arXiv subjects

Wioletta M. Ruszel

Publications and source records attributed to Wioletta M. Ruszel.

At least 19 recordsLinked to original sources

arXiv2026

Central limit theorems for non-linear functionals of Gaussian fields via Wiener chaos decomposition

We review and present some known results for non-linear functionals of Gaussian variables in the context of discrete Gaussian fields defined on the $d$ dimensional lattice. Our main result is a Central Limit Theorem in the spirit of the classical Breuer-Major theorem, together with applications to the powers of the Gaussian Free Field. Notably, we show that even powers of the discrete Gaussian Free Field converge to the Gaussian white noise, while odd powers converge to a continuous Gaussian Free Field with explicit covariance. The proofs are based on the Wiener chaos decomposition and the fourth moment theorem (Nualart-Peccati, 2005), and include a tightness result. Even if these tools are well-known in the literature, their application to Gaussian fields on the lattice appears to be new.

math.PR↗

arXiv2026

Topology, noise, and parallel updates in a model of circular opinion dynamics

We study a circular opinion dynamics model with local midpoint interactions, extended to allow parallel updates of multiple sites. On a ring, the dynamics admits twisted states associated with integer winding numbers. We investigate how bi-modal noise, which drives opinions toward two antipodal directions, affects these configurations. Numerically, we find that noise both destabilizes winding states and induces a flip--flop regime, characterized by macroscopic switching between preferred orientations. We introduce order parameters that distinguish topological trapping from symmetry breaking, providing a simple macroscopic description of the dynamics.

math.DS↗

arXiv2026

On minimal shapes and isoperimetric constants in hyperbolic lattices

We fully characterize the set of finite shapes with minimal perimeter on hyperbolic lattices given by regular tilings of the hyperbolic plane whose tiles are regular $p$-gons meeting at vertices of degree $q$, with $1/p+1/q<\frac{1}{2}$. In particular, we prove that the ratio between the perimeter and the area (i.e., the number of vertices) of this set of minimal shapes converges to the isoperimetric constant computed in Häggström-Jonasson-Lyons. In fact, our balls which are constructed via layers and not combinatorial balls, will realize the isoperimetric constant for any fixed number of vertices.

math.CO↗

arXiv2026

Topological trapping in circular midpoint opinion dynamics

We study a discrete-time asynchronous midpoint dynamics on the circle in which, at each step, a uniformly chosen neighboring pair moves to the midpoint along the shortest arc. Although the update rule is locally contractive, we show that the global relaxation mechanism depends sharply on the boundary topology. Under open boundary conditions the system converges almost surely to consensus through pure contraction. Under periodic boundary conditions the graph contains a single cycle, and the wrapped edge increments define an integer-valued winding number. While consensus remains the unique absorbing state for every fixed system size, we show that topology profoundly reshapes the transient dynamics. We prove that branch-crossings are the only mechanism capable of modifying the winding number and compute explicitly their probability for disordered initial data. Local averaging rapidly suppresses large gradients and drives the system into a no-branch-crossing regime where the winding number freezes. Inside a fixed winding sector we construct an adaptive co-moving frame in which the dynamics becomes an exact Euclidean midpoint process and establish strict contraction toward a twisted linear profile determined by the winding number. Our results isolate a minimal mechanism by which a single cycle induces sector locking and escape, even though the final equilibrium remains unchanged.

math.PR↗

arXiv2026

Wick theorem for analytic functions of Gaussian fields

We compute the correlation of analytic functions of general Gaussian fields in terms of multigraphs and Feynman diagrams on the lattice Z^d. Then, we connect its scaling limit to tensors of the correlation functionals of Fock space fields. Afterwards, we investigate the relation with fermionic Gaussian field states for even functions. For instance, we characterize the correlation functionals of the exponential of a continuous Gaussian Free Field or general analytic functions of fractional Gaussian fields as limits of quantities constructed via a sequences of discrete fields. Finally, we show that the duality between even powers of bosonic Gaussian fields and "complex" fermionic Gaussian fields can be reformulated in terms of a principal minors assignment problem of the corresponding covariance matrices.

