Extensions of Rayleigh–Bloch waves above the cutoff frequency are studied via discrete spectrum a transfer operator for channel containing single cylinder with quasi-periodic side-wall conditions. Above cutoff, wavenumber becomes complex valued and an additional appears. For small- to intermediate-radius values, extended shown connect Neumann Dirichlet trapped modes before embedding in continuo...

We show that on every Ramanujan graph G, the simple random walk exhibits cutoff: when G has n vertices and degree d, the total-variation distance of the walk from the uniform distribution at time t = d d−2 logd−1 n + s √ logn is asymptotically P(Z > c s) where Z is a standard normal variable and c = c(d) is an explicit constant. Furthermore, for all 1 ≤ p ≤ ∞, d-regular Ramanujan graphs minimiz...

In the Gilbert–Shannon–Reeds shuffle, a deck of N cards is cut into two approximately equal parts which are riffled together uniformly at random. Bayer and Diaconis (Ann. Appl. Probab. 2 294–313) famously showed that this Markov chain undergoes cutoff in total variation after 3log(N) 2log(2) shuffles. We establish for more general asymmetric riffle shuffles one cuts differently sized parts. The...