PreprintarXiv licencearXiv
Linear Cotype Bounds for Nonlinear Spectral Gaps of Scalar Expanders
Abstract
Let G be a finite simple d-regular graph, d≥ 3, whose second largest adjacency eigenvalue satisfies λ2(G)≤ (1−δ)d, where 0<δ≤ 1. We prove that the Poincaré constant satisfies γ(G,|·|X)≲ qCK/δ whenever X is a real Banach space with a K-unconditional Schauder basis and cotype q with constant C. This constant compares the mean distance over all vertex pairs with the mean distance across edges. For real Banach lattices the bound is O(qC/δ). The proof estimates where a random walk first leaves each coordinate level set. As a consequence of the bound, if n-vertex graphs in a fixed-degree expander family embed with bounded distortion into spaces with bounded C,K, then q must grow at least as fast as log n. This order is attained in ℓqn.
Related
- Sharp Metric Cotype Inequalities for L1 via Nonlinear Cut Smoothing
- Quantum expanders and dimension-free commutator bounds
- Optimal Spectral Lower Bounds and Nonradial Nonlinear Asymptotic Stability of a Family of Three-Dimensional Keller--Segel Self-Similar Blow-Up Solutions
- A Counterexample to Talagrand's Operator Cotype Problem
- Quotients of L1 by subsequences of the Haar system have cotype 2
BibTeX
@misc{polavarapu2026linear,
title = {{Linear Cotype Bounds for Nonlinear Spectral Gaps of Scalar Expanders}},
author = {Achintya Raya Polavarapu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05649},
eprint = {2611.05649},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05649},
}
RIS
TY - UNPB
TI - Linear Cotype Bounds for Nonlinear Spectral Gaps of Scalar Expanders
AU - Achintya Raya Polavarapu
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05649
UR - https://arxiv.org/abs/2611.05649
AB - Let G be a finite simple d-regular graph, d≥ 3, whose second largest adjacency eigenvalue satisfies λ_2(G)≤ (1−δ)d, where 0<δ≤ 1. We prove that the Poincaré constant satisfies γ(G,|·|_X)≲ qCK/δ whenever X is a real Banach space with a K-unconditional Schauder basis and cotype q with constant C. This constant compares the mean distance over all vertex pairs with the mean distance across edges. For real Banach lattices the bound is O(qC/δ). The proof estimates where a random walk first leaves each coordinate level set. As a consequence of the bound, if n-vertex graphs in a fixed-degree expander family embed with bounded distortion into spaces with bounded C,K, then q must grow at least as fast as log n. This order is attained in ℓ_q^n.
ER -