We will discuss results relating the behavior of random walks on planar graphs and the relation to the geometric properties of a circle packing of the graph. For example, if the graph has bounded degrees, the simple random walk is recurrent if and only if the boundary of the circle packing embedding is a polar set (that is, Brownian motion misses it almost surely). Another result is that when a discrete harmonic function on a circle packing of a graph approximates the continuous harmonic function with the same boundary conditions when all the circles are small.
No knowledge beyond basic probability will be assumed.