arXiv subjects

Balint Toth

Publications and source records attributed to Balint Toth.

At least 19 recordsLinked to original sources

arXiv2024

Diffusion of the random Lorentz process in a magnetic field

Consider the motion of a charged, point particle moving in the complement of a Poisson distribution of hard sphere scatterers in two dimensions under the effect of a fixed magnetic field. Building on, and extending a coupling method established by the authors, we show that this 'magnetic Lorentz gas' satisfies an invariance principle in an intermediate scaling limit. That is, we apply the low-density (Boltzmann-Grad) limit and simultaneously take the limit as time goes to infinity, then prove convergence of the rescaled trajectory to a Brownian motion in this limit.

math.PR↗

arXiv2024

CH(A) Radical Formation in Coulomb Explosion from Butane Seeded Plasma Generated with Chirp-Controlled Ultrashort Laser Pulses

We experimentally studied the formation of CH(A) radicals in butane seeded plasma generated with chirp-controlled ultrashort laser pulses ($\sim 760 \, \mu \text{J}/\text{pulse}$, 890 nm, 1 kHz, 8 fs). The focused beam with high peak intensity ($\sim 10^{14} - 10^{16} \, \text{W/cm}^2$) caused Coulomb explosion (CE). The time dependent emission spectra were observed with the Fourier-transform Visible spectroscopy (FTVis) step-scan method. The average signal intensity decreased with the chirp in the Ar$^+$ > C$_2$ > H-$\alpha$ $\sim$ CH(A) order with a plateau for CH(A) in the $-200$ to $-100 \, \text{fs}^2$ range. The short rise time of the CH(A) emission signal, the monoexponential emission decay, and the nearly constant rotational and vibrational temperatures of the CH(A) radicals ($\sim 3000 \, \text{K}$ and $\sim 3800 \, \text{K}$) all support their formation as a primary product. Our TDDFT calculations predict that CH and many other fragments can be formed beyond CE at $\sim 7 \times 10^{14} \, \text{W/cm}^2$ intensity. The average charge of CH (+0.6) and its relative abundance (0.5\%) support the formation of detectable CH(A) within 120 fs.

physics.chem-ph↗

arXiv2018

Diffusion and superdiffusion in lattice models of colliding particles with stored momentum

We introduce two discrete models of a collection of colliding particles with stored momentum and study the asymptotic growth of the mean-square displacement of an active particle. We prove that the models are superdiffusive in one dimension (with power law correction) and diffusive in three and higher dimensions. In two dimensions we demonstrate superdiffusivity (with logarithmic correction) for certain anisotropic initial conditions.

math.PR↗

arXiv2015

Invariance principle for the periodic Lorentz gas in the Boltzmann-Grad limit

In earlier work we showed that the particle displacement for the multidimensional periodic Lorentz gas, in the limit of low scatterer density (Boltzmann-Grad limit), satisfies a central limit theorem with superdiffusive scaling. The present paper extends this result to a functional central limit theorem, i.e., the weak convergence of the particle trajectory to Brownian motion.

math-ph↗

arXiv2014

Superdiffusion in the periodic Lorentz gas

We prove a superdiffusive central limit theorem for the displacement of a test particle in the periodic Lorentz gas in the limit of large times $t$ and low scatterer densities (Boltzmann-Grad limit). The normalization factor is $\sqrt{t\log t}$, where $t$ is measured in units of the mean collision time. This result holds in any dimension and for a general class of finite-range scattering potentials. We also establish the corresponding invariance principle, i.e., the weak convergence of the particle dynamics to Brownian motion.

math-ph↗

arXiv2013

Overlapping modularity at the critical point of k-clique percolation

One of the most remarkable social phenomena is the formation of communities in social networks corresponding to families, friendship circles, work teams, etc. Since people usually belong to several different communities at the same time, the induced overlaps result in an extremely complicated web of the communities themselves. Thus, uncovering the intricate community structure of social networks is a non-trivial task with great potential for practical applications, gaining a notable interest in the recent years. The Clique Percolation Method (CPM) is one of the earliest overlapping community finding methods, which was already used in the analysis of several different social networks. In this approach the communities correspond to k-clique percolation clusters, and the general heuristic for setting the parameters of the method is to tune the system just below the critical point of k-clique percolation. However, this rule is based on simple physical principles and its validity was never subject to quantitative analysis. Here we examine the quality of the partitioning in the vicinity of the critical point using recently introduced overlapping modularity measures. According to our results on real social- and other networks, the overlapping modularities show a maximum close to the critical point, justifying the original criteria for the optimal parameter settings.