math.PR↗

arXiv2026

Complex abstract Wiener spaces

Real abstract Wiener spaces (AWS) were originally defined by Gross using measurable norms, as a generalisation of the theory of advanced integral calculus in infinite dimensions as introduced by Cameron and Martin. In this paper we present a rigorous, complete and self-contained general framework for $\mathbb{K}$-AWS, where $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ using the language of characteristic functions instead of measurable norms. In particular, we will prove that $X$ is a centred resp. proper $H$-valued Gaussian field over $\mathbb{K}$ iff the covariance function can be written in terms of some non-negative, self-adjoint trace class operator, and that the existence and uniqueness of $X$ is equivalent to the $\mathbb{K}$-AWS. Finally we will relate the $\mathbb{C}$-AWS to the $\mathbb{R}$-AWS by way of a real structure, which is a real linear, complex anti-linear involution on a complex vector space. This allows for a commutative relation between the real and complex Gaussian fields and the real and complex abstract Wiener spaces. We will construct specific examples which fall under this framework like the complex Brownian motion, complex Feynman-Kac formula and complex fractional Gaussian fields.

math.PR↗

arXiv2025

The fermionic DGFF and its scaling limit logCFT

In this paper, we identify the scaling limit of the fermionic discrete Gaussian free field (fDGFF) as a logarithmic conformal field theory (CFT) in two dimensions. We first establish a one-to-one correspondence between the space of local observables of the fDGFF and the space of local fields of the symplectic fermions CFT, a logarithmic CFT with central charge $c = - 2$. This correspondence is meaningful in the sense that, when appropriately renormalised, the fDGFF correlation functions converge to corresponding CFT correlation functions in the scaling limit. As an application to these results, we interpret (the scaling limit of) certain local observables in the uniform spanning tree and the Abelian sandpile model as local fields of the symplectic fermions.

math-ph↗

arXiv2025

Emergence of metastability on the hyperbolic lattice: Effects of boundary conditions

We investigate the Ising model on finite subgraphs of the hyperbolic lattice under minus boundary conditions and in the presence of a positive external field $h$. Interpreting the boundary as frozen or cold wall conditions, we show that, for small values of $h$, the system exhibits metastable behaviour. Our result is very surprising, since non-amenable graphs, such as hyperbolic lattices, feature exponentially growing boundaries, which typically destabilize local energy minima. In particular, we identify the unique metastable state and characterize the exit time from it. Furthermore, we establish asymptotic results for the distribution of the first hitting time and provide estimates for the spectral gap. Finally, we analyze the energy landscape and describe the nucleation mechanism for values of $h$ outside the metastable regime.

math.PR↗

arXiv2024

Absence of shift-invariant Gibbs states (delocalisation) for one-dimensional $\mathbb Z$-valued fields with Long-Range interactions

We show that a modification of the proof of our paper [CvELNR18], in the spirit of [FP81], shows delocalisation in the long-range Discrete Gaussian Chain, and generalisations thereof, for any decay power $α>2$ and at all temperatures. The argument proceeds by contradiction: any shift-invariant and localised measure (in the $L^1$ sense), is a convex combination of ergodic localised measures. But the latter cannot exist: on one hand, by the ergodic theorem, the average of the field over growing boxes would be almost surely bounded ; on the other hand the measure would be absolutely continuous with respect to its height-shifted translates, as a simple relative entropy computation shows. This leads to a contradiction and answers, in a non-quantitative way, an open question stated in a recent paper [G23] of C.Garban.

math.PR↗

arXiv2023

Properties of the gradient squared of the discrete Gaussian free field

In this paper we study the properties of the centered (norm of the) gradient squared of the discrete Gaussian free field in $U_ε=U/ε\cap \mathbb{Z}^d$, $U\subset \mathbb{R}^d$ and $d\geq 2$. The covariance structure of the field is a function of the transfer current matrix and this relates the model to a class of systems (e.g. height-one field of the Abelian sandpile model or pattern fields in dimer models) that have a Gaussian limit due to the rapid decay of the transfer current. Indeed, we prove that the properly rescaled field converges to white noise in an appropriate local Besov-Hölder space. Moreover, under a different rescaling, we determine the $k$-point correlation function and cumulants on $U_ε$ and in the continuum limit as $ε\to 0$. This result is related to the analogue limit for the height-one field of the Abelian sandpile (\citet{durre}), with the same conformally covariant property in $d=2$.

math.PR↗

arXiv2023

Stochastic homogenization of Gaussian fields on random media

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $Ξ^{g,{\bf a}} $ and bi-Laplacian fields $Ξ^{b,{\bf a}}$. They can be characterized as follows: for $f=δ$ the solution $u$ of $\nabla \cdot \mathbf{a} \nabla u =f$, ${\bf a}$ is a uniformly elliptic random environment, is the covariance of $Ξ^{g,{\bf a}}$. When $f$ is the white noise, the field $Ξ^{b,{\bf a}}$ can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain $D\subset \mathbb{R}^d$, or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator $Δ$, we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator $\ahom Δ$, with constant $\ahom$ depending on the law of the environment ${\bf a}$. The proofs are based on the results found in \cite{Armstrong2019} and \cite{gloria2014optimal}.

