From tissues, to canyon landscapes, to optimal control : a view from high dimension

Simple models of confluent tissues can be defined from the tessellation of space into cells. The cell-cell interaction is then described by a cost function in which each cell tries to optimize its target shape (surface, volume or both). It has been pointed out that when the target shape of the cells is properly tuned, one can encounter a rigidity transition that separates a liquid phase, where cells can diffuse, from a glassy/solid phase, where cells are stuck in a solid amorphous elastic medium. I will argue that the energy landscape of the liquid phase can be understood as a canyon landscape, namely a zero energy manifold that constraints the dynamics of individual cells. Using an abstract mapping I will construct a simple high-dimensional, mean field model having a rigidity transition of the same kind as the finite dimensional models and discuss how the canyon landscape behaves across the control parameter space of the model. I will also discuss what happens if one adds quantum fluctuations and show that in the region around the rigidity transition, moderate quantum fluctuations can in fact promote glassiness rather than open new thermalization channels. Finally I will argue why the quantum model is interesting as a first step in studying high dimensional optimal control problems.

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