The Square of the Dirichlet-to-Neumann map equals minus Laplacian
  The Dirichlet-to-Neumann maps connect boundary values of harmonic functions. It is an amazing fact that the square of the non-local Dirichlet-to-Neumann map for the uniform conductivity 1 on the unit disc equals minus the local(!) Laplace operator on the boundary circle. To establish a new connection between discrete and continuous Dirichlet-to-Neumann maps and for the approximations I construct a finite and an infinite graphs which Dirichlet-to-Neumann map have the same property: \Lambda^2(1) = - d^2/d \theta^2. The construction gives a new continued fraction identity. It is interesting to consider the geometric and probabilistic (trajectories of the random walk) consequences of this localizing identity unifying discrete and continuous equations for potentials. 
David V. Ingerman