A note on mixing times of planar random walks

Abstract

We present an infinite family of finite planar graphs $\{X_n\}$ with degree at most five and such that for some constant $c>0$, $$ λ_1(X_n) \geq c(\frac{\log \diam(X_n)}{\diam(X_n)})^2\,, $$ where ${\lambda }_1$ denotes the smallest non-zero eigenvalue of the graph Laplacian. This significantly simplifies a construction of Louder and Souto. We also remark that such a lower bound cannot hold when the diameter is replaced by the average squared distance: There exists a constant $c>0$ such that for any family $\{X_n\}$ of planar graphs we have $${\lambda }_1(X_n)\le c(\frac{1}{|X_n{|}^2}\sum _{x,y\in X_n}d(x,y{)}^2{)}^{-1},$$ where $d$ denotes the path metric on $X_n$.

References in corpus (1)

• Diameter and spectral gap for planar graphs