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

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

for all $i \neq j$. I have two questions

(a)  Consider the  moment generating function  function

 $$ F_m(x) = E_{GOE}\bigl(   e^{ tr( xA) }  \bigr), $$


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

(b)   Similarly, consider the characteristic function

 $$ G_m(x) = E_{GOE}\bigl(   e^{ \sqrt{-1} tr( xA) }  \bigr), $$


 $x$ real. What can one say about the behavior of  $G_m(x)$  as $m\rightarrow \infty$.



Any   help will be greatly appreciated.