Loewner--Kufarev entropy and large deviations of the Hastings--Levitov model
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918228513521664 |
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| author | Berestycki, Nathanaël Guskov, Vladislav Viklund, Fredrik |
| author_facet | Berestycki, Nathanaël Guskov, Vladislav Viklund, Fredrik |
| contents | 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:
\[
H(ρ) = \frac{1}{2π}\iint \barρ_t(θ) \log \barρ_t(θ) dθdt,
\]
whenever $ρ= \barρ_t dθdt/2π$ with $\int_{S^1} \barρ_t dθ/2π= 1$.
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02855 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Loewner--Kufarev entropy and large deviations of the Hastings--Levitov model Berestycki, Nathanaël Guskov, Vladislav Viklund, Fredrik Probability Complex Variables 60F10, 60J67, 82C24, 30C55 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: \[ H(ρ) = \frac{1}{2π}\iint \barρ_t(θ) \log \barρ_t(θ) dθdt, \] whenever $ρ= \barρ_t dθdt/2π$ with $\int_{S^1} \barρ_t dθ/2π= 1$. 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. |
| title | Loewner--Kufarev entropy and large deviations of the Hastings--Levitov model |
| topic | Probability Complex Variables 60F10, 60J67, 82C24, 30C55 |
| url | https://arxiv.org/abs/2512.02855 |