Two-valued sets are local sets of the 2D Gaussian free field (GFF) that can be thought of as representing all points of the domain that may be connected to the boundary by a curve on which the GFF takes values only in $[-a,b]$. Two-valued sets exist whenever $a+b\geq 2\lambda ,$ where $\lambda$ depends explicitly on the normalization of the GFF. We prove that the almost sure Hausdorff dimension of the two-valued set ${\mathbb{A}}_{-a,b}$ equals $d=2-2\lambda ^2/(a+b)^2$. For the key two-point estimate needed to give the lower bound on dimension, we use the real part of a “vertex field” built from the purely imaginary Gaussian multiplicative chaos. We also construct a non-trivial $d$-dimensional measure supported on ${\mathbb{A}}_{-a,b}$ and discuss its relation with the $d$-dimensional conformal Minkowski content of ${\mathbb{A}}_{-a,b}$.
Two-valued sets are local sets of the two-dimensional Gaussian free field (GFF) that can be thought of as representing all points of the domain that may be connected to the boundary by a curve on which the GFF takes values only in [−a, b]. Two-valued sets exist whenever a + b 2λ, where λ depends explicitly on the normalization of the GFF. We prove that the almost sure Hausdorff dimension of the two-valued set A −a,b equals d = 2 − 2λ 2 /(a + b) 2 . For the two-point estimate, we use the real part of a "vertex field" built from the purely imaginary Gaussian multiplicative chaos. We also construct a non-trivial d-dimensional measure supported on A −a,b and discuss its relation with the d-dimensional conformal Minkowski content for A −a,b .
We show that the modulus of continuity of the SLE4 uniformizing map is given by (log δ −1 ) −1/3+o(1) as δ → 0. As a consequence of our analysis, we show that the Jones-Smirnov condition for conformal removability (with quasihyperbolic geodesics) does not hold for SLE4. We also show that the modulus of continuity for SLE8 with the capacity time parameterization is given by (log δ −1 ) −1/4+o(1) as δ → 0, proving a conjecture of Alvisio and Lawler.
We study $${{\,\mathrm{SLE}\,}}_\kappa (\rho )$$
SLE
κ
(
ρ
)
curves, with $$\kappa $$
κ
and $$\rho $$
ρ
chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed “angle” and determine the almost sure Hausdorff dimensions of these sets. This is done by studying the moments of the spatial derivatives of the conformal maps $$g_t$$
g
t
, by employing the Girsanov theorem and using imaginary geometry techniques to derive a correlation estimate.
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