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Vlad Margarint

Publications and source records attributed to Vlad Margarint.

At least 19 recordsLinked to original sources
arXiv2026

Numerical simulations of the spread from the mean of the SLE and Multiple SLE dynamics

The Schramm-Loewner Evolution (SLE) describes a family of fractal curves that arise in the study of the scaling limits of many planar Statistical Physics models. These curves are modeled using the Loewner Differential Equation for the conformal maps $g_t(z)$ with a Brownian motion driver. Using Euler's Method, in the current work we performed numerical experiments to study at a fixed time the quantities $|g_t(z) - \overline{g_t(z)}|$ and $Re(g_t(z)) - Re(\overline{g_t(z)})$, where $Re$ denotes the real part and $\overline{g_t(z)}$ refers to the sample average. These random variables measure the 'spread' of the dynamics from the average behavior at fixed time. One of the scopes of this work is to give numerical predictions for future theoretical investigations on these quantities. When investigating these quantities in the SLE case our experiments predict that the distribution is bimodal when the dynamics started close to the origin, and it can become bell-shaped if the dynamics is started further from the origin. In the second part, we performed experiments for a Multiple SLE model whose driver is Dyson Brownian Motion. Due to singularity in the dynamics of the drivers and the many data points needed, this part is challenging from a computational perspective. In the multiple SLE case, our experiments predict that the distribution is bell-shaped in all cases. In addition, we check the changes in the distributions as we vary the parameter $\kappa$ in the SLE case and $\beta$ in the Multiple SLE case.

cond-mat.stat-mech↗
arXiv2026

Neural Networks and Schramm-Loewner Evolutions

In this manuscript, we explore the application of neural networks to predict the natural parameter $\kappa \geq 0$ of Schramm-Loewner Evolution (SLE$_\kappa$) theory. SLE$_\kappa$ is a family of random fractal curves that has significant implications in Statistical Mechanics and Conformal Field Theory. This parameter $\kappa \geq 0$ plays an important role in the theory as there are models of Planar Statistical Physics that are proven to have SLE as scaling limits as well as others that are conjectured to have this limit for various choices of the parameter $\kappa \geq 0$. In addition, there are three different statistical behaviors of the SLE curves as the parameter $\kappa$ changes in $[0, \infty).$ Leveraging the powerful pattern recognition capabilities of neural networks, this study aims to develop a predictive model that can estimate the $\kappa$ parameter with good accuracy.

cond-mat.dis-nn↗
arXiv2025

Splitting algorithm and normed convergence for drawing the random Loewner curves

Recent advances in Schramm-Loewner evolution have driven increasing interest in non-standard Loewner flows. In this work, we propose a novel splitting algorithm to simulate random Loewner curves with rigorous convergence analysis in sup-norm and $L^p$. The algorithm is further extended to explore fractional SLE, driven by fractional Brownian motion, and noise-reinforced SLE, incorporating the effect on long-term memory. These exploratory and numerical extensions enable theoretical predictions on fractal dimensions and other statistical phenomena, providing new insights into such dynamics and opening directions for future research.

math.PR↗
arXiv2024

On the cover time of Brownian motion on the Brownian continuum random tree

Upon almost-every realisation of the Brownian continuum random tree (CRT), it is possible to define a canonical diffusion process or `Brownian motion'. The main result of this article establishes that the cover time of the Brownian motion on the Brownian CRT (i.e.\ the time taken by the process in question to visit the entire state space) is equal to the infimum over the times at which the associated local times are strictly positive everywhere. The proof of this result depends on the recursive self-similarity of the Brownian CRT and a novel version of the first Ray-Knight theorem for trees, which is of independent interest. As a consequence, we obtain that the suitably-rescaled cover times of simple random walks on critical, finite variance Galton-Watson trees converge in distribution with respect to their annealed laws to the cover time of Brownian motion on the Brownian CRT. Other families of graphs that have the Brownian CRT as a scaling limit are also covered. Additionally, we partially confirm a 1991 conjecture of David Aldous regarding related cover-and-return times.

math.PR↗
arXiv2024

On the analytic extension of Random Riemann Zeta Functions for some probabilistic models of the primes

The first step in the formulation and study of the Riemann Hypothesis is the analytic continuation of the Riemann Zeta Function (RZF) in the full Complex Plane with a pole at $s=1$. In the current work, we study the analytic continuation of two random versions of RZF using, for $Re s>1$, the Euler representation of ZF in terms of the product of functions over primes. In the first case, we substitute in the Euler product pseudo-prime numbers from the famous Cram\'er Model. In the second case, we use pseudo-primes with local symmetries. We show that in the Cram\'er case analytic continuation is possible $\mathbb{P}$-a.s. for $Res>1/2$, but not through the critical line $Re s=1/2.$ In the second case, we show that the analytic continuation is possible in a larger domain. We also study for the Cram\'er pseudo-primes several problems from Additive Number Theory.

