Appendix: Section 2 Proofs

In appendix, we provide a proof for Lemma 2.2, and then apply it to prove Lemma 2.8.

1.1 Proof of Lemma 2.2

Before beginning the main proof, we need the following preliminary results.

Lemma A.1

Let \(\_^n\) be eigenvalues of a scaled LUE or LOE matrix \(\frac M_\). Assume s is such that \(s>C\) for some \(C>0\) and \(s=o(n^)\) as \(n\rightarrow \infty \). The following statements hold for \(\mathcal _s:=\#\,\infty ) \}\).

$$\begin \mathbb \mathcal _s&=\fracd_+}s^+O(s^n^). \end$$

(A.1)

$$\begin \,}}(\mathcal _s)&=\frac\log (s)(1+o(1)). \end$$

(A.2)

The lemma is the analog of Proposition 6.5 from [40], which bounds the expectation and variance of the counting function in the case of GOE matrices. There, the result was obtained by applying the corresponding result for GUE matrices by Gustavsson [33], and the relation between eigenvalues of Gaussian orthogonal and unitary ensembles in [30]. The proof of [40] works in our case, up to translating from Gaussian to Laguerre Ensembles. For completeness, we reproduce it here, first proving for LUE matrices using a result in [51], and then extend to LOE matrices using the following result.

Theorem A.2

(Theorem 5.2 of [30]). For independent eigenvalue point processes \(\text _\), \(\text _\),

$$\begin \text (\text _\cup \text _)=\text _, \end$$

where the notation \(\text (\cdot )\) denotes the set containing only the even numbered elements among the ordered list of elements in the original set.

Proof of Lemma A.1

In the case of LUE matrix, the lemma follows from the results of Su in [51]. Namely, the first inequality holds by Lemma 1 of [51], which states that

$$\begin \mathbb \#\&=n\int _^}p_\,}}}(x)dx\\&= \frac-\alpha _}}}n(\beta _-t_n)^+ O(n(\beta _-t_n)^). \end$$

As the matrix in [51] is scaled by 1/n instead of 1/m as in this paper, our interval of interest \([d_+-sn^, \infty )\) corresponds to \(t_n=\beta _-\fracn^\) in [51]. Meanwhile, the inequality for variance directly follows from Lemma 4 there.

We now consider the case of LOE matrix. Let \(M^_\), \(M^_\) be independent LOE matrices of size \(n\times m\) and \((n+1)\times (m+1)\), respectively, and let \(M^_\) be a LUE matrix of size \(n\times m\). Set \(X^_\) to be the number of eigenvalues of \(M^_\) that are at least \(m\left( d_+-sn^}\right) \). We define \(X^_\) and \(X^_\) similarly, for the two LOE matrices. Theorem A.2 implies that there is a random variable Y and a random variable \(Z\in [0,1]\) such that

$$\begin X^_\limits ^}Y, \quad Y-Z=\frac\left( X^_+X^_\right) . \end$$

The estimates (A.1) and (A.2) hold for Y by the previous paragraph. The estimate (A.2) for Y, together with boundedness of Z and the fact \(X^_\) and \(X^_\) are independent implies that (A.2) holds for the \(X^\)’s as well. Now,

$$\begin \mathbb [X^_]=\frac\left( \mathbb [X^_]+\mathbb [X^_]\right) +c, \quad \text c\in [0,1]. \end$$

(A.3)

From the tridiagonal form of Laguerre Ensembles, the top left \(n\times n\) minor of \(M^_\) has the same distribution as \(M^_\). The eigenvalues of this minor interlace those of \(M^_\), which implies there is a random variable \(\tilde^_\) with the same distribution as \(X^_\) and satisfies

$$\begin |\tilde^_-X^_|\le 1. \end$$

We then obtain (A.1) for \(X^_\) and \(X^_\), using (A.1) for \(X^_\), (A.3) and the above inequality.\(\square \)

We now have the needed tools to prove Lemma 2.2.

