We discuss the distribution of the largest eigenvalue of a random Hermitian matrix. Utilising results from the quantum gravity and string theory literature it is seen that the orthogonal polynomials approach, first introduced by Majumdar and Nadal, can be extended to calculate both the left and right tail large deviations of the maximum eigenvalue. This framework does not only provide computational advantages when considering the left and right tail large deviations for general potentials, as is done explicitly for the first multi-critical potential, but it also offers an interesting interpretation of the results. In particular, it is seen that the left tail large deviations follow from a standard perturbative large expansion of the free energy, while the right tail large deviations are related to the non-perturbative expansion and thus to instanton corrections. Considering the standard interpretation of instantons as tunnelling of eigenvalues, we see that the right tail rate function can be identified with the instanton action which in turn can be given as a simple expression in terms of the spectral curve. From the string theory point of view these non-perturbative corrections correspond to branes and can be identified with FZZT branes.

Matrix Models Integrable Equations in Physics Solitons Monopoles and Instantons 2D Gravity

Article funded by SCOAP3

Introduction

a \end{array} \right. \label{scalingleftright} \eea where the functions $\psi_\pm$, known as the left and right rate functions, are given by, \bea \psi_{-}(z)&=&\frac{z^2}{3}-\frac{z^4}{108}-\sqrt{z^2+6} \frac{\left(z^3+15 z\right)}{108}- \frac{1}{2}\ln\left[\frac{\sqrt{z^2+6}+z}{\sqrt{2}}\right]+\frac{\ln 3}{2},\label{psi-lit}\\ \psi_{+}(z)&=&\frac{z \sqrt{z^2-2}}{2} +\ln\left[\frac{z-\sqrt{z^2-2}}{\sqrt{2}}\right]. \eea An obvious question is; can expressions for the left and right tail be obtained for arbitrary potentials and hence for the multi-critical cases? It is the purpose of this paper to address this issue within the context of the orthogonal polynomial approach begun in~\cite{SN}. We note here that the topological recursion approach of~\cite{LargeDevTR1,LargeDevTR2} is in principle powerful enough to answer this question; however only explicit results were given for the Gaussian case in these studies. We will find in the course of this paper that the orthogonal polynomial approach can be related to the instanton effects studied in~\cite{MarinoSaddle,MarinoPolynomials,MarinoMulticut,MarinoReview} and via this we make contact with some results of~\cite{LargeDevTR2}. Another perspective on the work here is that there are three well developed approaches to computing in random matrix theory; saddle point methods, loop equations and orthogonal polynomials. In the context of large deviations both the saddle point method and the loop equation method have been fully developed, however the orthogonal polynomial method has only been considered in~\cite{SN}. It is an aim of this paper to fully develop this approach. The plan of this paper is as follows: in section~\ref{Sec2} we review the orthogonal polynomial approach to gap probabilities which are given by the free energy in the presence of a hard wall. We describe the string equations in absence and presence of this hard wall. In section~\ref{Sec3} we describe how to use this approach to obtain the large deviations of the maximum eigenvalue. It is seen that the large deviations in the left tail can be obtained from the perturbative expansion of the free energy, while for the large deviations in the right tail non-perturbative effects and thus instantons effects are relevant. We apply this machinery both to the case of the Gaussian matrix model as well as to a case of a multi-critical potential. Having made the link between large deviations in the right tail and instantons effects we proceed in section~\ref{Sec5} by giving an interpretation in terms of eigenvalue tunnelling. Calculating the eigenvalue tunnelling using a saddle point method, enables one to obtain an expression for the right tail in terms of the spectral curve for an arbitrary potential. Furthermore, this provides interesting links to a string theoretic picture. We summarise our results and conclude in section~\ref{Sec6}. Appendix~\ref{app1} provides some useful results regarding properties of the family of multi-critical potentials considered in section~\ref{Sec3}. ]]>

The orthogonal polynomial approach to gap probabilities

Orthogonal polynomials for matrix models in presence of a hard wall

1$, allowing us to conclude that$Q$can be written as, \beq Q_{nm} = \sqrt{q_{n+1}}\delta_{n+1,m} + s_n\delta_{nm} + \sqrt{q_{n}} \delta_{n-1,m} \eeq where$q_n$and$s_n$are some yet to be determined functions$z$,$t$, and the potential. In fact by considering the quantity$\braket{\pi_n}{x \pi_{n-1}}$and$\braket{\pi_n}{x \pi_{n}}$one can show$q_n = r_n$and$s_n = -\frac{g}{N} \partial_{t_1} \log h_n$respectively. In applications of random matrix theory to string theory, in which recursion relations for$r_n$were first obtained, these recursion relations go by the name of string equations. In~\cite{SN} recursion relations for$r_n$were obtained in the case of a Gaussian potential. Unfortunately they had the unpleasant property of containing derivatives of$z$which makes them difficult to use for potentials of arbitrary order. One of the advances made in~\cite{HigherTW} was to obtain a set of purely algebraic recursion relations for any potential, which is what we will refer to as the string equations. Before reviewing the form of these string equations we will first consider the case of no hard-wall, i.e.\ when$z = \infty$. ]]> String equations in absence of a hard wall String equations in presence of a hard wall Large deviations from string equations and instantons Perturbative and non-perturbative expansion of the string equations 0$ leads to an expansion of the form, \beq \left[\frac{R^{(l)}(t;z)}{R^{(0)}(t;z)}\right]_c = e^{-l A(t)/g_s} \sum^\infty_{k=0} g_s^{k} c_{l,k+1}(z;t), \eeq and it can be shown~\cite{MarinoPolynomials} that~\eqref{Fleqn} can be solved order by order with the result, \beq \label{Fl1eqn} F^{(l)}_{1}(z;t) = \frac{1}{4} c_{l,1} \mathrm{cosech}^2 \left( l A'(t)/2 \right). \eeq The solution for higher order $F^{(l)}_{k}(z;t)$ can be found in~\cite{MarinoPolynomials}. We now have all the technical tools at hand to tackle the computation of the large deviations in the left and right tail in a number of examples. ]]>

Large deviations in the case of Gaussian potential

Large deviations in the left tail

Large deviations in the right tail

Large deviations in the case of a multi-critical potential

Large deviations in the left tail

Large deviations in the right tail

Eigenvalue tunnelling and the spectral curve

Eigenvalue tunnelling and the qualitative behaviour of \texorpdfstring{$\BB{P}_N(z ;t)$}{P(N)(z;t)}

V(x_p)$where$x_p$is the position of the next deepest minima to the right of the support. When$V(z) > V(x_p)$the large deviation probability is nearly independent of$z$until$z$reaches the saddle point at$x_p$. One can think of this as being due to the fact that the most likely place to find an eigenvalue far from the bulk is in the next deepest minimum. Hence the cumulative probability does not increase until we have included this minima in our integration. We now turn our attention to the multi-critical potentials~\eqref{MCV}. These potential have only a single turning point on the real axis and therefore when no hard wall is present, the contour$\gamma$in~\eqref{Zgamma} coincides with the real axis, going from$-\infty$to$\infty$. When a hard wall is present to the right of the cut, then$\gamma$consists of the previous contour going from$-\infty$to$+\infty$along the real axis followed by a contour going from$+\infty$back along the real axis to$z\$; see figure~\ref{fig4}. \begin{figure}[t] \centering \includegraphics[scale=0.9]{Saddle-m.pdf} ]]>

The rate function and the spectral curve