Stabilization of Percolation Probability in Supercritical Regimes: A Measure‐Theoretic Approach

Abstract

Richard Murdoch Mongomery

In this paper, we present a rigorous proof demonstrating the stabilization of percolation probability in supercritical regimes for percolation models. We analyze a sequence of expanding compact balls in Rn , incorporating correction terms that vanish asymptotically, and show that the percolation probability within these regions converges to a finite, non�?�zero value as the balls expand to cover the entire space. Our approach combines key concepts from percolation theory with measure�?�theoretic tools, such as the Monotone Convergence Theorem and Fatouʹs Lemma, to rigorously establish the existence and uniqueness of the limiting percolation probability. The result extends classical results on percolation probability in lattice models and provides a new framework for understanding convergence in infinite systems. The non�?�triviality of the limit is demonstrated in the supercritical regime, where percolation occurs with positive probability. The framework introduced here could serve as a bridge between discrete and continuous models in statistical physics.

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