Random walk to $ϕ^4$ and back
In this paper we establish an exact relationship between the asymptotic probability distributions $ν_0$ and $ν_2$ of the multiple point range of the planar random walk and the proper functions $Γ^{[0]}$ and $Γ^{[2]}$ respectively of the planar, complex $ϕ^4$-theory, setting the number of components $m=0$: The characteristic functions $Φ_0$ and $Φ_2$ of $ν_0$ and $ν_2$ have simple integral transforms $ζ^{[0]}$ and $ζ^{[2]}$ respectively which turn out to be the extensions of the proper functions $Γ^{[0]}$ and $Γ^{[2]}$ onto a Riemann surface (with infinitely many sheets) in the coupling constant $g$ and are well defined mathematically. $ζ^{[0]}$ and $ζ^{[2]}$ restricted to a specific sheet have a (sectorwise) uniform asymptotic expansion in $g=0$. The standard perturbation series of $Γ^{[0]}$ and $Γ^{[2]}$ in $g$ have expansion coefficients $Γ^{[0],pt}_r$ and $Γ^{[2],pt}_r$ which are polynomials in $m$. Order by order the lowest nontrivial polynomial coefficient in $m$: $Γ^{[0],pt}_{r,1} = ζ^{[0]}_{r}$ and $Γ^{[2],pt}_{r,0} = ζ^{[2]}_{r}$ where $ζ^{[0]}_{r}$ and $ζ^{[2]}_{r}$ are the coefficients of the asymptotic series of $ζ^{[0]}$ and $ζ^{[2]}$ around $g=0$ respectively. $Φ_0$ and $Φ_2$ turn out to be modified Borel type summations of those series. \\ As an application we derive the rising edge behaviour of $ν_0$ and $ν_2$ 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.