Universal phenomena in statistics of random growth models by Kevin Yang

Kevin Yang

Stanford

In 1986, the physicists Kardar, Parisi, and Zhang conjectured a universal behavior in the statistics of seemingly unrelated random growth models including the patterns in surfaces of bacterial growth colonies, the fronts of wildfires, and the shape of tumor growth. Confirming the proposed universality for models beyond a few incredibly special models has posed a difficult mathematical challenge, and the aim of my research is to make progress in this direction.

ABSTRACT

The physics community, led originally by Kardar, Parisi, and Zhang in the 80s, conjectured the universality of statistics of random growth models including patterns in tumor growth and wildfire fronts. Verifying this conjecture with laboratory data turns out to be quite difficult, and theoretical approaches have failed beyond models for which statistics can be “solved for” or computed almost explicitly. Confirming this proposed universality rigorously outside a modest number of “solvable” growth models poses a longstanding mathematical problem, reflecting difficulties in experimental verification of the universality. The primary aim of my research is to make progress in this direction and confirm the proposed universality for non-solvable models. The key approach we take is through a connection between random growth models and statistical mechanics models for classical fluids. Analyzing the statistics of these fluid dynamics models via model-independent “universal” philosophies has then led to confirming the original conjectured universality of Kardar, Parisi, and Zhang for a number of non-solvable models which had been previously inaccessible by other means.

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