Proof of Lemma 2.2

For \(j=1,\dots , n^\) and \(t>0\), by definition,

$$\begin \begin \mathbb (A_j\ge t)=\mathbb \left( \mu _j\ge d_+-\left( \left( C^\star j\right) ^-t\right) n^\right) = \mathbb (\mathcal _T\ge j), \end \end$$

(A.4)

where \(C^\star =\tfrac \pi \lambda ^d_+\) and \(T=T(j,t):=\left( C^\star j\right) ^-t\). If \(\mathbb \mathcal _T<j\), then

$$\begin \mathbb (\mathcal _T\ge j)\le \mathbb (|\mathcal _T-\mathbb \mathcal _T|\ge j-\mathbb \mathcal _T)\le \frac\,}}\mathcal _T}\mathcal _T)^2}. \end$$

(A.5)

In order to make use of this inequality, we need to know what values of t (depending on j) satisfy \(\mathbb \mathcal _T<j\). By Lemma A.1, there exist \(K,c_0>0\) such that, for any \(c_1>0\) and any sufficiently large n, if \(K\le j\le n^\) and \(0<t<(C^\star j)^-c_1\), then

$$\begin \begin j-\mathbb \mathcal _T&\ge j-\tfrac((C^\star j)^-t)^-c_0j^n^\\&\ge j-j\left( 1-\frac}\right) ^-c_0\; \ge \; \frac}}-c_0. \end\end$$

(A.6)

In particular, this means that \(\mathbb \mathcal _<j\) is satisfied (along with the conditions of Lemma A.1) when \(c_0(C^\star )^ j^<t<(C^\star j)^-c_1\) and \(K\le j\le n^\) (note that one should choose \(K>c_0\)). Thus, for t, j satisfying these conditions, we combine (A.4)–(A.6) with the variance bound from Lemma A.1 to conclude that, for some \(c_2>0\) and sufficiently large n,

$$\begin \mathbb (A_j\ge t)\le \fractj^-c_0)^2}. \end$$

(A.7)

Next, taking \(T'=\left( C^\star j\right) ^+t\) we can follow the same argument to bound \(\mathbb (A_j<-t)\). This time, we find that \(\mathbb \mathcal _\ge j\) is satisfied (along with the conditions of Lemma A.1) when \(c_0(C^\star )^ j^<t\ll n^\) and \(K\le j\le n^\). Then, for t, j satisfying these conditions, and for some \(c_3>0\) with sufficiently large n,

$$\begin \begin \mathbb (A_j\le -t)&=\mathbb (\mu _j <d_+-T'n^)\le \mathbb (|\mathcal _-\mathbb \mathcal _|> \mathbb \mathcal _-j)\\&\le c_3\fractj^-c_0)^2}. \end \end$$

(A.8)

Thus, for j, t satisfying \(K\le j\le n^\) and \(c_0(C^\star )^ j^<t<(C^\star j)^-c_1\), we have

$$\begin \mathbb (|A_j|\ge t)=O\left( \fractj^-c_0)^2}\right) . \end$$

Taking \(t=\lambda j^\), then for all \(k\ge K\),

$$\begin \mathbb \left( \bigcup _}\left\\right\} \right) =O\left( \sum _^}\frac\right) =O\left( \frac\right) . \end$$

This bound holds uniformly for \(K\le k\le n^\). Taking \(k\rightarrow \infty \) (for example \(k=n^\)), we obtain (2.18).

It remains to prove the second part of the lemma. Set \(t^*=c_0(C^\star )^j^\). For \(K\le j\le n^\), we have

$$\begin \mathbb \left[ \mathbbm _(\mu _j-d_+)\le -C\}}\left| A_j\right| \right]&\le \int _0^\mathbb (A_j\ge t)\textrmt+\int _0^\mathbb (-A_j\le -t)\textrmt\\&\le \left( t^*+\int _^-C}\mathbb (A_j>t)\textrmt+0\right) \\&\quad +\left( t^*+\int _^-\delta }}\mathbb (-A_j\le -t)\textrmt+o(n^)\right) \\&\le 2t^*+C' \int _^\fractj^-c_0)^2}\textrmt\\&\le 2t^*+C''\frac} = O\left( \frac}\right) , \end$$

where, in the second line, we obtained \(\int _-C}^\infty \mathbb (A_j\ge t)\textrmt=0\) from the indicator in the expectation, and \(\int _-\delta }}^\infty \mathbb (-A_j\le -t)\textrmt=o(n^)\) from eigenvalue rigidity.\(\square \)

