Moderate Deviations for the Capacity of the Random Walk range in dimension four

In this paper, we find a natural four dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen and Rosen \cite{BCR} concerning the volume of the random walk range for $d=2$. We find that the deviation statistics of the capacity of the random walk can be related to the following constant of generalized Gagliardo-Nirenberg inequalities, \begin{equation*} \label{eq:maxineq} \inf_{f: |\nabla f|{L^2}<\infty} \frac{|f|^{1/2}{L^2} |\nabla f|^{1/2}{L^2}}{ [\int{($\mathbb{R}$^4)^2} f^2(x) G(x-y) f^2(y) $\text{d}$x $\text{d}$y]^{1/4}}. \end{equation*}