Scaling limits for planar aggregation with subcritical fluctuations - PowerPoint PPT Presentation

Scaling limits for planar aggregation with subcritical fluctuations Amanda Turner Lancaster University and University of Geneva Joint work with James Norris and Vittoria Silvestri (Cambridge) Amanda Turner Lancaster University and University of Geneva Scaling limits for planar aggregation with subcritical fluctuations

Physical motivation Bacterial growth in increasingly stressed conditions Source: https://users.math.yale.edu/public html/People/frame/Fractals/Panorama/Biology/Bacteria/Bacteria.html Amanda Turner Lancaster University and University of Geneva Scaling limits for planar aggregation with subcritical fluctuations

Physical motivation Bacterial growth in increasingly stressed conditions Source: https://users.math.yale.edu/public html/People/frame/Fractals/Panorama/Biology/Bacteria/Bacteria.html Amanda Turner Lancaster University and University of Geneva Scaling limits for planar aggregation with subcritical fluctuations

Conformal models for planar random growth Conformal mapping representation of single particle Let D 0 denote the exterior unit disk in the complex plane C and P denote a particle of logarithmic capacity c and attachment angle θ . Use the unique conformal mapping f θ c : D 0 → D 0 \ P that fixes ∞ as a mathematical description of the particle. Amanda Turner Lancaster University and University of Geneva Scaling limits for planar aggregation with subcritical fluctuations

Conformal models for planar random growth Conformal mapping representation of a cluster Suppose P 1 , P 2 , . . . is a sequence of particles, where P n has capacity c n and attachment angle θ n , n = 1 , 2 , . . . . Set Φ 0 ( z ) = z . Recursively define Φ n ( z ) = Φ n − 1 ◦ f θ n c n ( z ), for n = 1 , 2 , . . . . This generates a sequence of conformal maps Φ n : D 0 → K c n , where K n − 1 ⊂ K n are growing compact sets, which we call clusters. By varying the sequences { θ n } and { c n } , it is possible to describe a wide class of growth models. Amanda Turner Lancaster University and University of Geneva Scaling limits for planar aggregation with subcritical fluctuations

Conformal models for planar random growth Cluster formed by iteratively composing conformal mappings Amanda Turner Lancaster University and University of Geneva Scaling limits for planar aggregation with subcritical fluctuations