Consistency on cubic lattices for determinants of arbitrary orders

TL;DR

This paper introduces a new consistency condition for discrete equations on cubic lattices, focusing on determinants of field values on NxN squares, and proves this consistency for a class of such equations.

Contribution

It proposes a novel type of consistency condition on cubic lattices for discrete equations based on determinant vanishing on NxN squares, extending previous concepts.

Findings

Established consistency for arbitrary N on cubic lattices

Connected consistency conditions to bending elementary NxN squares

Proved the main theorem for determinant-based discrete equations

Abstract

We consider a special class of two-dimensional discrete equations defined by relations on elementary NxN squares, N>2, of the square lattice Z^2, and propose a new type of consistency conditions on cubic lattices for such discrete equations that is connected to bending elementary NxN squares, N>2, in the cubic lattice Z^3. For an arbitrary N we prove such consistency on cubic lattices for two-dimensional discrete equations defined by the condition that the determinants of values of the field at the points of the square lattice Z^2 that are contained in elementary NxN squares vanish.