Incomplete Determinantal Processes: From Random Matrix to Poisson Statistics

Abstract

We study linear statistics of a class of determinantal processes which interpolate between Poisson and GUE/Ginibre statistics in dimension 1 or 2. These processes are obtained by performing an independent Bernoulli percolation on the particle configuration of a log-gas confined in a general potential. We show that, depending on the expected number of deleted particles, there is a universal transition for mesoscopic linear statistics. Namely, at small scales, the point process behave according to random matrix theory, while, at large scales, it needs to be renormalized because the variance of any linear statistic diverges. The crossover is explicitly characterized as the superposition of a $H^{1}$- or $H^{1/2}$-correlated Gaussian noise depending on the dimension and an independent Poisson process. The proof consists in computing the limits of the cumulants of linear statistics using the asymptotics of the correlation kernel of the process.

Appendices

Appendix A: Incomplete Determinantal Processes

In this appendix, we review some background material on determinantal processes and we give an alternative proof of the results of Bohigas and Pato, [9], that the incomplete process is determinantal with correlation kernel $p_NK^N_V(z,w)$. There are many excellent surveys about determinantal processes and we refer to [11, 33, 36, 47] for further details and an overview of the main examples.

Let ${\mathfrak {X}}$ be a complete separable metric space equipped with a Radon measure $\mu$. The configuration space ${\mathscr {Q}}({\mathfrak {X}})$ is the set of integer-valued locally finite Borel measure on ${\mathfrak {X}}$ equipped with the topology which is generated by the maps

$$\begin{aligned} {\mathscr {Q}}({\mathfrak {X}}) \ni \Xi \mapsto \Xi _A := \int _A d\Xi \end{aligned}$$

for every Borel set $A\subseteq {\mathfrak {X}}$. A point process ${\mathbb {P}}$ is a Borel probability measure on the configuration space ${\mathscr {Q}}({\mathfrak {X}})$. A point process can be characterized by its correlation functions $(\rho _n)_{n=1}^\infty$. If it exists, $\rho _n$ is a symmetric function on ${\mathfrak {X}}^n$ which satisfies the identity

$$\begin{aligned} {\mathbb {E}}\left[ {\Xi _{A_1} \atopwithdelims ()k_1} \cdots {\Xi _{A_\ell } \atopwithdelims ()k_\ell } \right] = \frac{1}{k_1!\cdots k_\ell !} \int _{A_1^{k_1} \times \cdots \times A_\ell ^{k_\ell }} \rho _n(x_1,\dots , x_n) d\mu ^n(x) \end{aligned}$$

(A.1)

for all compositions $\mathbf {k}\vdash n$ and for all disjoint Borel sets $A_1,\dots , A_\ell \subseteq {\mathfrak {X}}$. A point process is called determinantal if all its correlation functions exist and are given by

$$\begin{aligned} \rho _n(x_1,\dots , x_n) = \det _{n\times n}\big [ K(x_i, x_j) \big ] . \end{aligned}$$

(A.2)

The function $K: {\mathfrak {X}}\times {\mathfrak {X}}\rightarrow {\mathbb {C}}$ is called the correlation kernel. For instance, given random points $(\lambda _1,\dots , \lambda _N)$ with the joint density $G_N(x)=e^{-\beta {\mathscr {H}}^N_V(x)}/Z^N_V$ on ${\mathfrak {X}}^N$, see (1.2), the random measure $\sum _{k=1}^N \delta _{\lambda _k}$ defines a point process and its correlation functions satisfy $\rho _N = N! G_N$ and

$$\begin{aligned} \rho _k(x_1, \dots , x_k) = \frac{N!}{(N-k)!} \int G_N(x_1, \dots , x_N) dx_{k+1}\cdots dx_N \end{aligned}$$

(A.3)

for all $k<N$. When $\beta =2$, it is easy to verify that

$$\begin{aligned} G_N(x) = \frac{1}{N!} \det _{N\times N} \big [K^N_V(x_i, x_j) \big ] \end{aligned}$$

and that, by formula (A.3), the process $\Xi$ is determinantal with the correlation kernel $K^N_V$ given by (1.3). In general, if K is a continuous, Hermitian symmetric, function which satisfies property (2.1), then there exists a determinantal process on ${\mathfrak {X}}$ with correlation kernel K; c.f. [47, Theorem 3].

