  Graph-limit theory focuses on the convergence of sequences of graphs when the
number of nodes becomes arbitrarily large. This framework defines a continuous
version of graphs allowing for the study of dynamical systems on very large
graphs, where classical methods would become computationally intractable.
Through an approximation procedure, the standard system of coupled ordinary
differential equations is replaced by a nonlocal evolution equation on the unit
interval. In this work, we adopt this methodology to explore the continuum
limit of random walks, a popular model for diffusion on graphs. We focus on two
classes of processes on dense weighted graph, in discrete and in continuous
time, whose dynamics are encoded in the transition matrix and the random-walk
Laplacian. We also show that previous works on the discrete heat equation,
associated to the combinatorial Laplacian, fall within the scope of our
approach. Finally, we apply the spectral theory of operators to characterize
the relaxation time of the process in the continuum limit.