physics.soc-ph↗

arXiv2012

Comment on a theorem of M. Maxwell and M. Woodroofe

We present a direct derivation of the theorem of M. Maxwell and M. Woodroofe (Ann. Probab. 28 (2000) 713-724), on martingale approximation of additive functionals of stationary Markov processes, from the non-reversible version of the Kipnis-Varadhan theorem.

math.PR↗

arXiv2012

Relaxed sector condition

In this note we present a new sufficient condition which guarantees martingale approximation and central limit theorem a la Kipnis-Varadhan to hold for additive functionals of Markov processes. This condition which we call the relaxed sector condition (RSC) generalizes the strong sector condition (SSC) and the graded sector condition (GSC) in the case when the self-adjoint part of the infinitesimal generator acts diagonally in the grading. The main advantage being that the proof of the GSC in this case is more transparent and less computational than in the original versions. We also hope that the RSC may have direct applications where the earlier sector conditions don't apply. So far we don't have convincing examples in this direction.

math.PR↗

arXiv2011

Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction

We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle compared to the center of mass of the system. The rates are higher for those left behind, and lower for those ahead of the center of mass, providing an attractive interaction keeping the particles together. We prove that in the fluid limit, as the number of particles goes to infinity, the evolution of the system is described by a mean field equation that exhibits traveling wave solutions. A connection to extreme value statistics is also provided.

math.PR↗

arXiv2010

Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d=2

We consider two models of random diffusion in random environment in two dimensions. The first example is the self-repelling Brownian polymer, this describes a diffusion pushed by the negative gradient of its own occupation time measure (local time). The second example is a diffusion in a fixed random environment given by the curl of massless Gaussian free field. In both cases we show that the process is superdiffusive: the variance grows faster than linearly with time. We give lower and upper bounds of the order of t log log t, respectively, t log t. We also present computations for an anisotropic version of the self-repelling Brownian polymer where we give lower and upper bounds of t (log t)^{1/2}, respectively, t log t. The bounds are given in the sense of Laplace transforms, the proofs rely on the resolvent method. The true order of the variance for these processes is expected to be t (log t)^{1/2} for the isotropic and t (log t)^{2/3} for the non-isotropic case. In the appendix we present a non-rigorous derivation of these scaling exponents.

math.PR↗

arXiv2010

Some locally self-interacting walks on the integers

We study certain self-interacting walks on the set of integers, that choose to jump to the right or to the left randomly but influenced by the number of times they have previously jumped along the edges in the finite neighbourhood of their current position (in the present paper, typically, we will discuss the case where one considers the neighbouring edges and the next-to-neighbouring edges). We survey a variety of possible behaviours, including some where the walk is eventually confined to an interval of large length. We also focus on certain "asymmetric" drifts, where we prove that with positive probability, the walks behave deterministically on large scale and move like a constant times the square root of time, or like a constant times the logarithm of time.

math.PR↗

arXiv2010

Stuck Walks

We investigate the asymptotic behaviour of a class of self-interacting nearest neighbour random walks on the one-dimensional integer lattice which are pushed by a particular linear combination of their own local time on edges in the neighbourhood of their current position. We prove that in a range of the relevant parameter of the model such random walkers can be eventually confined to a finite interval of length depending on the parameter value. The phenomenon arises as a result of competing self-attracting and self-repelling effects where in the named parameter range the former wins.

math.PR↗

arXiv2010

On the range, local times and periodicity of random walk on an interval

The range, local times, and periodicity of symmetric, weakly asymmetric and asymmetric random walks at the time of exit from a strip with $N$ locations are considered. Several results on asymptotic distributions are obtained.

math.PR↗

arXiv2010

Diffusive limits for "true" (or myopic) self-avoiding random walks and self-repellent Brownian polymers in d >= 3

