# Random walk to $\phi^4$ and back

## TL;DR

This paper establishes a precise mathematical relationship between the asymptotic distributions of the multiple point range of planar random walks and the proper functions of planar complex $^4$-theory, revealing deep connections between probability and quantum field theory.

## Contribution

It introduces integral transforms linking random walk distributions to $^4$-theory functions, extending them onto Riemann surfaces and analyzing their asymptotic behavior.

## Key findings

- Derived the asymptotic edge behavior of the distributions as Gamma distributions.
- Established the integral transform relationship between probability distributions and quantum field theory functions.
- Connected perturbation series coefficients to asymptotic series coefficients of the proper functions.

## Abstract

In this paper we establish an exact relationship between the asymptotic probability distributions $\nu_0$ and $\nu_2$ of the multiple point range of the planar random walk and the proper functions $\Gamma^{[0]}$ and $\Gamma^{[2]}$ respectively of the planar, complex $\phi^4$-theory, setting the number of components $m=0$: The characteristic functions $\Phi_0$ and $\Phi_2$ of $\nu_0$ and $\nu_2$ have simple integral transforms $\zeta^{[0]}$ and $\zeta^{[2]}$ respectively which turn out to be the extensions of the proper functions $\Gamma^{[0]}$ and $\Gamma^{[2]}$ onto a Riemann surface (with infinitely many sheets) in the coupling constant $g$ and are well defined mathematically. $\zeta^{[0]}$ and $\zeta^{[2]}$ restricted to a specific sheet have a (sectorwise) uniform asymptotic expansion in $g=0$. The standard perturbation series of $\Gamma^{[0]}$ and $\Gamma^{[2]}$ in $g$ have expansion coefficients $\Gamma^{[0],pt}_r$ and $\Gamma^{[2],pt}_r$ which are polynomials in $m$. Order by order the lowest nontrivial polynomial coefficient in $m$: $\Gamma^{[0],pt}_{r,1} = \zeta^{[0]}_{r}$ and $\Gamma^{[2],pt}_{r,0} = \zeta^{[2]}_{r}$ where $\zeta^{[0]}_{r}$ and $\zeta^{[2]}_{r}$ are the coefficients of the asymptotic series of $\zeta^{[0]}$ and $\zeta^{[2]}$ around $g=0$ respectively. $\Phi_0$ and $\Phi_2$ turn out to be modified Borel type summations of those series. \\ As an application we derive the rising edge behaviour of $\nu_0$ and $\nu_2$ from the large order estimates of Lipatov \citep{lipatov}. It turns out to be of the form of a Gamma distribution with parameters known numerically.