# Probability to return to the origin for a uniform random walk

Consider a uniform random walk on $$\mathbb{R}$$, with stepsize chosen uniformly from the interval $$(-1,1)$$. The random walk start at $$x=0$$. Denote by $$\rho_p dx$$ the probability that the random walk returns to the interval $$(0,dx)$$ after $$p$$ steps, for infinitesimally small $$dx$$. For $$p=1$$ and for $$p=2$$ one has $$\rho_p=1/2$$, for $$p=3$$ one has $$\rho_p=3/8$$, for large $$p$$ the diffusion approximation gives $$\rho_p=(2\pi p/3)^{-1/2}$$.

Q: Is $$\rho_p$$ known exactly for any $$p$$?

Motivation: this would give the small-$$a$$ limit of the box integral in https://mathoverflow.net/a/331993/11260 , $$\lim_{a\rightarrow 0}a^{1-p}f_p(a)=2^p\rho_p$$, see also fedja's comment in https://math.stackexchange.com/q/3233160/87355

Looking at fedja's comment that you mentioned, we can formalize your description as follows: $$\rho_p=g_p(0),$$ where $$g_p$$ is the pdf of $$T_p:=\sum_{i=1}^p V_i=2S_p-p,$$ $$S_p:=\sum_{i=1}^p U_i$$, the $$U_i$$'s are i.i.d. random variables (r.v.'s) uniformly distributed on $$[0,1]$$, and $$V_i:=2U_i-1$$, so that the $$V_i$$'s are i.i.d. r.v.'s uniformly distributed on $$[-1,1]$$.
Note that $$g_p(t)=\frac12\,f_p(\frac{p+t}2)$$ for real $$t$$, where $$f_p$$ is the pdf of $$S_p$$. So, by the Irwin–Hall formula, $$\rho_p=g_p(0)=\frac12\,f_p\Big(\frac p2\Big) =\frac1{2(p-1)!}\sum_{k=0}^{\lfloor p/2\rfloor}(-1)^{k-1}\binom pk\Big(\frac p2-k\Big)^{p-1}.$$
In particular, the respective values of $$\rho_p$$ for $$p=1,2,3,4,5$$ are $$\frac{1}{2},\frac{1}{2},\frac{3}{8},\frac{1}{3},\frac{115}{384}$$.