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Updated: Jan 1, 2026

Induction of Microstreaming by Nonspherical Bubble Oscillations in an Acoustic Levitation System
Published on: May 9, 2021
Percolation Is Odd
Stephan Mertens1,2, Cristopher Moore2
1Institut für Physik, Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany.
Abstract:
We prove a remarkable combinatorial symmetry in the number of spanning configurations in site percolation: for a large class of lattices, the number of spanning configurations with an odd or even number of occupied sites differs by ±1. In particular, this symmetry implies that the total number of spanning configurations is always odd, independent of the size or shape of the lattice. The class of lattices that share this symmetry includes the square lattice and the hypercubic lattice in any dimension, with a wide variety of boundary conditions.
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