arXiv subjects

Bálint Tóth

Publications and source records attributed to Bálint Tóth.

At least 19 recordsLinked to original sources

arXiv2026

Microscopic sorting in a multispecies exclusion process

We study a one-dimensional multispecies TASEP model with each particle type having an associated speed parameter and faster particles overtaking slower ones with the rate being the positive part of their speed difference. After expressing the sorting time of a finite system given a reverse-speed initial condition as the maximum of sorting times for all different fast-slow cuts, we prove that forbidding fast-fast and slow-slow overtakes does not alter the law of the sorting time. This allows for an LPP formulation and an analytic expression of the sorting time.

math.PR↗

arXiv2026

Central limit theorem for random walk in degenerate divergence-free random environment: $\mathcal H_{-1}$ reloaded with relaxed ellipticity

This paper enhances the result of the work [G. Kozma, B. Tóth, Ann. Probab. vol. 45 (2017) 4307-4347] . We prove the central limit theorem (in probability w.r.t. the environment) for the displacement of a random walker in divergence-free (or, doubly stochastic) random environment, with substantially relaxed ellipticity assumptions. Integrability of the reciprocal of the symmetric part of the jump rates is only assumed (rather than their boundedness, as in previous works on this type of RWRE). Relaxing ellipticity involves substantial changes in the proof, making it conceptually elementary in the sense that it does not rely on Nash's inequality in any disguise.

math.PR↗

arXiv2025

Finetuning and Quantization of EEG-Based Foundational BioSignal Models on ECG and PPG Data for Blood Pressure Estimation

Blood pressure (BP) is a key indicator of cardiovascular health. As hypertension remains a global cause of morbidity and mortality, accurate, continuous, and non-invasive BP monitoring is therefore of paramount importance. Photoplethysmography (PPG) and electrocardiography (ECG) can potentially enable continuous BP monitoring, yet training accurate and robust machine learning (ML) models remains challenging due to variability in data quality and patient-specific factors. Recently, multiple research groups explored Electroencephalographic (EEG)--based foundation models and demonstrated their exceptional ability to learn rich temporal resolution. Considering the morphological similarities between different biosignals, the question arises of whether a model pre-trained on one modality can effectively be exploited to improve the accuracy of a different signal type. In this work, we take an initial step towards generalized biosignal foundation models by investigating whether model representations learned from abundant EEG data can effectively be transferred to ECG/PPG data solely with fine-tuning, without the need for large-scale additional pre-training, for the BP estimation task. Evaluations on the MIMIC-III and VitalDB datasets demonstrate that our approach achieves near state-of-the-art accuracy for diastolic BP (mean absolute error of 1.57 mmHg) and surpasses by 1.5x the accuracy of prior works for systolic BP (mean absolute error 2.72 mmHg). Additionally, we perform dynamic INT8 quantization, reducing the smallest model size by over 3.5x (from 13.73 MB down to 3.83 MB) while preserving performance, thereby enabling unobtrusive, real-time BP monitoring on resource-constrained wearable devices.

eess.SP↗

arXiv2025

Semi-Quenched Invariance Principle for the Random Lorentz Gas -- Beyond the Boltzmann-Grad Limit

By synchronously coupling multiple Lorentz trajectories exploring the same environment consisting of randomly placed scatterers in R^3 we upgrade the annealed invariance principle proved in [C. Lutsko, B. Tóth, Commun. Math. Phys. 379 589-632 (2020)] to quenched setting (that is, valid for almost all realizations of the environment) along sufficiently fast extractor sequences.

math.PR↗

arXiv2025

Central limit theorem for random walks in divergence free random drift field -- revisited

In [Kozma-Toth, Ann. Probab. v 45, pp 4307-4347 (2017)] the weak CLT was established for random walks in doubly stochastic (or, divergence-free) random environments, under the following conditions: 1. Strict ellipticity assumed for the symmetric part of the drift field. 2. $H_{-1}$ assumed for the antisymmetric part of the drift field. The proof relied on a martingale approximation (a la Kipnis-Varadhan) adapted to the non-self-adjoint and non-sectorial nature of the problem. The two substantial technical components of the proof were: 1. A functional analytic statement about the unbounded operator formally written as $|L+L^*|^{-1/2}(L-L^*)|L+L^*|^{-1/2}$, where $L$ is the infinitesimal generator of the environment process, as seen from the position of the moving random walker. 2. A diagonal heat kernel upper bound which follows directly from Nash's inequality, or, alternatively, from the "evolving sets" arguments of [Morris-Peres, Probab. Theory Rel. Fields. v. 133 pp 245-266 (2005)] valid only under the assumed strict ellipticity. In this note we present a partly alternative proof of the same result which relies only on functional analytic arguments and not on the diagonal heat kernel upper bound provided by Nash's inequality. This alternative proof is relevant since it can be naturally extended to non-elliptic settings pushed to the optimum, which will be presented in a forthcoming paper. The goal of this note is to present the argument in its simplest and most transparent form.

