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Peano modes at the D=2 delocalization transition

We study the relaxation of the Peano chain's deformations in the presence of gaussian noise. The D=2 spring network which models the chain's elasticity has no bulk, but only boundaries:its continuum version would be a flat and thin elastic strip covering the plane, smoothly curved so as to follow the Peano pattern. We find normal modes, named Peano modes: in the same way as crystal waves are sustained by the matching of translations with rotations,the Peano modes correspond to the matching of the eightfold symmetry with discrete dilatations. The eightfold symmetry is shared by the Peano chain with the octagonal quasicrystal, but the latter has a much higher connectivity. The relaxation process which starts from the Peano chain is found to mark the saddle point separating the solid from the liquid, the phase separation being driven by anisotropy. For the equivalent quantum mechanical problem this corresponds to the D=2 quantum delocalization transition.Various experimental observations, regarding a priori uncorrelated systems, come together within the viewpoint proposed here, at least as far as their qualitative behavior is concerned. 1)Two features observed in experiments on D=2 colloids can be understood within the model: the bosonic peak occurring at low frequencies in the phononic spectrum and the time evolution and spatial structure of excitations: if initially concentrated in small regions, they progressively expand towards larger size. 2) The so-called inverse energy cascade of D=2 turbulence 3)The recently observed liquid to quasicrystal transition in bilayer water 4) Self-similar DNA conformations are connected here with structures lying at the boundary between a liquid and a solid, such as colloids and quasicrystals.