Slide show of the cluster expansions for KPZ equation

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Time flows backwards (                 )  from left to right. Generally speaking, colors indicate active zones being expanded. 

n-valent vertices (indicated by circles) produce n gradient (∇) fields out of a single one, with n≥2 (in all examples shown, n=2). Boxes have a priori open boundaries (dotted lines). Fat calligraphic lines                         separate the scales.

 

Dirichlet cluster expansion. Blue circles               are placed at every end of a link. Each Dirichlet derivative produces a link                  . Setting s=0 instead closes                 a boundary between two boxes.

Noise. Boxes where the noisy nonlinear flow KPZ    is expanded are in light red.    Noise contractions                       are produced by the noise cluster expansion.