Related papers: On Multiple Schramm-Loewner Evolutions

This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a…

Exploring the genetic basis of heritable traits remains one of the central challenges in biomedical research. In simple cases, single polymorphic loci explain a significant fraction of the phenotype variability. However, many traits of…

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…

We show that, for $\kappa\in(0,8)$, the integral of the laws of two-sided radial SLE$_\kappa$ curves through different interior points against a measure with SLE$_\kappa$ Green function density is the law of a chordal SLE$_\kappa$ curve,…

Evolutionary agentic systems intensify the trade-off between computational efficiency and reasoning capability by repeatedly invoking large language models (LLMs) during inference. This setting raises a central question: how can an agent…

In this paper we define and prove of the existence of the multi-point Green's function for SLE - a normalized limit of the probability that an $SLE_{\kappa}$ curve passes near to a pair of marked points in the interior of a domain. When…

Recently, Lenski et al [Elena,Lenski,Travisano] have carried out several experiments on bacterial evolution. Their findings support the theory of punctuated equilibrium in biological evolution. They have further quantified the relative…

It is widely believed that the scaling limit of self-avoiding walks (SAWs) at the critical temperature is (i) conformally invariant, and (ii) describable by Schramm-Loewner Evolution (SLE) with parameter $\kappa = 8/3.$ We consider SAWs in…

The question of whether classically conformal modifications of the standard model are consistent with experimental obervations has recently been subject to renewed interest. The method of Gildener and Weinberg provides a natural framework…

We present a way to study the conformal structure of random planar maps. The main idea is to explore the map along an SLE (Schramm--Loewner evolution) process of parameter $ \kappa = 6$ and to combine the locality property of the SLE_{6}…

We formulate a classification conjecture for conformally invariant families of measures on simple loops that builds on a conjecture of Kontsevich and Suhov. The main example in this class of objects was constructed by Werner as boundaries…

Large language models (LLMs) have achieved widespread success on a variety of in-context few-shot tasks, but this success is typically evaluated via correctness rather than consistency. We argue that self-consistency is an important…

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 study SLE$_\kappa(\rho)$ curves, with $\kappa$ and $\rho$ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed "angle" and determine the almost sure…

The scaling limit of the spin cluster boundaries of the Ising model with domain wall boundary conditions is SLE with kappa=3. We hypothesise that the three-state Potts model with appropriate boundary conditions has spin cluster boundaries…

The general tendency for species number (S) to increase with sampled area (A) constitutes one of the most robust empirical laws of ecology, quantified by species-area relationships (SAR). In many ecosystems, SAR curves display a power-law…

We derive some geometric properties of chordal SLE$(\kappa;\vec{\rho})$ processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE$(\kappa;\vec{\rho})$…

Generalizing some popular sequences like Catalan's number, Schr\"oder's number, etc, we consider the sequence $s_n$ with $s_0=1$ and for $n\ge 1$, \begin{multline*} s_n=\sum_{x_1+\dots+x_{\ell_1}=n-1} \kappa_1 s_{x_1}\dots s_{x_{\ell_1}} +…

Lam suggested that the PMNS matrix (or at least some of its elements) can be predicted by embedding the residual symmetry of the leptonic mass terms into a bigger symmetry. We analyze the possibility that the residual symmetries consist of…

We consider collections of $N$ chordal random curves obtained from a critical lattice model on a planar graph, in the limit when a fine-mesh graph approximates a simply-connected domain. We define and study candidates for such limits in…