There will certainly exist a Weyl law for random planar graphs. However, the nature of the law will depend very sensitively on exactly which model of random graphs one takes. One can start to see this by considering the first two moments of the eigenvalues of the graph Laplacian. The first moment is the average degree of a vertex, while the second moment is the average of $d(d+1)$, for $d$ the degree. This means that from the Weyl law we can determine the mean and variance of the degree distribution. For a model of random large planar graphs the mean and variance of the degree will typically converge to some constants as the size of the graph goes to $\infty$, but the constants depend on the model. More generally the $k$th moment of the eigenvalue distribution of the graph Laplacian is the trace of the $k$th power of the graph Laplacian which can be expressed as a sum over loops of length $\leq k$ in the graph. For each vertex $v$ of a random graph, the sum over loops of length $\leq k$ starting and ending at $v$ is an integer-valued random variable. In a reasonable model of large random planar graphs we would expect this random variable to converge to a limiting distribution as the size of the graph goes to $\infty$, and we would expect variables associated to points a distance significantly more than $k$ from each other to be approximately independent, so a law of large numbers implies that the $k$th moment of the eigenvalue distribution converges almost surely to the expectation of the limiting distribution. But this variable depends finely on the local structure of the graph, and so its distribution depends on the random model. I don't think there is any kind of universal distribution that applies to an arbitrary planar graph. ----- > In her applications, the low frequencies correspond to large-scale spatial patterns, whereas the higher eigenfunctions tend to be localized (i.e. they are very small except at a few vertices) and oscillate rapidly. The fact that low frequencies correspond to large-scale spatial patterns should be possible to verify rigorously. If a function $f$ is an eigenvector of the Laplacian $L$ with eigenvalue $\lambda$ small then $|| L f||_2 = || \lambda f ||_2 = \lambda ||f||_2$ is small. But $||L f||_2^2 $ is just the sum over edges connecting vertices $v_1,v_2$ of $(f(v_1)-f(v_2))^2$. So $f(v_1)-f(v_2)$ must be small for most edges, meaning the function $f$ does not change rapidly. The fact that high frequencies oscillate rapidly is also generic, applying to almost any graph, and can be established rigorously, by a reverse argument: For $f$ an eigenvector with large eigenvalue (compared to the average degree of vertices where $f$ is large) the differences $f(v_1)-f(v_2)$ must be large for many pairs of $v_1,v_2$ connected by an edge, which is only possible if $f$ oscillates rapidly. The last fact, that higher eigenfunctions are localized, seems most interesting. I don't think this is not true for generic graphs or even for typical planar graphs, e.g. I am pretty sure this is not true for the grid graph. This must reflect a special feature of your colleague's graphs. One possible explanation is that her graphs contain clusters with a large number of edges within the cluster and a small number of edges connecting the cluster outside - I mean with a larger discrepancy between these two figures than you would get in a uniform planar graph. In such a graph I think you might see eigenfunctions localized on the clusters.