Let $q\in (0,1)$ and $p=1-q$. Recall that X is a Binomial random variable with parameter p and $N\in {\mathbb {N}}$ if for all $k \in {\mathbb {N}}_0$,

$$\begin{aligned} {\mathbb {E}}\left[ {X \atopwithdelims ()k} \right] = p^k {N \atopwithdelims ()k} . \end{aligned}$$

(A.4)

By convention, ${N \atopwithdelims ()k} = 0$ if $k>N$. Let $\Xi$ be a random (point) configuration and let $\widehat{\Xi }$ be the configuration obtained after performing a Bernoulli percolation on $\Xi$. By construction, for any disjoint Borel set $A\subseteq {\mathfrak {X}}$, the conditional distribution of the random variable $\widehat{\Xi }_A$ given $\Xi$ has a Binomial distribution with parameters p and $\Xi _A$ and it is statistically independent of $\Xi _B$ for any Borel set B disjoint of A. By formula (A.4), this implies that

$$\begin{aligned} {\mathbb {E}}\left[ {{\widehat{\Xi }}_{A_1} \atopwithdelims ()k_1} \cdots {{\widehat{\Xi }}_{A_\ell } \atopwithdelims ()k_\ell } \bigg | \Xi _A=N_1, \dots , \Xi _A=N_\ell \right] = p^n {N_1 \atopwithdelims ()k_1} \cdots {N_\ell \atopwithdelims ()k_\ell } \end{aligned}$$

for any composition $\mathbf {k}\vdash n$ and for all disjoint Borel sets $A_1,\dots , A_\ell \subseteq {\mathfrak {X}}$. Hence, we obtain

$$\begin{aligned} {\mathbb {E}}\left[ {{\widehat{\Xi }}_{A_1} \atopwithdelims ()k_1} \cdots {{\widehat{\Xi }}_{A_\ell } \atopwithdelims ()k_\ell } \right] = p^n {\mathbb {E}}\left[ {\Xi _{A_1} \atopwithdelims ()k_1} \cdots {\Xi _{A_\ell } \atopwithdelims ()k_\ell } \right] \end{aligned}$$

so that, by formula (A.1), the correlation functions of the incomplete process$\widehat{\Xi }$ are given by $p^n \rho _n(x_1,\dots , x_n)$ for all $n \ge 1$. In particular, we deduce from formula (A.2) that, if $\Xi$ is a determinantal process with a correlation kernel K, then the point process $\widehat{\Xi }$ is also determinantal with kernel pK.

Appendix B: Off-Diagonal Decay of the Correlation Kernel $K^N_V$ in Dimension 2

In this section, we review some classical estimates for the correlation kernel (1.3) which have been used in [2] to prove the CLT (1.10). Then, we prove Lemma 3.3 and an analogous result for the cumulants of the $\infty$-Ginibre process. For completeness, we also give the proof of Lemma 3.2. We will use the formulation of Sect. 5 in [2] but the estimates (B.1) and (B.2) go back to the papers [7] and [1]. Suppose that the potential $V:{\mathbb {C}}\rightarrow {\mathbb {R}}$ is real-analytic and satisfies the condition (1.1). Then, there a function $\phi _V : {\mathbb {C}}\rightarrow {\mathbb {R}}^+$ such that $\phi _V(z) \ge \nu \log |z|^2$ as $|z| \rightarrow \infty$ and some constants $C, c, \delta >0$ such that

$$\begin{aligned} \big | K_V^N(z,w) \big | \le C N e^{- c\sqrt{N} (|z-w|\wedge \delta )} , \end{aligned}$$

(B.1)

and

$$\begin{aligned} K_V^N(z,z) \le C N e^{-N \phi _V(z)} , \end{aligned}$$

(B.2)

for all $w \in {\mathscr {S}}$ and $z\in {\mathbb {C}}$.