math.PR↗
arXiv2024

Scaling Limits of Disorder Relevant Non-Binary Spin Systems

In [7], Caravenna, Sun and Zygouras gave general criteria for the partition functions of binary valued spin systems with a relevant random field perturbation to have non-trivial continuum and weak disorder limits. In this work, we show how these criteria can be extended to non-binary valued spin systems.

math.PR↗
arXiv2024

Splitting algorithm and normed convergence for drawing the random fractal Loewner curves

In the first part of the paper we propose and study the approximation of the $SLE_κ$ trace via the Ninomiya-Victoir splitting algorithm. We prove the uniform convergence in probability with respect to the sup-norm to the distance between the $SLE_κ$ trace and the output of the Ninomiya-Victoir splitting algorithm when applied in the context of the Loewner differential equation. Further investigations on the $L^p$-norm convergence is also exhibited, shedding light on the more delicate convergence structure. In the second part we show the uniform convergence of the approximation of the $SLE_κ$ trace obtained using a different scheme that is based on the linear interpolation of the Brownian driving force.

math.PR↗
arXiv2024

Local Central Limit Theorem for unbounded long-range potentials

We prove the equivalence between the integral central limit theorem and the local central limit theorem for two-body potentials with long-range interactions on the lattice $\mathbb{Z}^d$ for $d\ge 1$. The spin space can be an arbitrary, possibly unbounded subset of the real axis with a suitable a-priori measure. For general unbounded spins, our method works at high-enough temperature, but for bounded spins our results hold for every temperature. Our proof relies on the control of the integrated characteristic function, which is achieved by dividing the integration into three different regions, following a standard approach proposed forty years ago by Campanino, Del Grosso and Tirozzi. The bounds required in the different regions are obtained through cluster-expansion techniques. For bounded spins, the arbitrariness of the temperature is achieved through a decimation ("dilution") technique, also introduced in the later reference.

math-ph↗
arXiv2023

Law of the SLE tip

We analyze the law of the SLE tip at a fixed time in capacity parametrization. We describe it as the stationary law of a suitable diffusion process, and show that it has a density which is a unique solution of a certain PDE. Moreover, we identify the phases in which the even negative moments of the imaginary value are finite. For the negative second and negative fourth moments we provide closed-form expressions.

math.PR↗
arXiv2023

Rate of Convergence in Multiple SLE using Random Matrix Theory

We provide an order of convergence for a version of the Carathéodory convergence for the multiple SLE model with a Dyson Brownian motion driver towards its hydrodynamic limit, for $β=1$ and $β=2$. The result is obtained by combining techniques from the field of Schramm-Loewner Evolutions with modern techniques from random matrices. Our approach shows how one can apply modern tools used in the proof of universality in random matrix theory, in the field of Schramm-Loewner Evolutions.

math.PR↗
arXiv2022

Drivers, hitting times, and weldings in Loewner's equation

In addition to conformal weldings $φ$, simple curves $γ$ growing in the upper half plane generate driving functions $ξ$ and hitting times $τ$ through Loewner's differential equation. While the Loewner transform $γ\mapsto ξ$ and its inverse $ξ\mapsto γ$ have been carefully examined, less attention has been paid to the maps $ξ\mapsto τ\mapsto φ$. We study their continuity properties and show that uniform driver convergence implies uniform hitting time convergence and uniform welding convergence, even when the corresponding curves do not converge. Welding convergence implies neither hitting time nor driver convergence, while hitting time convergence implies driver convergence in (at least) the case of constant drivers. As an application, we show that a curve $γ$ of finite Loewner energy can be well approximated by an energy minimizer that matches $γ$'s welding on a sufficiently-fine mesh.

math.CV↗
arXiv2022

Local Central Limit Theorem for Long-Range Two-Body Potentials at Sufficiently High Temperatures

Dobrushin and Tirozzi [14] showed that, for a Gibbs measure with the finite-range potential, the Local Central Limit Theorem is implied by the Integral Central Limit Theorem. Campanino, Capocaccia, and Tirozzi [7] extended this result for a family of Gibbs measures for long-range pair potentials satisfying certain conditions. We are able to show for a family of Gibbs measures for long-range pair potentials not satisfying the conditions given in [7], that at sufficiently high temperatures, if the Integral Central Limit Theorem holds for a given sequence of Gibbs measures, then the Local Central Limit Theorem also holds for the same sequence. We also extend [7] when the state space is general, provided that it is equipped with a finite measure.

