What does this research mean for the field?

The deviation statistics for the capacity of a four-dimensional random walk range are directly related to a specific constant derived from generalized Gagliardo-Nirenberg inequalities. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

This paper aims to extend moderate deviation results for the capacity of random walks to four dimensions.

July 3, 2026The Annals of Probability1 citationsOpen Access

Moderate deviations for the capacity of the random walk range in dimension four

AAArka AdhikariTokyo Metropolitan Komaba High School IOIzumi OkadaChiba University

Key Points

This paper aims to extend moderate deviation results for the capacity of random walks to four dimensions.
Developed a four-dimensional analog of existing moderate deviation results for random walks.
Analyzed the relationship between deviation statistics and Gagliardo–Nirenberg inequalities.
Built on previous work by Bass, Chen, and Rosen regarding two-dimensional cases.
Identified a new relationship between the capacity of random walk and generalized inequalities.
Presented a formula linking deviations to constants within Gagliardo–Nirenberg inequalities.

Abstract

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 (Mem. Amer. Math. Soc. 198 (2009) viii+82) 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: inff:‖∇f‖L2<∞ ‖f‖L21/2‖∇f‖L21/2 ∫(R4)2f2(x)G(x−y)f2(y)dxdy1/4.

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