Asymptotic classification of solutions to the first discrete Painlevé equation (Q1357989)

Studies in the theory of matrix models of two-dimensional gravity employ orthogonal polynomials with the weighted function \(w(x)= \exp[-V(x)]\), where \(V(x)\) is a polynomial of even powers. The equations \[ c^{1/2}_n[V'(L)]_{n,n-1}= n,\tag{1} \] where \(L_{nm}= c^{1/2}_m\delta_{n+1,m}+ c^{1/2}_m\delta_{n-1,m}\), \(n\in\mathbb{Z}\), are usually referred to as discrete string equations. The simplest nontrivial string equation corresponds to the case \(V(x)= g_2x^2+ g_4x^4\). For \(g_2=0\), asymptotic formulas for \(c_n\) (as \(n\to\infty\)) were written down by P. Nevai. The main objective of the present article is to study the general case \((g_2\neq 0)\) when (1) has the form \[ c_n+ 4gc_n(c_{n-1}+ c_n+c_{n+1})= \varepsilon n+\nu,\quad n=0,1,2,\dots, \tag{2} \] with \(g\), \(\nu\), and \(\varepsilon>0\) constant real parameters and \(c_n\) an unknown sequence. Also, there is some procedure of passing to a continuous limit which takes (2) into the first Painlevé equation: \(c_n= \rho[1-2ay(x)]\), where \(\varepsilon^{-1}= Ba^{5/2}\), \(\varepsilon n= A(1+\delta a^2x)\), \(B=-72g\), \(A=\rho/2\), \(\delta=2/3\), and \(\rho= (24g)^{-1}\), as \(a\to 0\). Therefore, (2) is usuall referred to as the first discrete Painlevé equation.