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TY - JOUR
T1 - SLE scaling limits for a Laplacian random growth model
AU - Higgs, Frankie
PY - 2022/8/31
Y1 - 2022/8/31
N2 - We consider a model of planar random aggregation from the ALE$(0,\eta)$ family where particles are attached preferentially in areas of low harmonic measure. We find that the model undergoes a phase transition in negative $\eta$, where for sufficiently large values the attachment distribution of each particle becomes atomic in the small particle limit, with each particle attaching to one of the two points at the base of the previous particle. This complements the result of Sola, Turner and Viklund for large positive $\eta$, where the attachment distribution condenses to a single atom at the tip of the previous particle.As a result of this condensation of the attachment distributions we deduce thatin the limit as the particle size tends to zero the ALE cluster converges to a Schramm--Loewner evolution with parameter $\kappa = 4$ (SLE$_4$).We also conjecture that using other particle shapes from a certain family, we have a similar SLE scaling result, and can obtain SLE$_\kappa$ for any $\kappa \geq 4$.
AB - We consider a model of planar random aggregation from the ALE$(0,\eta)$ family where particles are attached preferentially in areas of low harmonic measure. We find that the model undergoes a phase transition in negative $\eta$, where for sufficiently large values the attachment distribution of each particle becomes atomic in the small particle limit, with each particle attaching to one of the two points at the base of the previous particle. This complements the result of Sola, Turner and Viklund for large positive $\eta$, where the attachment distribution condenses to a single atom at the tip of the previous particle.As a result of this condensation of the attachment distributions we deduce thatin the limit as the particle size tends to zero the ALE cluster converges to a Schramm--Loewner evolution with parameter $\kappa = 4$ (SLE$_4$).We also conjecture that using other particle shapes from a certain family, we have a similar SLE scaling result, and can obtain SLE$_\kappa$ for any $\kappa \geq 4$.
KW - scaling limits
KW - Schramm-Loewner evolution
KW - conformal aggregation
KW - harmonic measure
U2 - 10.1214/21-AIHP1217
DO - 10.1214/21-AIHP1217
M3 - Journal article
VL - 58
SP - 1712
EP - 1739
JO - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
JF - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
SN - 0246-0203
IS - 3
ER -