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Register a new userLarge deviations of radial SLE$_{infty}$
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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.
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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 SLE$_kappa$ when the $kappa$ parameter goes to zero in the chordal and multichordal case and to infinity in the radial case. The rate functions, namely Loewner and Loewner-Kufarev energies, are closely related to the Weil-Petersson class of quasicircles and real rational functions.
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 $d=1+4/kappa$. We propose a different approach which is based on the analysis of the Bessel SDE with $d=1-4/kappa$. This not only gives a new perspective, but also allows to describe the formation of the SLE `bubbles for $kappa>4$.
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 $delta$ tends to infinity. The process $X^{(delta)}$ is a solution of a stochastic differential equation with a degenerate diffusion coefficient. Our approach is based upon the introduction of exponential martingales. We give some applications to large deviations for functionals of the Wishart processes, for example the set of eigenvalues.
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 setting, a conjecture of van den Berg, Bolthausen and den Hollander [BBH04], who analyzed this question for the Wiener sausage in finite-time horizon. The proof builds on their result (which was resumed in the discrete setting by Phetpradap [Phet12]), and combines it with a series of tools that were developed in recent works of the authors [AS17, AS19a, AS20]. Moreover, we show that most of the intersection occurs in a single box where both walks realize an occupation density of order one.
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 asymptotics of the so-called rectangular spherical integrals as $m,n$ go to infinity while $m/n$ converges.
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