Investigations concerning random Morse functions led me to the following problem. Consider  the  classical   GOE of $m\times m$ real symmetric matrices $A$  with independent   Gaussian entries with zero means and variances

$$ \boldsymbol{E}(a_{ii}^2)=2 \boldsymbol{E}(a_{ij}^2)= 2 $$

for all $i \neq j$. 

(a)  Consider the   function

 $$ F_m(x, y) = \boldsymbol{E}_{GOE}\bigl( |\det(y+A)|  e^{ -x(tr A)^2 }  \bigr), $$


 $x,y$ real, $x>0$. What can one say about the behavior of  $F_m(x,y)$  as $m\rightarrow \infty$.

Equivalently  we can consider the  Gaussian ensemble $\mathcal{S}(m,x)$ of  symmetric $m\times m$  real matrices    with probability density
 
$$ dP(A)=\frac{1}{Z_{m,x}} e^{-\frac{1}{2}tr(A^2)-x(tr A)^2} \prod_{i\leq j} da_{ij}, $$

$x>0$,  and then ask for the  bevavior as $m\rightarrow \infty$ of the expectation

$$ \boldsymbol{E}_{\mathcal{S}(m,x)}\left( |\det(A+y)|\right). $$



Observe that  

$$GOE= \mathcal{S}(m,x)_{x=0}.$$

 The  normalizing constant $Z_{m,x}$ can be explicitly computed for  any $x$   and thus

 $$ F_m(x,y)= \frac{Z_{m,x}}{Z_{m,0}} \boldsymbol{E}_{\mathcal{S}(m,x)}\left( |\det(A+y)|\right).  $$

In the geometric problem I am interested  $x=\frac{1}{4}$.  In this case the ensemble $\mathcal{S}_m:=\mathcal{S}(m, \frac{1}{4})$  can be described as the ensemble of  real, symmetric $m\times m$ matrices whose   entries   are mean zero Gaussian variables satisfying the covariance equalities

$$ \boldsymbol{E}\left( a_{ij} a_{k\ell}\right)=-\frac{2}{2+m}\delta_{ij}\delta_{k\ell} +\left( \delta_{ik}\delta_{j\ell}+ \delta_{i\ell}\delta_{jk}\right).$$

Note that as $m\rightarrow \infty$ this ensemble resembles    more  and more the classical  GOE which satisfies the covariance     equalities

$$ \boldsymbol{E}\left( a_{ij} a_{k\ell}\right)= \left(\delta_{ik}\delta_{j\ell}+ \delta_{i\ell}\delta_{jk}\right).$$




Finally, I want to explain how is this related to Morse theory.    To put things in perspective observe that if $A$ is a symmetric $m\times m$ matrix, then its spectrum  can be identified with the set of critical values of the restriction to the unit sphere in $\mathbb{R}^m$ of the quadratic polynomial

$$\mathbb{R}^m\ni x\mapsto q_A(X)=(Ax,x).$$

To a  Morse function $f$ on a  compact smooth manifold $M$ of dimension  $m$ we can associate two measures.

(a) A measure $K_f$ on $M$ defined as the sum of   Dirac delta's concentrated at the critical points of $f$

$$K_f=\sum_{df(p)=0}\delta_p.$$

(b) A measure $\Delta_f$ on $\mathbb{R}$ supported on the set of critical values of $f$ and defined as the pushforward of $K_f$ via $f$, 

$$\Delta_f:=f_*(K_f).$$

In other words, $\Delta_f$ counts the critical values with multiplicity. Note that when $f$ is the restriction to the unit sphere of the quadratic form $q_A$ then $\Delta_f$ coincides with the spectral measure of $A$.  

Fix a Riemann metric  $g$ on $M$     and an orthonormal $(\Psi_k)_{k\geq 0}$ basis of $L^2(M)$ consisting of eigenfunctions  of the Laplacian

$$ \Delta \Psi_k=\lambda_k \Psi_k. $$

Fix  i.i.d. standard Gaussian random variables  $(x_k)_{k\geq 0}$ and for every $L >0$ define  the random function

$$f_L=\sum_{\lambda_k\leq L^2}x_k\Psi_k. $$


The function $f_L$ is roughly speaking  a random polynomial of  large degree.  Equivalently one should think of $f_L$ as  a random element in the space $U_L$ spanned by the eigenfunctions corresponding to eigenvalues  $\leq L^2$   and equipped with the standard Gaussian measure.  The large $L$ behavior of $\dim U_L$ is governed by Weyl's asymptotic formula

$$ \dim U_L \sim const. L^m.$$



To $f_L$ we associate two random measures

$$  K_{f_L},\;\; \Delta_{f_L} $$

that have  normalized expectations


$$ K_L:=\frac{1}{\dim U_L} \boldsymbol{E}( K_{f_L} ), $$

$$ \Delta_L:=\frac{1}{\dim U_L} \boldsymbol{E}( \Delta_{f_L} ). $$


Above, $K_L$ is a measure on $M$ and $\Delta_L$ is a measure on $\mathbb{R}$. I can show that as $L\to\infty$  the measure $K_L$ converges weakly to $C_m dV_g$, where $dV_g$ denotes the volume measure determined by the metric $g$, and $C_m$ is a certain  explicit constant that depends only on $m$ but not on $(M,g)$.   Thus     critical set of a random $f_L$, $L\gg 0$,  is   uniformly distributed on average.

As $L\to \infty$ the measure  a suitable rescaled version of  $\Delta_L$ converges to a measure on $\mathbb{R}$ that absolutely continuous with respect to the Lebesgue measure. Its  density  has the form

$$\rho_m(y) dy=  Const_m \times \boldsymbol{E}_{\mathcal{S}(m,1/4)}\left( \;|\det(A-y )|\;\right) e^{-\frac{y^2}{2 s_m}} dy,\;\;s_m=\frac{m+2}{m+4} $$