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Register a new userLarge deviations of Schramm-Loewner evolutions: A survey
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Yilin Wang
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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.
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Let $D={mathbb H}setminus bigcup_{j=1}^N C_j$ be a standard slit domain, where ${mathbb H}$ is the upper half plane and $C_j,1le jle N,$ are mutually disjoint horizontal line segments in ${mathbb H}$. A stochastic Komatu-Loewner evolution denoted by ${rm SKLE}_{alpha,b}$ has been introduced in cite{CF} as a family ${F_t}$ of random growing hulls with $F_tsubset D$ driven by a diffusion process $xi(t)$ on $partial {mathbb H}$ that is determined by certain continuous homogeneous functions $alpha$ and $b$ defined on the space ${cal S}$ of all labelled standard slit domains. We aim at identifying the distribution of a suitably reparametrized ${rm SKLE}_{alpha,b}$ with that of the Loewner evolution on ${mathbb H}$ driven by the path of a certain continuous semimartingale and thereby relating the former to the distribution of ${rm SLE}_{alpha^2}$ when $alpha$ is a constant. We then prove that, when $alpha$ is a constant, ${rm SKLE}_{alpha,b}$ up to some random hitting time and modulo a time change has the same distribution as ${rm SLE}_{alpha^2}$ under a suitable Girsanov transformation. We further show that a reparametrized ${rm SKLE}_{sqrt{6},-b_{rm BMD}}$ has the same distribution as ${rm SLE}_6$, where $b_{rm BMD}$ is the BMD-domain constant indicating the discrepancy of $D$ from ${mathbb H}$ relative to Brownian motion with darning (BMD in abbreviation). A key ingredient of the proof is a hitting time analysis for the absorbing Brownian motion on ${mathbb H}.$ We also revisit and examine the locality property of ${rm SLE}_6$ in several canonical domains. Finally K-L equations and SKLEs for other canonical multiply connected planar domains than the standard slit one are recalled and examined.
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.
In this paper we statistically analyze the Fokker-Planck (FP) equation of Schramm-Loewner evolution (SLE) and its variant SLE($kappa,rho_c$). After exploring the derivation and the properties of the Langevin equation of the tip of the SLE trace, we obtain the long and short time behaviors of the chordal SLE traces. We analyze the solutions of the FP and the corresponding Langevin equations and connect it to the conformal field theory (CFT) and present some exact results. We find the perturbative FP equation of the SLE($kappa,rho_c$) traces and show that it is related to the higher order correlation functions. Using the Langevin equation we find the long-time behaviors in this case. The CFT correspondence of this case is established and some exact results are presented.
The Schramm-Loewner evolution (SLE) is a powerful tool to describe fractal interfaces in 2D critical statistical systems. Yet the application of SLE is well established for statistical systems described by quantum field theories satisfying only conformal invariance, the so called minimal conformal field theories (CFTs). We consider interfaces in Z(N) spin models at their self-dual critical point for N=4 and N=5. These lattice models are described in the continuum limit by non-minimal CFTs where the role of a Z_N symmetry, in addition to the conformal one, should be taken into account. We provide numerical results on the fractal dimension of the interfaces which are SLE candidates for non-minimal CFTs. Our results are in excellent agreement with some recent theoretical predictions.
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.
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