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Here is a typical physicist's argument (Bogomolny and Schmidt). You want to study the nodal domains of a random Gaussian wave $F$ (the Fourier transform of the white noise on the unit sphere times the surface measure). Let's say, we are in dimension 2 and want just to know the typical number of nodal lines (components of the set $\{F=0\}$) per unit area. The stationary random function $F$ has only a power decay of correlations. However, we ignore that and model it with a square lattice that has the same length per unit area as $F$ (this is a computable quantity if you use some standard integral geometry tricks). Now, at each intersection of lattice lines, we choose one of the two natural ways to separate them (think of the intersection as of a saddle point with the crossing lines being the level lines at the saddle level). Then, we get a question (still unresolved on the mathematical level, by the way) about a pure percolation type model. Thinking by analogy once more, we get a numerical prediction.