Proof of Lemma 3.3

We use the convention $z_0=z_{n+1}$. Since the kernel $K_V^N$ is reproducing, by the Cauchy-Schwartz inequality,

$$\begin{aligned} \big | K_V^N (z,w) \big | \le \sqrt{ K_V^N (z,z) K_V^N (w,w) } , \end{aligned}$$

for all $z, w\in {\mathbb {C}}$, so that

$$\begin{aligned} \bigg | \prod _{j=0}^{n}K_V^N(z_j,z_{j+1}) \bigg | \le \big | K_V^N(z_0,z_{1}) \big |\sqrt{K_V^N(z_0,z_{0})K_V^N(z_1,z_{1})} \prod _{j=2}^{n}K_V^N(z_j,z_j) . \end{aligned}$$

Since $\displaystyle \int _{\mathbb {C}}K_V^N(z,z) d\mathrm {A}(z) = N$ and $K_V^N(z,z) \le C N$ [see the estimate (B.2)], we obtain

$$\begin{aligned} \underset{|z_1-z_{n+1}|> \epsilon _N}{\int _{{\mathbb {C}}^n\times {\mathscr {S}}}} \bigg |\prod _{j=0}^n K^N_V(z_{j}, z_{j+1})\bigg |\ d\mathrm {A}^{n+1}(\mathrm {z}) \le C N^n \underset{|z_1-z_0|> \epsilon _N}{\int _{{\mathscr {S}}\times {\mathbb {C}}}} \big | K_V^N(z_0 ,z_1) \big | d\mathrm {A}(z_0) d\mathrm {A}(z_1) . \end{aligned}$$

Then, it easy to check that the estimates (B.1) and (B.2) imply that

$$\begin{aligned} \underset{|z_1-z_0|> \epsilon _N}{\int _{{\mathscr {S}}\times {\mathbb {C}}}} \big | K_V^N(z_0 ,z_1) \big | d\mathrm {A}(z_0) d\mathrm {A}(z_1) \le C N e^{-c \sqrt{N} \epsilon _N } . \end{aligned}$$

Hence, if $\epsilon _N = \kappa N^{-1/2} \log N$ and $\kappa \ge (n+1)/ c$, we obtain

$$\begin{aligned} \underset{|z_1-z_{n+1}|> \epsilon _N}{\int _{{\mathbb {C}}^n \times {\mathscr {S}}}} \bigg | \prod _{j=0}^n K^N_V(z_{j}, z_{j+1})\bigg |\ d\mathrm {A}^{n+1}(\mathrm {z}) = \underset{N\rightarrow \infty }{O}(N^{-1}) . \end{aligned}$$

(B.3)

Moreover, since $\sup \big \{ |F_N(z_0, \mathrm {z})| : \mathrm {z}\in {\mathbb {C}}^{n} , N \ge N_0 \big \} \le C \mathbf {1}_{z_0 \in {\mathscr {S}}}$ by (3.6), the estimate (B.3) implies that

$$\begin{aligned}&\underset{z_0 = z_{n+1}}{\int _{{\mathbb {C}}^{n+1}}}\, F_N(\mathrm {z}) \prod _{j=0}^n K^N_V(z_{j}, z_{j+1}) d\mathrm {A}^{n+1}(\mathrm {z}) \nonumber \\&\quad = \underset{|z_1-z_{n+1}| \le \epsilon _N}{\int _{{\mathbb {C}}^n \times {\mathscr {S}}}} F_N(\mathrm {z}) \prod _{j=0}^n K^N_V(z_{j}, z_{j+1}) \ d\mathrm {A}^{n+1}(\mathrm {z}) + \underset{N\rightarrow \infty }{O}(N^{-1}) . \end{aligned}$$

(B.4)

Now, we can proceed by induction to get formula (3.7). If ${\mathscr {C}}_N = \{\mathrm {z}\in {\mathbb {C}}^{n+1} : z_{n+1} \in {\mathscr {S}}, |z_1-z_{n+1}| \le \epsilon _N \}$, the next step is to show that

$$\begin{aligned} \underset{|z_2-z_1|> \epsilon _N}{\int _{{\mathscr {C}}_N}} \bigg | \prod _{j=0}^n K^N_V(z_{j}, z_{j+1})\bigg |\ d\mathrm {A}^{n+1}(\mathrm {z}) = \underset{N\rightarrow \infty }{O}(N^{-1}) . \end{aligned}$$

(B.5)