The problems considered in the present paper have their roots in two different cultures. The 'true' (or myopic) self-avoiding walk model (TSAW) was introduced in the physics literature by Amit, Parisi and Peliti. This is a nearest neighbor non-Markovian random walk in Z^d which prefers to jump to those neighbors which were less visited in the past. The self-repelling Brownian polymer model (SRBP), initiated in the probabilistic literature by Durrett and Rogers (independently of the physics community), is the continuous space-time counterpart: a diffusion in R^d pushed by the negative gradient of the (mollified) occupation time measure of the process. In both cases, similar long memory effects are caused by a pathwise self-repellency of the trajectories due to a push by the negative gradient of (softened) local time. We investigate the asymptotic behaviour of TSAW and SRBP in the non-recurrent dimensions. First, we identify a natural stationary (in time) and ergodic distribution of the environment (the local time profile) as seen from the moving particle. The main results are diffusive limits. In the case of TSAW, for a wide class of self-interaction functions, we establish diffusive lower and upper bounds for the displacement and for a particular, more restricted class of interactions, we prove full CLT for the finite dimensional distributions of the displacement. In the case of SRBP, we prove full CLT without restrictions on the interaction functions. These results settle part of the conjectures, based on non-rigorous renormalization group arguments (equally 'valid' for the TSAW and SRBP cases). The proof of the CLT follows the non-reversible version of Kipnis-Varadhan theory. On the way to the proof, we slightly weaken the so-called graded sector condition.

math.PR↗

arXiv2010

Diffusive limit for the myopic (or "true") self-avoiding random walk in three and more dimension

The myopic (or `true') self-avoiding walk model (MSAW) was introduced in the physics literature by Amit, Parisi and Peliti (1983). It is a random motion in Z^d pushed towards domains less visited in the past by a kind of negative gradient of the occupation time measure. We investigate the asymptotic behaviour of MSAW in the non-recurrent dimensions. For a wide class of self-interaction functions, we identify a natural stationary (in time) and ergodic distribution of the environment (the local time profile) as seen from the moving particle and we establish diffusive lower and upper bounds for the displacement of the random walk. For a particular, more restricted class of interactions, we prove full CLT for the finite dimensional distributions of the displacement. This result settles part of the conjectures (based on non-rigorous renormalization group arguments) in Amit, Parisi and Peliti (1983). The proof of the CLT follows the non-reversible version of Kipnis-Varadhan-theory. On the way to the proof we slightly weaken the so-called graded sector condition (that is: we slightly enhance the corresponding statement).

math.PR↗

arXiv2009

Diffusive limit for self-repelling Brownian polymers in three and more dimensions

The self-repelling Brownian polymer model (SRBP) initiated by Durrett and Rogers in [Durrett-Rogers (1992)] is the continuous space-time counterpart of the myopic (or 'true') self-avoiding walk model (MSAW) introduced in the physics literature by Amit, Parisi and Peliti in [Amit-Parisi-Peliti (1983)]. In both cases, a random motion in space is pushed towards domains less visited in the past by a kind of negative gradient of the occupation time measure. We investigate the asymptotic behaviour of SRBP in the non-recurrent dimensions. First, extending 1-dimensional results from [Tarres-Toth-Valko (2009)], we identify a natural stationary (in time) and ergodic distribution of the environment (essentially, smeared-out occupation time measure of the process), as seen from the moving particle. As main result we prove that in three and more dimensions, in this stationary (and ergodic) regime, the displacement of the moving particle scales diffusively and its finite dimensional distributions converge to those of a Wiener process. This result settles part of the conjectures (based on non-rigorous renormalization group arguments) in [Amit-Parisi-Peliti (1983)]. The main tool is the non-reversible version of the Kipnis--Varadhan-type CLT for additive functionals of ergodic Markov processes and the graded sector condition of [Sethuraman-Varadhan-Yau (2000)].

math.PR↗

arXiv2009

Continuous time `true' self-avoiding random walk on Z

We consider the continuous time version of the `true' or `myopic' self-avoiding random walk with site repulsion in 1d. The Ray-Knight-type method which was applied to the discrete time and edge repulsion case, is applicable to this model with some modifications. We present a limit theorem for the local time of the walk and a local limit theorem for the displacement.

math.PR↗

arXiv2009

Erdos-Renyi random graphs + forest fires = self-organized criticality

We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings the system sticks to a permanent critical state.

math.PR↗