1.2 Proof of Lemma 2.8

We observe that

$$\begin S_2:= & \frac\sum _^n\frac-\int _^ \fracp_\,}}}(y)\textrmy\nonumber \\= & \sum _\int _^}\fracp_\,}}}(y)\textrmy. \end$$

(A.9)

The modulus of this sum satisfies

$$\begin\begin |S_2|&\le \sum _\int _^}\frac^ |\mu _i-y|}p_\,}}}(y)\textrmy\\&\le \sum _\int _^}\frac^}p_\,}}}(y)\textrmy. \end\end$$

We now split the sum as \(S_+S_\), summing over \(K\le i\le n^\) and \(i>n^\), respectively. First, consider \(K\le i\le n^\). By Lemma 2.2, given \(\varepsilon >0\), on the event \(\mathcal _\varepsilon \), there exists \(c>0\) such that, for sufficiently large n, \(n^(d_+-\mu _i)\ge ci^\) uniformly for all i in this range. Combining with the facts that \(\mathop }\limits z\ge d_+\) and \(d_+\ge \mu _i\) for \(i\ge K\) on \(\mathcal _\varepsilon \), we have

$$\begin n^|z-\mu _i|\ge \max \|z-d_+|,\; ci^\}. \end$$

(A.10)

Meanwhile, there exists \(C>0\), independent of n, such that \(C^i^\le n^(d_+-g_i)\le Ci^\) for all i (see, for example, [12]). Thus, (A.10) also holds for \(n^|z-y|\), uniformly for \(y\in (g_i,g_)\). For the numerator, we have \(n^(y-g_i) \le n^(g_-g_i)\le ci^\), using

$$\begin \frac = \int _^}p_\,}}}(y)\textrmy\ge c\sqrt(g_-g_i). \end$$

By (2.16), \(n^(\mu _i-g_i)= A_i+O\left( \frac}}\right) \), where \(A_i\) is given in (2.17). The term \(i^\) is of larger order than \(n^i^\) when \(K\le i \le n^\), and they have the same order when \(i=\Theta (n^)\). Thus,

$$\begin \mathbbm __\varepsilon }\frac^} & \le Cln^ l}\frac+|A_i|}(l+1)}+(n^}|z-d_+|)^},\nonumber \\ & \quad K\le i\le n^. \end$$

(A.11)

By Lemma 2.2,

$$\begin \begin \mathbb \left[ \mathbbm __\varepsilon }|S_|\right]&\le Cln^ l-1}\sum _}\frac+\mathbb \left[ \mathbbm __\varepsilon }|A_i|\right] }(l+1)}+(n^}|z-d_+|)^} \\ &\le C'ln^ l-1}\sum _}\frac\log i}(l+1)}+(n^}|z-d_+|)^}. \end\nonumber \\ \end$$

(A.12)

Next, we consider two separate cases and conclude that, for some \(C''>0,\)

$$\begin \mathbb \left[ \mathbbm __\varepsilon }|S_|\right] \le C''n^ l-1}\frac|z-d_+|)}|z-d_+|)^} & K^<n^|z-d_+|,\\ C''n^ l-1} & K^\ge n^|z-d_+|.\end\right. } \end$$

The bound in the first case is obtained by evaluating the right-hand side of (A.12) separately for \(i^<n^|z-d_+|\) and \(i^>n^|z-d_+|\). The bound in the second case follows from the convergence of \(\sum _^\infty i^ l-1}\log i\) for all \(l\ge 1\). Thus, we obtain

$$\begin \mathbb \left[ \mathbbm __\varepsilon }|S_|\right] =O\left( n^ l-1}\cdot \min \left\|z-d_+|)}|z-d_+|)^l}\right| ,\;1\right\} \right) . \end$$

(A.13)

Lastly, for \(S_\), we bound the numerator (which is \(l\cdot |\mu _i-y|\)) using rigidity and bound \(n^|z-y|\ge ci^\) by (A.10), while \(|z-\mu _i| \ge \max \\), where, with high probability,

$$\begin d_+-\mu _i \ge c>0, &\quad i>n/2,\\ ci^n^, &\quad n^<i<n/2, \text \, \delta <\frac\hbox \\ (2.16). \end\right. } \end$$

We obtain

$$\begin \begin \mathbbm __\varepsilon }|S_|&\le Cln^ l-1+\delta }\sum _}\frac(l+1)}\min \, (n+1-i)^\}}\\&\le C'ln^ l-1+\delta }\left( \sum _}^i^ l-1}+\sum _\frac (l+1)}(n+1-i)^}\right) \\&= O(n^ l-1}\cdot n^), \end\nonumber \\ \end$$

(A.14)

which is \(o(\mathbb \left[ \mathbbm __\varepsilon }|S_|\right] )\), provided \(\delta <4l/15\). This completes our proof of Lemma 2.8.

