Related papers: Stochastic Loewner Evolution
This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the L\'{e}vy-type…
In this paper, we define and study Loewner chains and evolution families on finitely multiply-connected domains in the complex plane. These chains and families consist of conformal mappings on parallel slit half-planes and have one and two…
We investigate the relation between the Laplacian-$b$ motion and stochastic Komatu-Loewner evolution (SKLE) on multiply connected subdomains of the upper half-plane, both of which are analogues to SLE. In particular, we show that, if the…
Existence of Loewner trace is revisited. We identify finite energy paths (the "skeleton of Wiener measure") as natural class of regular drivers for which we find simple and natural estimates in terms of their (Cameron--Martin) norm.…
The shifted-l expansion technique (SLET) has been developed to get eigenvalues of Schrodinger equation in three (3D) and two dimensions (2D). SLET simply consists of 1/\bar{l} as a perturbation parameter, where \bar{l}=l-\beta, \beta is a…
The goal of this expository article is to explain how a fundamental functional on the space of Jordan curves arising from SLE - Loewner energy - is connected to a seemingly far apart subject: the K\"ahler geometry of universal Teichm\"uller…
In this study, the non-equilibrium free energy corresponding to the curve generated by a modified stochastic Loewner evolution (SLE), which is driven by the Langevin equation, is theoretically investigated. Under certain conditions, we…
This article focuses on the characterization of global multiple Schramm-Loewner evolutions (SLE). The chordal SLE describes the scaling limit of a single interface in various critical lattice models with Dobrushin boundary conditions, and…
Particular boundary correlation functions of conformal field theory are needed to answer some questions related to random conformally invariant curves known as Schramm-Loewner evolutions (SLE). In this article, we introduce a correspondence…
Square ice is a statistical mechanics model for two-dimensional ice, widely believed to have a conformally invariant scaling limit. We associate a Peano (space filling) curve to a square ice configuration, and more generally to a so-called…
Karhunen-Loeve expansions (KLE) of stochastic processes are important tools in mathematics, the sciences, economics, and engineering. However, the KLE is primarily useful for those processes for which we can identify the necessary…
One way to uniquely define Schramm-Loewner Evolution (SLE) in multiply connected domains is to use the restriction property. This gives an implicit definition of a $\sigma$-finite measure on curves; yet it is in general not clear how to…
Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting…
We review some recently completed research that establishes the scaling limit of Fomin's identity for loop-erased random walk on Z^2 in terms of the chordal Schramm-Loewner evolution (SLE) with parameter 2. In the case of two paths, we…
A quantum surface (QS) is an equivalence class of pairs $(D,H)$ of simply connected domains $D\subsetneq\mathbb{C}$ and random distributions $H$ on $D$ induced by the conformal equivalence for random metric spaces. This distribution-valued…
We present a novel probabilistic deep learning approach, the 'Stochastic Latent Transformer' (SLT), designed for the efficient reduced-order modelling of stochastic partial differential equations. Stochastically driven flow models are…
Dynamical scaling arises naturally in various many-body systems far from equilibrium. After a short historical overview, the elements of possible extensions of dynamical scaling to a local scale-invariance will be introduced.…
The conjecture that the scaling limit of the two-dimensional self-avoiding walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE) with $\kappa=8/3$ leads to explicit predictions about the SAW. A remarkable feature of…
Using results from Schramm Loewner evolution (SLE), we give the expression of the fluctuation-induced force exerted by a polymer on a small impenetrable disk, in various 2-dimensional domain geometries. We generalize to two polymers and…
For random collections of self-avoiding loops in two-dimensional domains, we define a simple and natural conformal restriction property that is conjecturally satisfied by the scaling limits of interfaces in models from statistical physics.…