Related papers: The dimension of the SLE curves

We compare, for L a Lie ring over the integers, its lower central series (\gamma_n(L))_{n>0} and its dimension series defined by \delta_n(L):=L\cap \varpi^n(L) in the universal enveloping algebra of L. We show that \gamma_n(L)=\delta_n(L)…

A recent SICOMP paper on classical and quantum algorithms for the shortest vector problem introduced a lattice-dependent parameter \(\gamma(L)\), bounded universally in the exponential sense by \(2^{0.402n+o(n)}\), and conjectured that this…

We consider bipolar oriented random planar maps with heavy-tailed face degrees. We show for each $\alpha \in (1,2)$ that if the face degree is in the domain of attraction of an $\alpha$-stable L\'evy process, the corresponding random planar…

We study a class of Schr\"odinger operators $H_{m,\lambda}$ with generalized Thue-Morse potential that generated by the substitution $\tau(a)=a^mb^m$, $\tau(b)=b^ma^m$ on two symbol alphabet $\Sigma=\{a,b\}$ for integer $m\ge 2$ and…

Let $X=\{X(t):t\geq0\}$ be an operator semistable L\'evy process in $\mathbb{R}^d$ with exponent $E$, where $E$ is an invertible linear operator on $\mathbb{R}^d$. For an arbitrary Borel set $B\subseteq\mathbb{R}_+$ we interpret the graph…

We use the interpretation of the Schramm-Loewner evolution as a limit of path measures tilted by a loop term in order to motivate the definition of $n$-radial SLE going to a particular point. In order to justify the definition we prove that…

We show that the Szeg\H{o} matrices, associated with Verblunsky coefficients $\{\alpha_n\}_{n\in\mathbb{Z}_+}$ obeying $\sum_{n = 0}^\infty n^\gamma |\alpha_n|^2 < \infty$ for some $\gamma \in (0,1)$, are bounded for values $z \in \partial…

This paper is concerned with restricted families of projections in $\mathbb{R}^{3}$. Let $K \subset \mathbb{R}^{3}$ be a Borel set with Hausdorff dimension $\dim K = s > 1$. If $\mathcal{G}$ is a smooth and sufficiently well-curved…

Sparse autoencoders (SAEs) decompose transformer residual streams into interpretable feature dictionaries, yet the relationship between SAE width and causal influence on model output has not been systematically characterised. We introduce…

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schr\"odinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$ on $L_2({\Bbb R})$. We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb R}…

In this paper, for $1<p<\infty$, we obtain the $L^p$-boundedness of the Hilbert transform $H^{\gamma}$ along a variable plane curve $(t,u(x_1, x_2)\gamma(t))$, where $u$ is a Lipschitz function with small Lipschitz norm, and $\gamma$ is a…

For given continuous functions $\gamma_{{}_{i}}: S^{n}\to \mathbb{R}_{+}$ (where $i=1, 2$), the functions $\gamma_{{}_{max}}$ and $\gamma_{{}_{min}}$ can be defined as natural way. In this paper, we show that the Wulff shape associated to…

Let $\mathbf{Y}=\mathbf{X}\bolds{\Theta}\mathbf{Z}'+\bolds{\mathcal {E}}$ be the growth curve model with $\bolds{\mathcal{E}}$ distributed with mean $\mathbf{0}$ and covariance $\mathbf{I}_n\otimes\bolds{\Sigma}$, where $\bolds{\Theta}$,…

We prove that for any two definable germs in a polynomially bounded o-minimal structure, there exists a critical threshold $\alpha_0 \in (0,1)$ such that if these germs are bi-$\alpha$-H"older equivalent for some $\alpha \ge \alpha_0$, then…

We construct radial stochastic Loewner evolution in multiply connected domains, choosing the unit disk with concentric circular slits as a family of standard domains. The natural driving function or input is a diffusion on the associated…

Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models.…

We consider 2-dimensional orientable self-shrinkers $\Sigma$ for the Mean Curvature Flow of polynomial volume growth immersed in $\mathbb R^n$. We look at closed one forms minimizing the norm $\int_\Sigma \eterm |\omega|^2$ in their…

We consider the boundary WZW model on a half-plane with a cut growing according to the Schramm-Loewner stochastic evolution and the boundary fields inserted at the tip of the cut and at infinity. We study necessary and sufficient conditions…

We prove existence (and simpleness) of the trace for both forward and backward Loewner chains under fairly general conditions on semimartingale drivers. As an application, we show that stochastic Komatu-Loewner evolutions SKLE$_{\alpha,b}$…

Similar to the well-known phases of SLE, the Loewner differential equation with Lip(1/2) driving terms is known to have a phase transition at norm 4, when traces change from simple to non-simple curves. We establish the deterministic analog…