math.PR↗

arXiv2023

Fractional Edgeworth expansions for one-dimensional heavy-tailed random variables and applications

In this article, we study a class of lattice random variables in the domain of attraction of an $α$-stable random variable with index $α\in (0,2)$ which satisfy a truncated fractional Edgeworth expansion. Our results include studying the class of such fractional Edgeworth expansions under simple operations, providing concrete examples; sharp rates of convergence to an $α$-stable distribution in a local central limit theorem; Green's function expansions; and finally fluctuations of a class of discrete stochastic PDE's driven by the heavy-tailed random walks belonging to the class of fractional Edgeworth expansions.

math.PR↗

arXiv2023

Random field induced order in two dimensions

In this article we prove that a classical $XY$ model subjected to weak i.i.d. random field pointing in a fixed direction exhibits residual magnetic order in $\mathbb{Z}^2$ and aligns perpendicular to the random field direction. The paper is a sequel to \cite{NC} where the three-dimensional case was treated. Our approach is based on a multi-scale Peierls contour argument developed in \cite{NC}. On the microscopic scale we extract energetic costs from the occurrence of contours, which themselves are defined on a macroscopic scale. The technical challenges in $\mathbb{Z}^2$ stem from difficulties controlling the size and roughness of the fluctuation fields which model the short length-scale oscillations of near-optimizers of the random field Hamiltonian.

math.PR↗

arXiv2022

Positive reinforced generalized time-dependent Pólya urns via stochastic approximation

Consider a generalized time-dependent Pólya urn process defined as follows. Let $d\in \mathbb{N}$ be the number of urns/colors. At each time $n$, we distribute $σ_n$ balls randomly to the $d$ urns, proportionally to $f$, where $f$ is a valid reinforcement function. We consider a general class of positive reinforcement functions $\mathcal{R}$ assuming some monotonicity and growth condition. The class $\mathcal{R}$ includes convex functions and the classical case $f(x)=x^α$, $α>1$. The novelty of the paper lies in extending stochastic approximation techniques to the $d$-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls any more.

math.PR↗

arXiv2019

Mean-field avalanche size exponent for sandpiles on Galton-Watson trees

We show that in abelian sandpiles on infinite Galton-Watson trees, the probability that the total avalanche has more than $t$ topplings decays as $t^{-1/2}$. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (2003), that was previously used by Lyons, Morris and Schramm (2008) to study uniform spanning forests on $\mathbb{Z}^d$, $d\geq 3$, and other transient graphs.

math.PR↗

arXiv2019

Non-criticality criteria for Abelian sandpile models with sources and sinks

We prove that the Abelian sandpile model on a random binary and binomial tree, as introduced in \cite{rrs}, is not critical for all branching probabilities $p<1$; by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps, we obtain some more information about the exponential tail of the avalanche radius. Next we study the sandpile model on $\mathbb{Z}^d$ with some additional dissipative sites: we provide examples and sufficient conditions for non-criticality; we also make a connection with the parabolic Anderson model. Finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non-criticality in the presence of a finite number of sources, using a connection with the homogeneous pinning model.

math.PR↗

arXiv2019

Scaling limits in divisible sandpiles: a Fourier multiplier approach

In this paper we complete the investigation of scaling limits of the odometer in divisible sandpiles on $d$-dimensional tori generalising the works Chiarini et al. (2018), Cipriani et al. (2017, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalised Gaussian fields in the limit by specifying the Fourier multiplier of their covariance kernel. In particular, using a Fourier multiplier approach, we can recover fractional Gaussian fields of the form $(-Δ)^{-(1+s)} W$ for $s>0$ and $W$ a spatial white noise on the $d$-dimensional unit torus.

math.PR↗

arXiv2019

Odometers of Divisible Sandpile Models: Scaling Limits, iDLA and Obstacle Problems. A Survey

The divisible sandpile model is a fixed-energy continuous counterpart of the Abelian sandpile model. We start with a random initial configuration and redistribute mass deterministically. Under certain conditions the sandpile will stabilize. The associated odometer function describes the amount of mass emitted from each vertex during stabilization. In this survey we describe recent scaling limit results of the odometer function depending on different initial configurations and redistribution rules. Moreover we review connections to the obstacle problem from potential theory, including the connection between odometers and limiting shapes of growth models such as iDLA. Finally we state some open problems.

math.PR↗