Appendix: Section 5 Proofs

In this section, we provide our proofs of Lemmas 5.4 and 5.5. The proof of Lemma 5.4 requires asymptotic bounds on \(|\rho _j^\pm |\) when \(\gamma =d_+\) and a few related quantities, which we state in the following two lemmas. Similar results were developed for the case \(\gamma >d_+\) in Lemmas 2.7 and 2.8 in [24].

Lemma B.1

The following asymptotic bounds hold, uniformly in \(i\ge 2\) (where i can be fixed or n-dependent):

(i)

\(|\rho _i^+|=\Theta (n)\), \(|\rho _i^-|=O(n)\),

(ii)

\(|\rho _i^+|-|\rho _i^-|=\Theta (n^(n-i+1)^)\),

(iii)

\(|\rho _i^-|-|\rho _^-|=O((\frac)^)\) and \(|\rho _^+|-|\rho _i^+|=O((\frac)^)\).

Proof

To show (i) for \(|\rho _i^-|\), observe that \(|\rho _i^-|\) is increasing in i, and

$$\begin |\rho _n^-|=\frac\left( 2\sqrt+1-2\sqrt+n+m-\tfrac}+O(1)\right) =O(n). \end$$

Similarly, part (i) for \(|\rho _i^+|\) holds since \(|\rho _i^+|\) is decreasing in i, \(|\rho _2^+|<2\sqrt+2n=\Theta (n)\), and

$$\begin |\rho _n^+|>\frac\left( d_+m-(m+n-1)\right) =\frac\left( 2\sqrt+1\right) =\Theta (n). \end$$

For part (ii), we have

$$\begin&|\rho _i^+|-|\rho _i^-|\\ &=\sqrt-1+2(n-i+1))^2-4(m-(n-i+1))(n-(n-i+1))}\\&=2\sqrt+1+(m+n+2\sqrt-1)(n-i+1)}=\Theta (n^(n-i+1)^). \end$$

Next, we verify (iii) by showing that \(|\rho _i^-|-|\rho _^-|+ |\rho _^+|-|\rho _i^+|=O((\frac)^)\). Indeed, the left-hand side can be written as

$$\begin \left( |\rho _^+|-|\rho _^-|\right) - \left( |\rho _i^+|-|\rho _i^-|\right) =\frac^+|-|\rho _^-|\right) ^2-\left( |\rho _i^+|-|\rho _i^-|\right) ^2}^+|-|\rho _^-|+ |\rho _i^+|-|\rho _i^-|}, \end$$

where numerator of the last ratio simplifies to \(4d_+m-4=\Theta (n)\) and the denominator is \(\Theta (n^(n-i+1)^)\) by part (ii). \(\square \)

Lemma B.2

There exist constants \(0<C_1<C_2\) such that, for sufficiently large n, and \(2\le i\le n\),

$$\begin C_1\left( \frac\right) ^\le 1-\omega _i\le C_2\left( \frac\right) ^. \end$$

Proof

We recall that \(\omega _i=\frac^+|}\). Using the bounds \(\frac^-|}^+|}<\omega _i<\frac\) we obtain

$$\begin \frac<1-\omega _i<\frac^+|-|\rho _^-|}^+|}. \end$$

Using Lemma B.1, the left and right sides of this inequality are both \(\Theta ((\fraci1})^)\), uniformly in i, which gives the desired bounds. \(\square \)

