Emergent Loewner Dynamics in Slime Mold Growth

Growth fronts of slime molds are characterized through a direct geometric analysis based on Loewner evolutions, using experimentally acquired time-resolved images. The associated Loewner driving functions reconstructed from expanding pseudopod boundaries display statistical properties consistent with Gaussian-like behavior. A geometric estimate of the diffusivity parameter~$κ$ is inferred from fractal scaling, while Brownian diagnostics are assessed on the reconstructed driving signal. These findings show that the boundaries of a growing living organism display statistical and geometric properties consistent with emergent Loewner dynamics over experimentally accessible scales. This study establishes a quantitative framework for analyzing biological growth interfaces and suggests new connections between morphogenesis, stochastic geometry, and network reorganization under varying environmental conditions. We provide, to our knowledge, the first explicit reconstruction of a Loewner driving function from a living growth interface, revealing an emergent Brownian-like conformal growth regime at expanding fronts.