an:02028570
Zbl 1053.15017
Bleher, Pavel; Eynard, Bertrand
Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations
EN
J. Phys. A, Math. Gen. 36, No. 12, 3085-3105 (2003).
00094034
2003
j
15B52 82B44 82B41 34M30 37K10
random matrices; double scaling limit; orthogonal polynomials; distribution of eigenvalues; phase transition
The authors consider the random matrix ensemble
\[
d\mu_N(M)=Z_N^{-1}\exp\left(-N Tr \,V(M)\right)\,dM
\]
on the space of Hermitian \(N\times N\) matrices \(M\), where \(V\) is a polynomial. In the limit \(N\to\infty\), the distribution of eigenvalues is known to be of the form
\[
d\nu_\infty(x)\propto h(x)\sqrt{\prod_{j=1}^q(x-a_j)(b_j-x)}\;1_{\{x\in \cup_{j=1}^q[a_j,b_j]\}}.
\]
\(d\nu_\infty(x)\) is said to be regular (otherwise singular) if \(h(x)\neq0\) for \(x\in \bigcup_{j=1}^q[a_j,b_j]\). If \(d\nu_\infty(x)\) is singular, the corresponding polynomial \(V(x)\) is said to be critical. In particular, the authors consider the case
\[
d\nu_\infty(x)\propto (x-c)^{2m}\sqrt{4-x^2},\quad m\geq1.
\]
To study the critical behavior in the vicinity of a critical polynomial \(V(x)\), one considers a parametric family \(V(x;t)\) such that \(V(x;t_c)=V(x)\) for some \(t_c\) and studies the limit \(t\to t_c\). As a first result, in the particular case \(m=1\) it is shown in the paper that the free energy has a third order phase transition at \(t_c\). The other problem is to analyze the asymptotics if the recurrence coefficients defining the orthogonal polynomials with respect to the measure \(\exp(-N V(x))\). The relevant limit is the one where \(n\) (the order of the polynomial) and \(N\) diverge with \(n/N\to t/t_c\). The ``double scaling limit'' consists in letting \(n/N\to 1\) with a suitable scaling for \(n-N\). In this regime, the authors formulate a scaling Ansatz for the recurrence coefficients, which is consistent with the known asymptotics for \(t<t_c\). Some consequences of this Ansatz are then discussed.
Fabio L. Toninelli (Lyon)