Random walks on Cayley graphs of complex reflection groups

Asymptotic properties of random walks on minimal Cayley graphs of complex reflection groups are investigated. The main result of the paper is theorem on fast mixing for random walks on Cayley graphs of complex reflection groups. Particularly, bounds of diameters and isoperimetric constants, a known result on fast fixing property for expander graphs play a crucial role to obtain the main result. A constructive way to prove a special case of Babai’s conjecture on logarithmic order of diameters for complex reflection groups is proposed. Basing on estimates of diameters and Cheeger inequality, there is obtained a non-trivial lower bound for spectral gaps of minimal Cayley graphs on complex reflection groups.

CUTOFF AT THE ENTROPIC TIME FOR RANDOM WALKS ON COVERED EXPANDER GRAPHS

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).

Ramanujan Graphs for Post-Quantum Cryptography

We introduce a cryptographic hash function based on expander graphs, suggested by Charles et al. ’09, as one prominent candidate in post-quantum cryptography. We propose a generalized version of explicit constructions of Ramanujan graphs, which are seen as an optimal structure of expander graphs in a spectral sense, from the previous works of Lubotzky, Phillips, Sarnak ’88 and Chiu ’92. We also describe the relationship between the security of Cayley hash functions and word problems for group theory. We also give a brief comparison of LPS-type graphs and Pizer’s graphs to draw attention to the underlying hard problems in cryptography.