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Given an integer $d$, let $\alpha_d(N)$ denote the number of symmetric integral positive definite matrices of size $d\times d$ with coefficients in $\lbrace -N,-N+1,\dots,N-1,N\rbrace$.

Asymptotically, the number $\alpha_d(N)$ is given by $\alpha_d(N)\sim c_dN^{d+1\choose 2}$ for some strictly positive real number $c_d$ (with $c_d\leq 2^{d+1\choose 2}$).

What is known on the constants $c_2,c_3,\dots$?

Added: $c_d$ is in fact equal to the proportion (with respect to the obvious Lebesgue measure) of $d\times d$ matrices with coefficients in $[-1,1]$ which are positive definite. It is also equal to $2^{d+1\choose 2}$ times the probability that a random matrix of size $d\times d$ (with respect to the uniform probability) with all coefficients in $[-1,1]$ is positive definite.

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  • $\begingroup$ Roland: nice question, as usual. $c_2=2/9$? $\endgroup$
    – Did
    Commented May 4, 2011 at 14:43
  • $\begingroup$ $c2\geq 1/4$ by considering $\left(\begin{array}{cc}a&b\\b&c\end{array}\right)$ with $a,c\in\lbrace N+1,\dots,2N\rbrace$ and $b\in \lbrace −N,…,N\rbrace$. $\endgroup$
    – Roland Bacher
    Commented May 4, 2011 at 14:48
  • $\begingroup$ In fact, $c_2=2\left(\int_0^1 t^{3/2}dt\right)^2=\frac{8}{25}$. $\endgroup$
    – Roland Bacher
    Commented May 4, 2011 at 15:13
  • $\begingroup$ I thought $|a|$ and $|c|$ were a priori at most $N$? And that the conditions for positive definiteness were $a\ge0$, $c\ge0$ and $b^2\le ac$. $\endgroup$
    – Did
    Commented May 4, 2011 at 15:49
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    $\begingroup$ For a symmetric 2 x 2 matrix with entries uniformly distributed in $[-1,1]$ the probability of being positive definite is $\frac{1}{8} \int_0^1 da \int_0^1 dc \int_{-\sqrt{ac}}^{\sqrt{ac}} db = \frac{1}{9}$. $\endgroup$
    – Robert Israel
    Commented May 4, 2011 at 17:48

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