On the spectral radius of the non-backtracking matrix of the configuration model

Summary · qwen2.5:32b

The study proves a concentration result for the leading eigenvalue of the non-backtracking matrix in the configuration model with uniformly bounded degrees, showing it converges to \(\frac{\mathbb{E}[P(P-1)]}{\mathbb{E}[P]}\) as vertices increase. This finding is significant as it relates to the mean offspring number in a branching process and can be applied to analyze subgroup growth rates in free groups.

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Learn about the spectral radius of the non-backtracking matrix in the context of configuration models and its implications for real-time rendering and free group subgroups.

Excerpt

arXiv:2404.07321v2 Announce Type: replace Abstract: We prove a concentration result for the leading eigenvalue of the non--backtracking matrix of the configuration model under the assumption of uniformly bounded degrees. Let $P$ denote the limiting degree distribution. Assuming polynomial approximation, we show that as the number of vertices tends to infinity, the leading eigenvalue of the non--backtracking matrix concentrates around \[ \frac{\mathbb{E}[P(P-1)]}{\mathbb{E}[P]}. \] This quantity corresponds to the mean offspring number of the excess--degree branching process associated with the local limit of the configuration model. As a byproduct of our work we explain how this result can be applied to prove the density of the growth rates of the subgroups of the free group.