We study protein diffusion in multicomponent lipid membranes close to a rigid
substrate separated by a layer of viscous fluid. The large-distance, long-time
asymptotics for Brownian motion are calculated using a nonlinear stochastic
Navier-Stokes equation including the effect of friction with the substrate. The
advective nonlinearity, neglected in previous treatments, gives only a small
correction to the renormalized viscosity and diffusion coefficient at room
temperature. We find, however, that in realistic multicomponent lipid mixtures,
close to a critical point for phase separation, protein diffusion acquires a
strong power-law dependence on temperature and the distance to the substrate
H, making it much more sensitive to cell environment, unlike the logarithmic
dependence on H and very small thermal correction away from the critical
point.Comment: 19 pages, 4 figure