Since the set ${\mathscr {S}}_V$ is open, there exists a compact set ${\mathscr {S}}' \subset {\mathscr {S}}_V$ such that ${\mathscr {S}}\subset {\mathscr {S}}'$ and ${\mathscr {C}}_N \subset \{\mathrm {z}\in {\mathbb {C}}^{n+1} : z_{n+1} , z_1 \in {\mathscr {S}}' \}$ when the parameter N is sufficiently large. Then, as before, we obtain

$$\begin{aligned} \underset{|z_2-z_{1}|> \epsilon _N}{\int _{{\mathscr {C}}_N}} \bigg |\prod _{j=0}^n K^N_V(z_{j}, z_{j+1})\bigg |\ d\mathrm {A}^{n+1}(\mathrm {z}) \le C N^n \underset{|z_2-z_1|> \epsilon _N}{\int _{{\mathscr {S}}'\times {\mathbb {C}}}} \big | K_V^N(z_1 ,z_2) \big | d\mathrm {A}(z_0) d\mathrm {A}(z_1) , \end{aligned}$$

and formula (B.5) also follows directly from the estimate (B.3). Hence, by formula (B.4), this implies that

$$\begin{aligned}&\underset{z_0 = z_{n+1}}{\int _{{\mathbb {C}}^{n+1}}}\, F_N(\mathrm {z}) \prod _{j=0}^n K^N_V(z_{j}, z_{j+1}) d\mathrm {A}^{n+1}(\mathrm {z}) \\&\quad = \underset{{\begin{array}{c} |z_1-z_{n+1}| \le \epsilon _N \\ |z_2-z_{1}| \le \epsilon _N \end{array}}}{\int _{{\mathbb {C}}^n \times {\mathscr {S}}}} F_N(\mathrm {z}) \prod _{j=0}^n K^N_V(z_{j}, z_{j+1}) \ d\mathrm {A}^{n+1}(\mathrm {z}) + \underset{N\rightarrow \infty }{O}(N^{-1}) . \end{aligned}$$

If we repeat this argument, we obtain (3.7). $\square$

Lemma 4.3

Let $n \in {\mathbb {N}}$, $w_0= w_{n+1} =0$, and let $H(\mathrm {w})$ be a polynomial of degree at most 2 in the variables $w_1,\dots , w_n, \overline{w_1},\dots , \overline{w_n}$. For any sequence $\delta _N \ge k \rho _N^{-1/2} \sqrt{\log \rho _N}$ with $k>0$, we have

$$\begin{aligned}&\int _{{\mathbb {C}}^n} H(\mathrm {w}) \prod _{j=0}^nK^\infty _{\rho _N}(w_{j}, w_{j+1})\ d\mathrm {A}^n(\mathrm {w}) \\&\quad = \int _{{\mathscr {A}}(0; \delta _N)} H(\mathrm {w}) \prod _{j=0}^nK^\infty _{\rho _N}(w_{j}, w_{j+1})\ d\mathrm {A}^n(\mathrm {w}) + \underset{N\rightarrow \infty }{O}(\rho _N^{1-k^2/2}) \end{aligned}$$

where the set ${\mathscr {A}}(0; \delta _N)$ is given by formula (3.5).

Proof

We will first show that

$$\begin{aligned} \underset{|w_1| > \delta _N}{\int _{{\mathbb {C}}^n}} H(\mathrm {w}) \prod _{j=0}^nK^\infty _{\rho _N}(w_{j}, w_{j+1}) \ d\mathrm {A}^n(\mathrm {w}) = \underset{N\rightarrow \infty }{O}\big ( \rho _N e^{- \rho _N \delta _N^2/2} \big ) . \end{aligned}$$

(B.6)

First, notice that if $H =1$, since $w_0=0$, by formula (3.14), we have

$$\begin{aligned} \underset{|w_1| > \delta _N}{\int _{{\mathbb {C}}^n}} \bigg | \prod _{j=0}^nK^\infty _{\rho _N}(w_{j}, w_{j+1})\bigg | \ d\mathrm {A}^n(\mathrm {w}) \le e^{- \rho _N \delta _N^2/2} \rho _N^{n+1} \underbrace{\int _{{\mathbb {C}}^n} \prod _{j=1}^n e^{-\rho _N |v_j|^2/2} d\mathrm {A}^n(\mathrm {v})}_{\displaystyle = \rho _N^{-n}} . \end{aligned}$$

In general, there exists a constant $C>0$ which only depends on the polynomial H so that

$$\begin{aligned} \big | H(\mathrm {w}) \big | \le C \big \{ 1+ |w_1|^2 +\cdots +|w_n|^2 \big \} \end{aligned}$$

or

$$\begin{aligned} \big | H(\mathrm {w}) \big | \le C \big \{ 1+ |w_2-w_1|^2 +\cdots +|w_{n-1} -w_n|^2 + |w_n|^2 \big \} . \end{aligned}$$

(B.7)