math.PR↗

arXiv2020

Invariance Principle for the Random Lorentz Gas -- Beyond the Boltzmann-Grad Limit

We prove the invariance principle for a \emph{random Lorentz-gas} particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius $r$, placed according to a Poisson point process of density $\varrho$, in the limit $\varrho\to\infty$, $r\to0$, $\varrho r^{2}\to1$ up to time scales of order $T=o(r^{-2}{|\log r|}^{-2})$. To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (1969), Spohn (1978) and Boldrighini-Bunimovich-Sinai (1983). The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simultaneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling. Similar results have been earlier obtained for the weak coupling limit of classical and quantum random Lorentz gas, by Komorowski-Ryzhik (2006), respectively, Erd\H os-Salmhofer-Yau (2007). However, the following are substantial differences between our work and these ones: (1) The physical setting is different: low density rather than weak coupling. (2)The method of approach is different: probabilistic coupling rather than analytic/perturbative. (3) Due to (2), the time scale of validity of our diffusive approximation -- expressed in terms of the kinetic time scale -- is much longer and fully explicit.

math.PR↗

arXiv2020

Split-and-Merge in Stationary Random Stirring on Lattice Torus

We show that in any dimension $d\ge1$, the cycle-length process of stationary random stirring (or, random interchange) on the lattice torus converges to the canonical Markovian split-and-merge process with the invariant (and reversible) measure given by the Poisson-Dirichlet law $\mathsf{PD(1)}$, as the size of the system grows to infinity. In the case of transient dimensions, $d\ge 3$, the problem is motivated by attempts to understand the onset of long range order in quantum Heisenberg models via random loop representations of the latter.

math.PR↗

arXiv2019

Invariance Principle for the Random Wind-Tree Process

Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes - the random wind-tree model introduced in Ehrenfest-Ehrenfest (1912). We show that, in the joint Boltzmann-Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to a Brownian motion in a particular scaling limit. In a previous paper (2019) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, the same strategy with some modification can be used to prove an invariance principle for the random wind-tree model.

math.PR↗

arXiv2018

Diffusive and Super-Diffusive Limits for Random Walks and Diffusions with Long Memory

We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ${\Bbb R}^d$ or ${\Bbb Z}^d$. The first class consists of random walks on ${\Bbb Z}^d$ in divergence-free random drift field, modelling the motion of a particle suspended in time-stationary incompressible turbulent flow. The second class consists of self-repelling random diffusions, where the diffusing particle is pushed by the negative gradient of its own occupation time measure towards regions less visited in the past. We establish normal diffusion (with square-root-of-time scaling and Gaussian limiting distribution) in three and more dimensions and typically anomalously fast diffusion in low dimensions (typically, one and two). Results are quoted from various papers published between 2012-2018, with some hints to the main ideas of the proofs. No technical details are presented here.

math.PR↗

arXiv2018

Random walk on the randomly-oriented Manhattan lattice

In the randomly-oriented Manhattan lattice, every line in $\mathbb{Z}^d$ is assigned a uniform random direction. We consider the directed graph whose vertex set is $\mathbb{Z}^d$ and whose edges connect nearest neighbours, but only in the direction fixed by the line orientations. Random walk on this directed graph chooses uniformly from the $d$ legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.

math.PR↗

arXiv2017

Quenched Central Limit Theorem for Random Walks in Doubly Stochastic Random Environment

We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger, $L^{2+\varepsilon}$ (rather than $L^2$) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the non-reversible, divergence-free drift case.

math.PR↗

arXiv2017

Central Limit Theorem for Random Walks in Doubly Stochastic Random Environment: $\mathcal{H}_{-1}$ Suffices

We prove a central limit theorem under diffusive scaling for the displacement of a random walk on ${\mathbb Z}^d$ in stationary and ergodic doubly stochastic random environment, under the $\mathcal{H}_{-1}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result of Komorowski, Landim and Olla (2012), where it is assumed that the stream tensor is in $\mathcal{L}^{\max\{2+δ, d\}}$, with $δ>0$. Our proof relies on an extension of the \emph{relaxed sector condition} of Horváth, Tóth and Vető (2012) and is technically rather simpler than existing earlier proofs of similar results by Oelschläger (1988) and Komorowski, Landim and Olla (2012)

math.PR↗

arXiv2016

Coexistence of shocks and rarefaction fans: complex phase diagram of a simple hyperbolic particle system

