arXiv subjects

Benedek Valko

Publications and source records attributed to Benedek Valko.

15 recordsLinked to original sources

arXiv2020

On a random entanglement problem

We study a model for the entanglement of a two-dimensional reflecting Brownian motion in a bounded region divided into two halves by a wall with three or more small windows. We map the Brownian motion into a Markov Chain on the fundamental groupoid of the region. We quantify entanglement of the path with the length of the appropriate element in this groupoid. Our main results are a law of large numbers and a central limit theorem for this quantity. The constants appearing in the limit theorems are expressed in terms of a coupled system of quadratic equations.

math.PR↗

arXiv2014

Matrix Dufresne Identity

We prove a version of the classical Dufresne identity for matrix processes. In particular, we show that the inverse Wishart laws on the space of positive definite r x r matrices can be realized by the infinite time horizon integral of M_t times its transpose in which t -> M_t is a drifted Brownian motion on the general linear group. This solves a problem in the study of spiked random matrix ensembles which served as the original motivation for this result. Various known extensions of the Dufresne identity (and their applications) are also shown to have analogs in this setting. For example, we identify matrix valued diffusions built from M_t which generalize in a natural way the scalar processes figuring into the geometric Levy and Pitman theorems of Matsumoto and Yor.

math.PR↗

arXiv2012

Random Schrodinger operators on long boxes, noise explosion and the GOE

It is conjectured that the eigenvalues of random Schrodinger operators at the localization transition in dimensions d>=2 behave like the eigenvalues of the Gaussian Orthogonal Ensemble (GOE). We show that there are sequences of n by m boxes with 1<<m<<n so that the eigenvalues in low disorder converge to Sine1, the limiting eigenvalue process of the GOE. For the GOE case, this is the first example where Wigner's famous prediction is proven rigorously: we exhibit a complex system whose eigenvalues behave like those of random matrices.

math.PR↗

arXiv2011

The scaling limit of the critical one-dimensional random Schrodinger operator

We consider two models of one-dimensional discrete random Schrodinger operators (H_n ψ)_l =ψ_{l-1}+ψ_{l +1}+v_l ψ_l, ψ_0=ψ_{n+1}=0 in the cases v_k=σω_k/\sqrt{n} and v_k=σω_k/ \sqrt{k}. Here ω_k are independent random variables with mean 0 and variance 1. We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem. In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the beta-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.

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

Interpolation of Gibbs measures with White Noise for Hamiltonian PDE

We consider the family of interpolation measures of Gibbs measures and white noise given by $$dQ_{0,\b}^{(p)} = Z_\b^{-1} \ind_{{\int_{\T} u^2\le K\b^{-1/2}\}} e^{-\int_{\T} u^2 +\b \int u^p} dP_{0,\b}$$ where $P_{0, \b}$ is the Wiener measure on the circle, with variance $β^{-1}$, conditioned to have mean zero. It is shown that as $β\to 0$, $Q_0^β$ converges weakly to mean zero Gaussian white noise $Q_0$. As an application, we present a straightforward proof that $Q_0$ is invariant for the Kortweg-de Vries equation (KdV). This weak convergence also shows that the white noise is a weak limit of invariant measures for the modified KdV and the cubic nonlinear Schrödinger equations.

math.PR↗

arXiv2009

Continuum limits of random matrices and the Brownian carousel

We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine_beta, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine_beta is continuous in the gap size and $β$, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at beta=2.

math.PR↗

arXiv2007

A note on the diffusivity of finite-range asymmetric exclusion processes on Z

The diffusivity $D(t)$ of finite-range asymmetric exclusion processes on $\mathbb Z$ with non-zero drift is expected to be of order $t^{1/3}$. Seppäläinen and Balázs recently proved this conjecture for the nearest neighbor case. We extend their results to general finite range exclusion by proving that the Laplace transform of the diffusivity is of the conjectured order. We also obtain a pointwise upper bound for $D(t)$ the correct order.

math.PR↗

arXiv2006

KdV Preserves White Noise

It is shown that white noise is an invariant measure for the Korteweg-deVries equation on $\mathbb T$. This is a consequence of recent results of Kappeler and Topalov establishing the well-posedness of the equation on appropriate negative Sobolev spaces, together with a result of Cambronero and McKean that white noise is the image under the Miura transform (Ricatti map) of the (weighted) Gibbs measure for the modified KdV equation, proven to be invariant for that equation by Bourgain.

math.AP↗

arXiv2006

$t^{1/3}$ Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on $\mathbb Z$

We consider finite-range asymmetric exclusion processes on $\mathbb Z$ with non-zero drift. The diffusivity $D(t)$ is expected to be of ${\mathcal O}(t^{1/3})$. We prove that $D(t)\ge Ct^{1/3}$ in the weak (Tauberian) sense that $\int_0^\infty e^{-λt}tD(t)dt \ge Cλ^{-7/3}$ as $λ\to 0$. The proof employs the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process, for which the result is a consequence of the scaling limit for the two-point function recently obtained by Ferrari and Spohn. In the nearest neighbor case, we show further that $tD(t)$ is monotone, and hence we can conclude that $D(t)\ge Ct^{1/3}(\log t)^{-7/3}$ in the usual sense.

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↗

arXiv2004

Hydrodynamic limit for perturbation of a hyperbolic equilibrium point in two-component systems

We consider one-dimensional, locally finite interacting particle systems with two conservation laws. The models have a family of stationary measures with product structure and we assume the existence of a uniform bound on the inverse of the spectral gap which is quadratic in the size of the system. Under Eulerian scaling the hydrodynamic limit for the macroscopic density profiles leads to a two-component system of conservation laws. The resulting pde is hyperbolic inside the physical domain of the macroscopic densities, with possible loss of hyperbolicity at the boundary. We investigate the propagation of small perturbations around a \emph{hyperbolic} equilibrium point. We prove that the perturbations essentially evolve according to two \emph{decoupled} Burgers equations. The scaling is not Eulerian: if the lattice constant is $n^{-1}$, the perturbations are of order $n^{-β}$ then time is speeded up by $n^{1+\b}$. Our derivation holds for $0<β< \frac15$. The proof relies on Yau's relative entropy method, thus it applies only in the regime of smooth solutions.

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↗

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↗