1.1 Proof of Lemma 5.4

Using \(\frac}=1-\frac}}=1-F_+\frac^2}}\) and the notations in (5.2) and (5.4), we have \(F_1=\frac-\alpha _1\), and for \(j=2,\dots , n-1\),

$$\begin F_j= & -1+\frac-\left( \alpha _j+\beta _j+\tau _j+\delta _j\right) \\ & -(\alpha _+\tau _)(\beta _j+\delta _j)\left( 1-F_+\frac^2}}\right) . \end$$

As \(1+\tau _j+\delta _j=\frac-\frac\), we re-arrange the terms to have

$$\begin F_j = \eta _j- \xi _j +\omega _j F_+\phi _j, \end$$

(B.1)

where we define

$$\begin \eta _j&=\frac,\end$$

(B.2)

$$\begin \phi _j&=-\omega _j+\frac-\alpha _\beta _j +(\alpha _\beta _j+\alpha _\delta _j+\tau _\beta _j)\frac}}-\omega _j\frac^2}}, \end$$

(B.3)

and \(\xi _j\) is given in (5.11). Note that, by Lemma B.1,

$$\begin 0<\omega _j-\frac= \frac\frac^+|-|\rho _j^+|}^+|}= O(n^}(n-j+1)^}). \end$$

(B.4)

Expanding the recurrence iteratively, we get

$$\begin \begin F_j&=\omega _j\dots \omega _2F_1+\left( \eta _j+\omega _j\eta _+\dots +\omega _j\dots \omega _3\eta _2\right) \\&\quad -\left( \xi _j+\omega _j\xi _+\dots +\omega _j\dots \omega _3\xi _2 \right) \\&\quad +\left( \phi _j+\omega _j\phi _+\dots +\omega _j\dots \omega _3\phi _2\right) . \end \end$$

(B.5)

On the event \(\mathcal ^_\), which holds with probability \(1-\varepsilon /6\) for some s, t depending on \(\varepsilon \), \(|\mu _1-d_+|\le tn^}\). As \(|\rho _i^+|/m=\Theta (1)\) for all \(i\le n\), we obtain

$$\begin \max _|\eta _j|=O\left( n^}\right) . \end$$

We recall that \(\alpha _j,\beta _j\) are the centered and scaled version of \(\chi \)-squared random variables \(a_j^2, b_^2\), respectively, and as such, they can be bounded using concentration of sub-gamma random variables (see, e.g., Theorem 2.3 of [21]). In particular, there exists some constant c such that, for all \(j\le n\) and for all \(t>0\),

$$\begin \mathbb \left( |\alpha _j|>c\left( \sqrt}+\tfrac\right) \right) \le 2e^ \end$$

(B.6)

and likewise for each \(\beta _j\), so we conclude that, for any \(\varepsilon \), with probability at least \(1-\varepsilon /6\),

$$\begin \max \\le cn^}\sqrt. \end$$

(B.7)

Thus, for some constant \(C_1>0\), with probability \(1-\varepsilon /3\),

$$\begin |\omega _j\dots \omega _2F_1|\le |F_1|=|\eta _1-\alpha _1|\le C_1n^}. \end$$

(B.8)

As \(\omega _j\) is increasing in j,

$$\begin 1+\omega _j+\omega _j\omega _+\dots +\omega _j\dots \omega _3\le 1+\omega _j+\omega _j^2+\dots =\frac. \end$$

By Lemma B.2, \(1-\omega _j=\Theta \left( \left( \frac\right) ^}\right) \). Thus, setting \(j_0:=\lfloor n-n^}(\log n)^3 \rfloor \), we observe that, for some constant \(C_2\), with probability \(1-\varepsilon /3\),

$$\begin \max _|\eta _j\omega _j\eta _\dots +\omega _j\dots \omega _3\eta _2|\le \max _\left( |\eta _j|\frac\omega _j}\right) \le C_2n^}(\log n)^}.\nonumber \\ \end$$

(B.9)

Having bounded the first line of (B.5), we turn to the second line and recall the definition of \(L_j\) in (5.10). We have

$$\begin \xi _j+\omega _j\xi _+\dots +\omega _j\dots \omega _3\xi _2 = L_j+\omega _j\dots \omega _3\xi _2. \end$$

Note that \(\max _|\xi _j|=O(n^}\sqrt)\) on the event (B.7). We also have, for some constant \(C_3>0\), with probability \(1-O(n^)\),

$$\begin \max _|L_j|=O(n^}(\log n)^}). \end$$

(B.10)