This paper investigates the non-equilibrium hydrodynamic behavior of a simple totally asymmetric interacting particle system of particles, antiparticles and holes on $\mathbb{Z}$. Rigorous hydrodynamic results apply to our model with a hydrodynamic flux that is exactly calculated and shown to change convexity in some region of the model parameters. We then characterize the entropy solutions of the hydrodynamic equation with step initial condition in this scenario which include various mixtures of rarefaction fans and shock waves. We highlight how the phase diagram of the model changes by varying the model parameters.

math.PR↗

arXiv2014

Cluster growth in the dynamical Erdős-Rényi process with forest fires

We investigate the growth of clusters within the forest fire model of Ráth and Tóth [22]. The model is a continuous-time Markov process, similar to the dynamical Erdős-Rényi random graph but with the addition of so-called fires. A vertex may catch fire at any moment and, when it does so, causes all edges within its connected cluster to burn, meaning that they instantaneously disappear. Each burned edge may later reappear. We give a precise description of the process $C_t$ of the size of the cluster of a tagged vertex, in the limit as the number of vertices in the model tends to infinity. We show that $C_t$ is an explosive branching process with a time-inhomogeneous offspring distribution and instantaneous return to $1$ on each explosion. Additionally, we show that the characteristic curves used to analyse the Smoluchowski-type coagulation equations associated to the model have a probabilistic interpretation in terms of the process $C_t$.

math.PR↗

arXiv2014

Central limit theorem for random walks in divergence-free random drift field: "H-minus-one" suffices

We prove central limit theorem under diffusive scaling for the displacement of a random walk on ${\mathbb Z}^d$ in stationary divergence-free random drift field, under the ${\mathcal H}_{-1}$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor be stationary and square integrable. This improves the best existing result of Komorowski, Landim and Olla (2012), where it is assumed that the stream tensor be in ${\mathcal L}^{\max\{2+δ,d\}}$, with $δ>0$. Our proof relies on the relaxed sector condition of Horváth, Tóth and Vető (2012), and is technically rather simpler than existing earlier proofs of similar results by Oelschläger (1988) and Komorowski, Landim, Olla (2012).

math.PR↗

arXiv2013

Modelling bursty time series

Many human-related activities show power-law decaying interevent time distribution with exponents usually varying between 1 and 2. We study a simple task-queuing model, which produces bursty time series due to the nontrivial dynamics of the task list. The model is characterised by a priority distribution as an input parameter, which describes the choice procedure from the list. We give exact results on the asymptotic behaviour of the model and we show that the interevent time distribution is power-law decaying for any kind of input distributions that remain normalizable in the infinite list limit, with exponents tunable between 1 and 2. The model satisfies a scaling law between the exponents of interevent time distribution (alpha) and autocorrelation function (beta): alpha + beta = 2. This law is general for renewal processes with power-law decaying interevent time distribution. We conclude that slowly decaying autocorrelation function indicates long-range dependency only if the scaling law is violated.

physics.soc-ph↗

arXiv2012

Marginal densities of the "true" self-repelling motion

Let X(t) be the true self-repelling motion (TSRM) constructed by B.T. and Wendelin Werner in 1998, L(t,x) its occupation time density (local time) and H(t):=L(t,X(t)) the height of the local time profile at the actual position of the motion. The joint distribution of (X(t),H(t)) was identified by B.T. in 1995 in somewhat implicit terms. Now we give explicit formulas for the densities of the marginal distributions of X(t) and H(t). The distribution of X(t) has a particularly surprising shape: It has a sharp local minimum with discontinuous derivative at 0. As a consequence we also obtain a precise version of the large deviation estimate of arXiv:1105.2948v3.

math.PR↗

arXiv2012

Diffusivity bounds for 1D Brownian polymers

We study the asymptotic behavior of a self-interacting one-dimensional Brownian polymer first introduced by Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337--349]. The polymer describes a stochastic process with a drift which is a certain average of its local time. We show that a smeared out version of the local time function as viewed from the actual position of the process is a Markov process in a suitably chosen function space, and that this process has a Gaussian stationary measure. As a first consequence, this enables us to partially prove a conjecture about the law of large numbers for the end-to-end displacement of the polymer formulated in Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337--349]. Next we give upper and lower bounds for the variance of the process under the stationary measure, in terms of the qualitative infrared behavior of the interaction function. In particular, we show that in the locally self-repelling case (when the process is essentially pushed by the negative gradient of its own local time) the process is super-diffusive.

math.PR↗