math-ph↗
arXiv2022

An asymptotic radius of convergence for the Loewner equation and simulation of $SLE_k$ traces via splitting

In this paper, we shall study the convergence of Taylor approximations for the backward Loewner differential equation (driven by Brownian motion) near the origin. More concretely, whenever the initial condition of the backward Loewner equation (which lies in the upper half plane) is small and has the form $Z_{0} = \varepsilon i$, we show these approximations exhibit an $O(\varepsilon)$ error provided the time horizon is $\varepsilon^{2+δ}$ for $δ> 0$. Statements of this theorem will be given using both rough path and $L^{2}(\mathbb{P})$ estimates. Furthermore, over the time horizon of $\varepsilon^{2-δ}$, we shall see that "higher degree" terms within the Taylor expansion become larger than "lower degree" terms for small $\varepsilon$. In this sense, the time horizon on which approximations are accurate scales like $\varepsilon^{2}$. This scaling comes naturally from the Loewner equation when growing vector field derivatives are balanced against decaying iterated integrals of the Brownian motion. As well as being of theoretical interest, this scaling may be used as a guiding principle for developing adaptive step size strategies which perform efficiently near the origin. In addition, this result highlights the limitations of using stochastic Taylor methods (such as the Euler-Maruyama and Milstein methods) for approximating $SLE_κ$ traces. Due to the analytically tractable vector fields of the Loewner equation, we will show Ninomiya-Victoir (or Strang) splitting is particularly well suited for SLE simulation. As the singularity at the origin can lead to large numerical errors, we shall employ the adaptive step size proposed by Tom Kennedy to discretize $SLE_κ$ traces using this splitting. We believe that the Ninomiya-Victoir scheme is the first high order numerical method that has been successfully applied to $SLE_κ$ traces.

math.PR↗
arXiv2022

Perturbations of Multiple Schramm-Loewner Evolution with Two Non-colliding Dyson Brownian Motions

In this article, we study multiple $SLE_κ$, for $κ\in(0,4]$, driven by Dyson Brownian motion. This model was introduced in the unit disk by Cardy in connection with the Calogero-Sutherland model. We prove the Carathéodory convergence of perturbed Loewner chains under different initial conditions and under different diffusivity $κ\in (0,4]$ for the case of $N=2$ driving forces. Our proofs use the analysis of Bessel processes and estimates on Loewner differential equation with multiple driving forces. In the last section, we estimate the Hausdorff distance of the hulls under perturbations of the driving forces, with assumptions on the modulus of the derivative of the multiple Loewner maps.

math.PR↗
arXiv2021

On Loewner chains driven by semimartingales and complex Bessel-type SDEs

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$_{\alpha,b}$ are generated by curves. As another application, motivated by a question of A. Sep\'{u}lveda, we show that for $\alpha >3/2$ and Brownian motion $B$, the driving function $|B_t|^\alpha$ generates a simple curve for small $t$. On a related note we also introduce a complex variant of Bessel-type SDEs and prove existence and uniqueness of strong solution. Such SDEs appear naturally while describing the trace of Loewner chains. In particular, we write SLE$_\kappa$, $\kappa <4$, in terms of stochastic flow of such SDEs.

math.PR↗
arXiv2020

Quasi-Sure Stochastic Analysis through Aggregation and SLE$_κ$ Theory

We study SLE$_κ$ theory with elements of Quasi-Sure Stochastic Analysis through Aggregation. Specifically, we show how the latter can be used to construct the SLE$_κ$ traces quasi-surely (i.e. simultaneously for a family of probability measures with certain properties) for $κ\in \mathcal{K}\cap \mathbb{R}_+ \setminus ([0, ε) \cup \{8\})$, for any $ε>0$ with $\mathcal{K} \subset \mathbb{R}_{+}$ a nontrivial compact interval, i.e. for all $κ$ that are not in a neighborhood of zero and are different from $8$. As a by-product of the analysis, we show in this language a version of the continuity in $κ$ of the SLE$_κ$ traces for all $κ$ in compact intervals as above.

math.PR↗
arXiv2020

Continuity of Zero-Hitting Times of Bessel Processes and Welding Homeomorphisms of SLE$_κ$

We consider a family of Bessel Processes that depend on the starting point $x$ and dimension $δ$, but are driven by the same Brownian motion. Our main result is that almost surely the first time a process hits $0$ is jointly continuous in $x$ and $δ$, provided $δ\le 0$. As an application, we show that the SLE($κ$) welding homeomorphism is continuous in $κ$ for $κ\in [0,4]$. Our motivation behind this is to study the well known problem of the continuity of SLE$_κ$ in $κ$. The main tool in our proofs is random walks with increments distributed as infinite mean Inverse-Gamma laws.

math.PR↗