} + \int \frac{\tilde{\rho }(x)}{x} \mathrm{d}x. \end{aligned}$$

(7.34)

For $x_1$ to give a “macroscopic” contribution to s, we must have $x_1 = \mathcal {O}(N^{-1})$. Then equation (7.33) imposes $\mu _1 = -x_1/\kappa + \mathcal {O}(p^{-2})$. Finally, at leading order in $p$, we get:

$$\begin{aligned}&\displaystyle \tilde{\rho }(x) = \rho _0^\star (x) + \mathcal {O}(p^{-1}),\end{aligned}$$

(7.35)

$$\begin{aligned}&\displaystyle x_1 = \frac{1}{p(s-\kappa )} + \mathcal {O}(p^{-2}),\end{aligned}$$

(7.36)

$$\begin{aligned}&\displaystyle \mu _1^\star = - \frac{1}{\kappa p} \frac{1}{s - \kappa } + \mathcal {O}(p^{-2}). \end{aligned}$$

(7.37)

1.2.1 Domain of Validity

This solution remains valid as long as the separate eigenvalue is away from the bulk, namely $x_1 < a_0$. This gives the condition

$$\begin{aligned} s > \kappa + \frac{1}{pa_0} = s_{II}(\kappa ). \end{aligned}$$

(7.38)

This corresponds to the upper domain represented on Fig. 10 (left). Note that $s_{II}(\kappa ) \rightarrow \kappa$ as $N \rightarrow \infty$.

1.2.2 Large Deviation Function

The energy can be computed analytically at leading order:

$$\begin{aligned} \mathcal {E}[\rho ^\star (x;\kappa ,s)] - \mathcal {E}[\rho _0^\star ] = \frac{1}{\kappa p} \ln [p(s - \kappa )] + \frac{\mathscr {C}}{p} + \mathcal {O}(p^{-2}). \end{aligned}$$

(7.39)

where $\mathscr {C}=-1 - 2 \ln 2$ was introduced above. The constant term is the same as the one appearing in Sect. 5 and Ref. [48]. It arises from corrections of order $p^{-1}$ to the density $\rho ^\star$ [26]. From the expression of the energy, we deduce the expression of the distribution of s, for $s > s_c$:

$$\begin{aligned} P_{N,\kappa }(s) \sim (s-\kappa )^{- N}. \end{aligned}$$

(7.40)

The simplification $p^2/(\kappa p)=N$ has thus produced the same exponent as for the tail of the distribution of the sum (truncated or not) of proper times. This is explained from the interpretation of Ref. [46], where it was shown that $\mathcal {Q}_{ii}$ coincides with the partial time delay for $\beta =2$.

Fig. 10

Left Domains of existence of each solution, for $N=200$. The upper solid line corresponds to $s = s_{I}(\kappa )$ and delimits the domain of existence of phase I. The lower solid line is $s = s_{II}(\kappa )$. Phase II exists above this line. As $N \rightarrow \infty$, it goes to the dashed line $s = \kappa$, where occurs the transition in the thermodynamic limit. Right phase diagram for the Coulomb gas

1.3 Summary

We obtained the following scalings for the distribution:

$$\begin{aligned} P_{N,\kappa }(s) \underset{N \rightarrow \infty }{\sim } {\left\{ \begin{array}{ll} \exp \left\{ - \kappa ^2 N^2 \varPsi _-(\kappa ;s) \right\} &{} \quad \text {for} \; s < s_{I}(\kappa ) \\ (\kappa N)^{-N} \exp \left\{ - N \varPsi _+(\kappa ;s) \right\} &{} \quad \text {for}\; s > s_{II}(\kappa ) \end{array}\right. } \end{aligned}$$

(7.41)

where the large deviation $\varPsi _+$ is given by:

$$\begin{aligned} \varPsi _+(\kappa ;s) = \ln (s - \kappa ) - \kappa (1 + 2 \ln 2), \end{aligned}$$

(7.42)

and $\varPsi _-$ has the following limiting behaviours:

$$\begin{aligned} \varPsi _-(\kappa ;s) \simeq {\left\{ \begin{array}{ll} \displaystyle \frac{1}{s} + \frac{\kappa +2}{2\kappa } \ln s + \mathcal {O}(1) &{} \quad \text {as}\; s \rightarrow 0 \\ \displaystyle \frac{(s-\kappa )^2}{2\kappa ^3(\kappa +1)} + \mathcal {O}((s-\kappa )^3) &{}\quad \text {as}\; s \rightarrow \kappa \end{array}\right. } \end{aligned}$$

(7.43)

The precise point $s_t$ where the transition between the two phases occurs can be obtained by matching the two large deviation functions. One can show that, for $N \rightarrow \infty$, $s_t \rightarrow \kappa$. Therefore for large N we have, for s close to $\kappa$:

$$\begin{aligned} \lim _{N \rightarrow \infty } - \frac{1}{N^2} \ln P_{N,\kappa }(s) \simeq \left\{ \begin{array}{cl} \displaystyle \frac{(s-\kappa )^2}{2\kappa ^3(\kappa +1)} &{} \text { for } s < \kappa \\ 0 &{} \text { for } s > \kappa \end{array} \right. \end{aligned}$$

(7.44)

This corresponds to a second order phase transition. This was already the case in Ref. [48] in the study of the full linear statistics ($\kappa =1$).