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Balint Toth

Publications and source records attributed to Balint Toth.

31 records · Page 2Linked to original sources
arXiv2008

Self-repelling random walk with directed edges on Z

We consider a variant of self-repelling random walk on the integer lattice Z where the self-repellence is defined in terms of the local time on oriented edges. The long-time asymptotic scaling of this walk is surprisingly different from the asymptotics of the similar process with self-repellence defined in terms of local time on unoriented edges. We prove limit theorems for the local time process and for the position of the random walker. The main ingredient is a Ray-Knight-type of approach. At the end of the paper, we also present some computer simulations which show the strange scaling behaviour of the walk considered.

math.PR↗
arXiv2007

Skorohod-reflection of Brownian Paths and BES^3

Let B(t), X(t) and Y(t) be independent standard 1d Borwnian motions. Define X^+(t) and Y^-(t) as the trajectories of the processes X(t) and Y(t) pushed upwards and, respectively, downwards by B(t), according to Skorohod-reflection. In a recent paper, Jon Warren proves inter alia that Z(t):= X^+(t)-Y^-(t) is a three-dimensional Bessel-process. In this note, we present an alternative, elementary proof of this fact.

math.PR↗
arXiv2007

Modeling the Epps effect of cross correlations in asset prices

We review the decomposition method of stock return cross-correlations, presented previously for studying the dependence of the correlation coefficient on the resolution of data (Epps effect). Through a toy model of random walk/Brownian motion and memoryless renewal process (i.e. Poisson point process) of observation times we show that in case of analytical treatability, by decomposing the correlations we get the exact result for the frequency dependence. We also demonstrate that our approach produces reasonable fitting of the dependence of correlations on the data resolution in case of empirical data. Our results indicate that the Epps phenomenon is a product of the finite time decay of lagged correlations of high resolution data, which does not scale with activity. The characteristic time is due to a human time scale, the time needed to react to news.

q-fin.ST↗
arXiv2006

On the zero mass limit of tagged particle diffusion in the 1-d Rayleigh-gas

We consider the M -> 0 limit for tagged particle diffusion in a 1-dimensional Rayleigh-gas, studied originaly by Sinai and Soloveichik (1986), respectively by Szasz and Toth (1986). In this limit we derive a new type of model for tagged paricle diffusion, with Calogero-Moser-Sutherland (i.e. inverse quadratic) interaction potential between the two central particles. Computer simulations on this new model reproduce exactly the numerical value of the limiting variance obtained by Boldrighini, Frigio and Tognetti (2002).

math.PR↗
arXiv2006

Random Trees and General Branching Processes

We consider a model of random tree growth, where at each time unit a new vertex is added and attached to an already existing vertex chosen at random. The probability with which a vertex with degree $k$ is chosen is proportional to $w(k)$, where the weight function $w$ is the parameter of the model. In the papers of B. Bollobas, O. Riordan, J. Spencer, G. Tusnady, and, independently, Mori, the asymptotic degree distribution is obtained for a model that is equivalent to the special case of ours, when the weight function is linear. The proof therein strongly relies on the linear choice of $w$. We give the asymptotical degree distribution for a wide range of weight functions. Moreover, we provide the asymptotic distribution of the tree itself as seen from a randomly selected vertex. The latter approach gives greater insight to the limiting structure of the tree. Our proof relies on the fact that considering the evolution of the random tree in continuous time, the process may be viewed as a general branching process, this way classical results can be applied.

math.PR↗
arXiv2003

Perturbation of singular equilibria of hyperbolic two-component systems: a universal hydrodynamic limit

We consider one-dimensional, locally finite interacting particle systems with two conservation laws which under Eulerian hydrodynamic limit lead to two-by-two systems of conservation laws: \pt ρ+\px Ψ(ρ, u)=0 \pt u+\px Φ(ρ,u)=0, with $(ρ,u)\in{\cal D}\subset\R^2$, where ${\cal D}$ is a convex compact polygon in $\R^2$. The system is typically strictly hyperbolic in the interior of ${\cal D}$ with possible non-hyperbolic degeneracies on the boundary $\partial {\cal D}$. We consider the case of isolated singular (i.e. non hyperbolic) point on the interior of one of the edges of ${\cal D}$, call it $(ρ_0,u_0)=(0,0)$ and assume ${\cal D}\subset\{ρ\ge0\}$. This can be achieved by a linear transformation of the conserved quantities. We investigate the propagation of small nonequilibrium perturbations of the steady state of the microscopic interacting particle system, corresponding to the densities $(ρ_0,u_0)$ of the conserved quantities. We prove that for a very rich class of systems, under proper hydrodynamic limit the propagation of these small perturbations are \emph{universally} driven by the two-by-two system \ptρ+ \px\big(ρu\big)=0 \pt u + \px\big(ρ+ γu^2\big) =0 where the parameter $γ:=\frac12 Φ_{uu}(ρ_0,u_0)$ (with a proper choice of space and time scale) is the only trace of the microscopic structure. The proof is valid for the cases with $γ>1$. [truncated]

math.PR↗
arXiv2003

Derivation of the Leroux system as the hydrodynamic limit of a two-component lattice gas

