We consider the nonlinear stochastic heat equation in one dimension. Under some conditions on the nonlinearity, we show that the "peaks" of the solution are rare, almost fractal like. We also provide an upper bound on the length of the "islands", the regions of large values. These results are obtained by analyzing the correlation length of the solution.
"Correlation-length bounds, and estimates for intermittent islands in parabolic SPDEs." Electron. J. Probab. 17 1 - 15, 2012. https://doi.org/10.1214/EJP.v17-2429