The details for this bound can be obtained using a similar argument to the one found in Section 6.2 of [24]. In particular, the bound (B.10) follows from line (6.17) of that paper (where the notations \(\alpha \) and \(Y_i\) can be translated as \(\alpha =2\) and \(Y_i=L_i+O(n^})\) in our context). Thus, for some constant \(C_3>0\), with probability \(1-\varepsilon /3\),

$$\begin \max _|\xi _j+\omega _j\xi _+\dots +\omega _j\dots \omega _3\xi _2|\le C_3n^}(\log n)^}. \end$$

(B.11)

Consider the event

$$\begin \mathcal :=\ (B.11) \text \}, \end$$

(B.12)

which holds with probability \(1-\varepsilon \), for sufficiently large n. We now show that on this event, the third line of (B.5) is \(o(n^})\). Since this quantity depends on \(F_l\)’s up to \(F_\), we can control it in the process of using induction to show

$$\begin \max _|F_j| = o(n^}) \quad \text \mathcal . \end$$

(B.13)

More specifically, we will show that \(\max _|F_j| < 2C_3n^}(\log n)^}\) where \(C_3\) is the constant from (B.11). The base case holds by (B.8). Assume \(\max _|F_| < 2C_3n^}(\log n)^}\). Then, by (B.3), (B.4) and (B.7),

$$\begin \max _|\phi _l|=o(n^}). \end$$

Note that the above maximum also includes \(\phi _j\). Thus, for some constant \(C_4>0\),

$$\begin |\phi _j+\omega _j\phi _+\dots +\omega _j\dots \omega _3\phi _2|\le \max _|\phi _l|\frac\le C_4n^}(\log n)^. \end$$

Finally, by (B.5), we have that on \(\mathcal \),

$$\begin |F_j| & \le C_1n^}(\log n) +C_2n^}(\log n)^}+C_3n^}(\log n)^}+C_4n^}(\log n)^ \\ & <2C_3n^}(\log n)^}. \end$$

This completes the induction step, and we obtain the lemma.

1.2 Proof of Lemma 5.5

Fix \(\varepsilon >0\). For \(j_0=\lfloor n-n^(\log n)^3 \rfloor \) and \(t=(e\log n)^2\), it suffices to show that, for sufficiently large n, each of the probabilities

$$\begin p_1&:=\mathbb \left( \max _a_jb_j<\sqrt-tn^\right) \le \mathbb \left( a_b_<\sqrt-tn^\right) \text \end$$

(B.14)

$$\begin p_2&:=\mathbb \left( \max _a_jb_j>\sqrt+tn^\right) =1-\prod _^\mathbb \left( a_jb_j<\sqrt+tn^\right) \end$$

(B.15)

is less than \(\varepsilon /2\). For any \(j=1,2,\dots , j_0\), observe that

$$\begin a_j^2b_j^2\limits ^}\left( \sum _^g_i^2\right) \left( \sum _^(g'_k)^2\right) , \end$$

(B.16)

where \(\limits ^}\) denotes equality in distribution, and \(g_1,\dots , g_, g'_1, \dots , g'_j\) are independent standard gaussian variables. This implies that \(\mathbb a_j^2b_j^2=j(m-n+j)\) and \(\,}}(a_j^2b_j^2)=2j(m-n+j)(m-n+2j+2)\). Viewing \(a_j^2b_j^2\) as a gaussian polynomial of degree 4 in \(m-n+2j\) variables \(g_i\)’s and \(g'_k\)’s, we have the following concentration result from [8] (see Corollary 5.49): For any \(s\ge (2e)^2\),

$$\begin & \mathbb \left( |a_j^2b_j^2-j(m-n+j)|\right. \nonumber \\ & \qquad \left. \ge s\sqrt\right) \le \exp \left( -2\sqrt/e\right) . \end$$

(B.17)

Apply this result to \((a_b_)^2\) with \(s=(e\log n)^2\), we obtain \(p_1\le n^\). At the same time, (B.17) implies \(\mathbb \left( a_jb_j<\sqrt+tn^\right) \ge 1-n^\) for all \(1\le j\le j_0\), which yields \(p_2\le 1-e^\) for some \(c>0\). This completes the proof of the lemma.