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's results concerning the volume of the random walk range for dimension 2. We find that the deviation statistics of the capacity of the random walk can be related to the optimal constant of the generalized Gagliardo–Nirenberg inequalities.

This is based on joint work with I. Okada.