turkmath.org

The Laplace operator is perhaps the most central object in physics - from the harmonic oscillator (the basis for "zeroth order" modelling at least half of physical phenomena) to diffusion (the other half!). Many phenomena of physical, biological or social interest take place on discrete lattices (networks) which are not embedded in metric spaces, and the graph Laplacian emerges as the appropriate operator here. I will illustrate this with a simple example from Statistical Physics: We normally expand fluctuations of some field in terms of the eigenfunctions of the Laplace operator (the harmonic functions on d-dimensional Euclidean spaces or lattices), and can systematically eliminate the "smaller wavelength fluctuations in order to probe certain properties of the physical system. The challenge is to be able to do a similar operation on arbitrary networks. However, obtaining the eigenvectors of the Laplace operator on even perfectly regular networks such as Cayley trees of arbitrarily large lattice size can be daunting. Some attempts in this direction and many open questions will be discussed.