It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ . Such a bound is obtained by comparing the walk on $G_n$ to the walk on d-regular tree $\mathcal{T}_d$ . If one can map another transitive graph $\mathcal{G} $ onto $G_n$ , then we can improve the strategy by using a comparison with the random walk on $\mathcal{G} $ (instead of that of $\mathcal{T} _d$ ), and we obtain a lower bound of the form $\frac {1}{\mathfrak{h} }\log n$ , where $\mathfrak{h} $ is the entropy rate associated with $\mathcal{G} $ . We call this the entropic lower bound.

It was recently proved that in the case $\mathcal{G} =\mathcal{T} _d$ , this entropic lower bound (in that case $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ ) is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibits cutoff at the entropic time. In this article, we provide a generalisation of the result by providing a sufficient condition on the spectra of the random walks on $G_n$ under which the random walk exhibits cutoff at the entropic time. It applies notably to anisotropic random walks on random d-regular graphs and to random walks on random n-lifts of a base graph (including nonreversible walks).

一个简单的事实很容易确定，在具有n个顶点的d正则图 $G_n$ 上的简单随机游走的混合时间从下面渐近地由 $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ 。通过比较 $G_n$ 上的行走与d正则树 $\mathcal{T}_d$ 上的行走来获得这样的界限。如果可以将另一个传递图 $\mathcal{G} $ 映射到 $G_n$ 上，那么我们可以通过与 $\mathcal{G} $ 上的随机游走（而不是 $\mathcal{ T}_d$ )，我们得到$\frac {1}{\mathfrak{h} }\log n$ 形式的下界，其中 $\mathfrak{h} $ 是与 $\mathcal{G} $ 相关的熵率. 我们称之为熵下限。

最近证明，在 $\mathcal{G} =\mathcal{T} _d$ 的情况下，这个熵下界（在那种情况下 $\frac {d }{d-2 } \frac {\log n}{ \log (d-1)}$ ) 当图的光谱半径最小时是锐利的，因此在这种情况下，随机游走在熵时间表现出截止。在本文中，我们通过提供 G_n 上随机游走光谱的充分条件来提供结果的概括，在该条件下，随机游走在熵时间表现出截止。它特别适用于随机d正则图上的各向异性随机游走和基础图的随机n提升上的随机游走（包括不可逆游走）。