# Exact null octagon

## TL;DR

This paper derives an exact, closed-form expression for a specific four-point correlation function in planar N=4 SYM theory in the light-like limit, using a Fredholm determinant representation and differential equations.

## Contribution

It introduces a novel exact formula for the null octagon as a Fredholm determinant, connecting it to integrable models and random matrix theory.

## Key findings

- Null octagon expressed as a Fredholm determinant.
- Exact computation via differential equations.
- Connection to integrable models and random matrices.

## Abstract

We consider the so-called simplest correlation function of four infinitely heavy half-BPS operators in planar N=4 SYM in the limit when the operators are light-like separated in a sequential manner. We find a closed-form expression for the correlation function in this limit as a function of the 't Hooft coupling and residual cross ratios. Our analysis heavily relies on the factorization of the correlation function into the product of null octagons and on the recently established determinant representation for the latter. We show that the null octagon is given by a Fredholm determinant of a certain integral operator which has a striking similarity to those previously encountered in the study of two-point correlation functions in exactly solvable models at finite temperature and of level spacing distributions for random matrices. This allows us to compute the null octagon exactly by employing a method of differential equations.