The long time behavior of a couple of interacting asymmetric exclusion processes of opposite velocities is investigated in one space dimension. We do not allow two particles at the same site, and a collision effect (exchange) takes place when particles of opposite velocities meet at neighboring sites. There are two conserved quantities, and the model admits hyperbolic (Euler) scaling; the hydrodynamic limit results in the classical Leroux system of conservation laws, \emph{even beyond the appearence of shocks}. Actually, we prove convergence to the set of entropy solutions, the question of uniqueness is left open. To control rapid oscillations of Lax entropies via logarithmic Sobolev inequality estimates, the symmetric part of the process is speeded up in a suitable way, thus a slowly vanishing viscosity is obtained at the macroscopic level. Following earlier work of the first author the stochastic version of Tartar--Murat theory of compensated compactness is extended to two-component stochastic models.

math.PR↗
arXiv2002

Onsager relations and Eulerian hydrodynamics for systems with several conservation laws

We present the derivation of the hydrodynamic limit under Eulerian scaling for a general class of one-dimensional interacting particle systems with two or more conservation laws. Following Yau's relative entropy method it turns out that in case of more than one conservation laws, in order that the system exhibit hydrodynamic behaviour, some particular identities reminiscent of Onsager's reciprocity relations must hold. We check validity of these identities for a wide class of models. It also follows that, as a general rule, the equilibrium thermodynamic entropy (as function of the densities of the conserved variables) is a globally convex Lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit. The Onsager relations arising in this context and its consequences seem to be novel. As concrete examples we also present a number of models modeling deposition (or domain growth) phenomena.

math.PR↗
arXiv2002

Between equilibrium fluctuations and Eulerian scaling: Perturbation of equilibrium for a class of deposition models

We investigate propagation of perturbations of equilibrium states for a wide class of 1D interacting particle systems. The class of systems considered incorporates zero range, $K$-exclusion, mysanthropic, `bricklayers' models, and much more. We do not assume attractivity of the interactions. We apply Yau's relative entropy method rather than coupling arguments. The result is \emph{partial extension} of T. Seppäläinen's recent paper. For $0<β<1/5$ fixed, we prove that, rescaling microscopic space and time by $N$, respectively $N^{1+β}$, the macroscopic evolution of perturbations of microscopic order $N^{-β}$ of the equilibrium states is governed by Burgers' equation. The same statement should hold for $0<β<1/2$ as in Seppäläinen's cited paper, but our method does not seem to work for $β\ge1/5$.

math.PR↗
arXiv2000

Hydrodynamic equation for a deposition model

We show that the two-component system of hyperbolic conservation laws $\partial_t ρ+ \partial_x (ρu) =0 = \partial_t u + \partial_x ρ$ appears naturally in the formally computed hydrodynamic limit of some randomly growing interface models, and we study some properties of this system. We show that the two-component system of hyperbolic conservation laws $\partial_t ρ+ \partial_x (ρu) =0 = \partial_t u + \partial_x ρ$ appears naturally in the formally computed hydrodynamic limit of some randomly growing interface models, and we study some properties of this system.

math.PR↗
arXiv2000

A signal-recovery system: asymptotic properties and construction of an infinite volume limit

We consider a linear sequence of `nodes', each of which can be in state 0 (`off') or 1 (`on'). Signals from outside are sent to the rightmost node and travel instantaneously as far as possible to the left along nodes which are `on'. These nodes are immediately switched off, and become on again after a recovery time. The recovery times are independent exponentially distributed random variables. We present properties for finite systems and use some of these properties to construct an infinite-volume extension, with signals `coming from infinity'. This construction is related to a question by D. Aldous and we expect that it sheds some light on, and stimulates further investigation of, that question.

math.PR↗
arXiv2000

Favourite sites of simple random walk

We survey the current status of the list of questions related to the favourite (or: most visited) sites of simple random walk on Z, raised by Pal Erdos and Pal Revesz in the early eighties.

math.PR↗