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...

The effect of wall proximity on the interfacial tension of ultra-thin liquid films of Lennard-Jones (12,6) fluid has been studied using Molecular Dynamics Simulation. The wall’s presence can potentially affect drop-wise condensation and other phase change phenomena such as boiling. Initially the LennardJones atoms are placed near a solid wall with FCC lattice structure in a doubly periodic box....

We develop a systematic multi-local expansion of the Polchinski-Wilson exact renormalization group (ERG) equation. Integrating out explicitly the non local interactions, we reduce the ERG equation obeyed by the full interaction functional to a flow equation for a function, its local part. This is done perturbatively around fixed points, but exactly to any given order in the local part. It is th...