Large Deviation Principle of Multidimensional Multiple Averages on $\mathbb{N}^d$

TL;DR

This paper proves the large deviation principle for multidimensional multiple averages on 5, extending previous work to higher dimensions and including boundary conditions, with explicit energy function formulas.

Contribution

It extends the large deviation principle for multiple averages to multidimensional lattices 5 and incorporates boundary conditions, broadening the scope of prior results.

Findings

Established LDP for 5 multiple averages

Derived explicit energy function formulas with boundary conditions

Extended techniques to weighted multiple averages

Abstract

This paper establishs the large deviation principle (LDP) for multiple averages on $\mathbb{N}^d$. We extend the previous work of [Carinci et al., Indag. Math. 2012] to multidimensional lattice $\mathbb{N}^d$ for $d\geq 2$. The same technique is also applicable to the weighted multiple average launched by Fan [Fan, Adv. Math. 2021]. Finally, the boundary conditions are imposed to the multiple sum and explicit formulae of the energy functions with respect to the boundary conditions are obtained.