Since, for any $k=1,\dots , n$,

$$\begin{aligned}&\underset{|w_1| > \delta _N}{\int _{{\mathbb {C}}^n}}|w_k-w_{k+1}|^2 \bigg | \prod _{j=0}^nK^\infty _{\rho _N}(w_{j}, w_{j+1}) \bigg | d\mathrm {A}^n(\mathrm {w}) \\&\quad \le e^{- \rho _N \delta _N^2/2} \underbrace{ \rho _N^{n+1} \int _{{\mathbb {C}}^n} |v_k|^2 \prod _{j=1}^n e^{-\rho _N | v_{j} |^2/2} d\mathrm {A}^n(\mathrm {v})}_{\displaystyle = 1} , \end{aligned}$$

the estimate (B.6) follows directly from (B.7) and the leading contribution comes from the constant term. If we use the estimate

$$\begin{aligned} \big | H(\mathrm {w}) \big | \le C \bigg \{ 1+ |w_1|^2 + \sum _{\begin{array}{c} j=1 \\ j \ne k \end{array}}^n |w_{j+1} -w_j|^2 +|w_n|^2 \bigg \} \end{aligned}$$

instead, the same argument shows that for any $k= 1,\dots n$,

$$\begin{aligned} \int \limits _{|w_k -w_{k+1}| > \delta _N } H(\mathrm {w}) \prod _{j=0}^nK^\infty _{\rho _N}(w_{j}, w_{j+1}) \ d\mathrm {A}^n(\mathrm {w}) = \underset{N\rightarrow \infty }{O}\big ( \rho _N e^{- \rho _N \delta _N^2/2} \big ) . \end{aligned}$$

Hence, the Lemma follows from applying a union bound and from the choice of the sequence $\delta _N$. $\square$

Proof of Lemma 3.2

The map $(z, w) \mapsto \Psi (z,w)$ is bi-holomorphic in a neighborhood of $(x_0, \overline{x_0})$, so there exists $0<\epsilon <1$ such that for all $|u|, |v| \le \epsilon$,

$$\begin{aligned} \Psi (x_0 + u, \overline{x_0} + v) = \sum _{k, j \ge 0} a_{kj} u^k v^j . \end{aligned}$$

By definition, $\overline{\Psi (z,w)} = \Psi ({\overline{w}},{\overline{z}})$, so that the coefficients of the previous power series are Hermitian-symmetric: $a_{kj} = \overline{a_{jk}}$ for all $k, j \ge 0$. Moreover, by definition, we have

$$\begin{aligned} a_{11} = \partial _z \overline{\partial }_z V |_{z=x_0} = \Delta V (x_0) = \frac{b_0(x_0, \overline{x_0})}{2} . \end{aligned}$$

(B.8)

Let

$$\begin{aligned} {\mathfrak {h}}(u) = i \sum _{k>0} \big \{ a_{k0} u^k - a_{0k} {\overline{u}}^k \big \} = -2 \mathfrak {I}\bigg \{ \sum _{k>0} a_{k0} u^k \bigg \} . \end{aligned}$$

Since $V(x_0 + u) = \Psi (x_0 + u , \overline{x_0} + {\overline{u}} )$, we see that for any $|u|, |v| \le \epsilon$,

$$\begin{aligned}&2\Psi (x_0 + u, \overline{x_0} +{\overline{v}}) -V(x_0+u)- V(x_0+v) \\&\quad = - i\{ {\mathfrak {h}}(u) - {\mathfrak {h}}(v)\} + a_{11} \big ( 2 u {\overline{v}} - |{\overline{u}}|^2 -|{\overline{v}}|^2 \big ) + O( \epsilon ^3) . \end{aligned}$$

By formula (3.1), this implies that for any $|u|, |v| \le \epsilon _N = \log (N^\kappa ) N^{-1/2}$,

$$\begin{aligned} \frac{B^N(x_0 + u , x_0 +v)e^{i N {\mathfrak {h}}(u)}}{e^{i N {\mathfrak {h}}(v)}}= N b_0(x_0, \overline{x_0}) e^{N a_{11} ( 2 u {\overline{v}} - |{\overline{u}}|^2 -|{\overline{v}}|^2) } \left\{ 1 + \underset{N\rightarrow \infty }{O}\big ( (\log N)^{2}\epsilon _N \big ) \right\} . \end{aligned}$$

By formula (B.8) and the definition of the $\infty$-Ginibre kernel, it completes the proof. $\square$