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We derive the large deviation principle for radial Schramm-Loewner evolution ($operatorname{SLE}$) on the unit disk with parameter $kappa rightarrow infty$. Restricting to the time interval $[0,1]$, the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures ${phi_t^2 (zeta), dzeta}_{t in [0,1]}$ on the unit circle and equals $int_0^1 int_{S^1} |phi_t|^2/2,dzeta ,dt$. Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.
These notes survey the first and recent results on large deviations of Schramm-Loewner evolutions (SLE) with the emphasis on interrelations among rate functions and applications to complex analysis. More precisely, we describe the large deviations of
It is well know that $SLE_kappa$ curves exhibit a phase transition at $kappa=4$. For $kappale 4$ they are simple curves with probability one, for $kappa>4$ they are not. The standard proof is based on the analysis of the Bessel SDE of dimension $d=1+
Let $X^{(delta)}$ be a Wishart process of dimension $delta$, with values in the set of positive matrices of size $m$. We are interested in the large deviations for a family of matrix-valued processes ${delta^{-1} X_t^{(delta)}, t leq 1 }$ as $delta$
We prove a Large Deviations Principle for the number of intersections of two independent infinite-time ranges in dimension five and more, improving upon the moment bounds of Khanin, Mazel, Shlosman and Sina{i} [KMSS94]. This settles, in the discrete
In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain the asympto