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The answer lies in the fact that, in graphene, there is an effective long range interaction mediated by the inverse biharmonic operator (which in 2D goes as $x^2\ln(x)$ and is extremely long-ranged) coupling the gaussian curvature at any two points on the sheet. Due to this, any static ripples or thermally produced dynamic ripples interact at arbitrary ...

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This is not quite a full answer to your homework problem here (our homework policy actually forbids detailed answers), but it should be enough to guide you to your answer. Note first that, formally, the solution to $$cx^2+ca^2-2cx^2=0$$ is $$x=\color{red}{\pm} a$$ This indicates that there are two local minima/maxima points. Inserting these into ...

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Yes, and I think it is even better. This might not be a full proof, but I think it gives a good idea. I will consider $n$ point sources. We know that the potential is harmonic, so there is no maximum and no minimum in the space we are interested in. All critical points, if they exist are saddle points. We are looking for those saddle points, or for points ...

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