We establish a large deviation principle for chordal SLEκ parametrized by capacity, as the parameter κ→0+, in the topology generated by uniform convergence on compact intervals of the positive real line. The rate function is shown to equal the Loewner energy of the curve. This strengthens the recent result of Peltola and Wang who obtained the analogous statement using the Hausdorff topology.

We consider the Hastings--Levitov HL(0) model in the small particle scaling limit and prove a large deviation principle. The rate function is given by the relative entropy of the driving measure ρ for the Loewner--Kufarev equation: 

H(ρ)=12π∬ρ¯t(θ)logρ¯t(θ)dθdt,

whenever

ρ=ρ¯tdθdt/2π with ∫S1ρ¯tdθ/2π=1. 

We investigate the class of shapes that can be generated by finite entropy Loewner evolution and show that it contains all Weil-Petersson quasicircles, all Becker quasicircles, a Jordan curve with a cusp, and a non-simple curve. We also consider the problem of finding a measure of minimal entropy generating a given shape as well as a simplified version of the problem for a related transport equation.