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More on the exact solution of the O(n) model on a random lattice and an investigation of the case |n|>2

B. Eynard, C. Kristjansen

TL;DR

This work completes the exact solution of the $O(n)$ model on a random lattice by obtaining the auxiliary function $G^{(0)}(p)$ in closed form as a combination of theta-functions, enabling exact, nonperturbative results for the eigenvalue distribution and observables. The authors apply the solution to the Gaussian version to derive a closed-form eigenvalue density for general $n$, recovering the Wigner law at $n=0$ and revealing a universal structure for the string susceptibility across $n$. They extend the analysis to $|n|>2$, showing the model remains well-defined in a coupling region, with new critical points at which $\gamma_{str}=+\tfrac{1}{2}$ for $n>2$ and no new critical points for $n<-2$. The work also highlights potential connections to 2D quantum gravity with $c>1$ and to integrable hierarchies, suggesting directions for future exploration of the continuum limit and higher-genus extensions.

Abstract

For $n\in [-2,2]$ the $O(n)$ model on a random lattice has critical points to which a scaling behaviour characteristic of 2D gravity interacting with conformal matter fields with $c\in [-\infty,1]$ can be associated. Previously we have written down an exact solution of this model valid at any point in the coupling constant space and for any $n$. The solution was parametrized in terms of an auxiliary function. Here we determine the auxiliary function explicitly as a combination of $θ$-functions, thereby completing the solution of the model. Using our solution we investigate, for the simplest version of the model, hitherto unexplored regions of the parameter space. For example we determine in a closed form the eigenvalue density without any assumption of being close to or at a critical point. This gives a generalization of the Wigner semi-circle law to $n\neq 0$. We also study the model for $|n|>2$. Both for $n<-2$ and $n>2$ we find that the model is well defined in a certain region of the coupling constant space. For $n<-2$ we find no new critical points while for $n>2$ we find new critical points at which the string susceptibility exponent $γ_{str}$ takes the value $+\frac{1}{2}$.

More on the exact solution of the O(n) model on a random lattice and an investigation of the case |n|>2

TL;DR

This work completes the exact solution of the model on a random lattice by obtaining the auxiliary function in closed form as a combination of theta-functions, enabling exact, nonperturbative results for the eigenvalue distribution and observables. The authors apply the solution to the Gaussian version to derive a closed-form eigenvalue density for general , recovering the Wigner law at and revealing a universal structure for the string susceptibility across . They extend the analysis to , showing the model remains well-defined in a coupling region, with new critical points at which for and no new critical points for . The work also highlights potential connections to 2D quantum gravity with and to integrable hierarchies, suggesting directions for future exploration of the continuum limit and higher-genus extensions.

Abstract

For the model on a random lattice has critical points to which a scaling behaviour characteristic of 2D gravity interacting with conformal matter fields with can be associated. Previously we have written down an exact solution of this model valid at any point in the coupling constant space and for any . The solution was parametrized in terms of an auxiliary function. Here we determine the auxiliary function explicitly as a combination of -functions, thereby completing the solution of the model. Using our solution we investigate, for the simplest version of the model, hitherto unexplored regions of the parameter space. For example we determine in a closed form the eigenvalue density without any assumption of being close to or at a critical point. This gives a generalization of the Wigner semi-circle law to . We also study the model for . Both for and we find that the model is well defined in a certain region of the coupling constant space. For we find no new critical points while for we find new critical points at which the string susceptibility exponent takes the value .

Paper Structure

This paper contains 16 sections, 120 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Model
  3. Determination of the function $G^{(0)}(p)$
  4. Reformulation of the problem
  5. The explicit expression for $G^{(0)}_+(p)$
  6. Rational case
  7. Explicit solution of a given model
  8. The string susceptibility
  9. The gaussian potential
  10. n $\in ]-2,2[$
  11. $n=\pm 2$
  12. $n=-2$
  13. $n=2$
  14. $n>2$
  15. $n<-2$
  16. ...and 1 more sections

Figures (3)

  • Figure 1: The variation of $a$ (lower curves) and $b$ (upper curves) as a function of $T$ for $\nu=1/10$ (full line), $\nu=1/2$ (dashed line) and $\nu=2/3$ (dotted line).
  • Figure 2: The parametric curves $(T(k),a(k))$ and $(T(k),b(k))$ for $k\in ]0,1]$ and $\bar{\nu}=1.2$.
  • Figure 3: The parametric curves $(T(k),b(k))$ and $(T(k),a(k))$ for $k\in [0.02;0.38]$ and $\bar{\nu}=1.2$.