arXiv subjects

Eytan Katzav

Publications and source records attributed to Eytan Katzav.

At least 55 records · Page 3Linked to original sources

arXiv2017

The distribution of first hitting times of random walks on directed Erdős-Rényi networks

We present analytical results for the distribution of first hitting times of random walkers (RWs) on directed Erdős-Rényi (ER) networks. Starting from a random initial node, a random walker hops randomly along directed edges between adjacent nodes in the network. The path terminates either by the retracing scenario, when the walker enters a node which it has already visited before, or by the trapping scenario, when it becomes trapped in a dead-end node from which it cannot exit. The path length, namely the number of steps, $d$, pursued by the random walker from the initial node up to its termination, is called the first hitting time. Using recursion equations, we obtain analytical results for the tail distribution of first hitting times, $P(d>\ell)$. The distribution $P(d>\ell)$ can be expressed as a product of an exponential distribution and a Rayleigh distribution. We obtain expressions for the mean, median and standard deviation of this distribution in terms of the network size and its mean degree. We also calculate the distribution of last hitting times, namely the path lengths of self-avoiding walks on directed ER networks, which do not retrace their paths. The last hitting times are found to be much longer than the first hitting times. The results are compared to those obtained for undirected ER networks. It is found that the first hitting times of RWs in a directed ER network are much longer than in the corresponding undirected network. This is due to the fact that RWs on directed networks do not exhibit the backtracking scenario, which is a dominant termination mechanism of RWs on undirected networks. It is shown that our approach also applies to a broader class of networks, referred to as semi-ER networks, in which the distribution of in-degrees is Poisson, while the out-degrees may follow any desired distribution with the same mean as the in-degree distribution.

cond-mat.dis-nn↗

arXiv2017

The distribution of first hitting times of non-backtracking random walks on Erdős-Rényi networks

We present analytical results for the distribution of first hitting times of non-backtracking random walks on finite Erdős-Rényi networks of $N$ nodes. The walkers hop randomly between adjacent nodes on the network, without stepping back to the previous node, until they hit a node which they have already visited before or get trapped in a dead-end node. At this point, the path is terminated. The length, $d$, of the resulting path, is called the first hitting time. Using recursion equations, we obtain analytical results for the tail distribution of first hitting times, $P(d>\ell)$, $\ell=0,1,2,\dots$, of non-backtracking random walks starting from a random initial node. It turns out that the distribution $P(d>\ell)$ is given by a product of a discrete Rayleigh distribution and an exponential distribution. It is found that the paths of non-backtracking random walks, up to their termination at the first hitting time, are longer, on average, than those of the corresponding simple RWs. However, they are shorter than those of SAWs on the same network, which terminate at the last hitting time. We obtain analytical results for the probabilities, $p_{\rm ret}$ and $p_{\rm trap}$, that a path will terminate by retracing, namely stepping into an already visited node, or by trapping, namely entering a node of degree $k=1$, which has no exit link, respectively. It is shown that in dilute networks the dominant termination scenario is trapping while in dense networks most paths terminate by retracing. We obtain expressions for the conditional tail distributions of path lengths, $P(d>\ell|{\rm ret})$ and $P(d>\ell|\rm{trap})$, for those paths which terminate by retracing or by trapping, respectively. We also study a class of generalized non-backtracking random walk models which not only avoid the backtracking step into the previous node but avoid stepping into the last $S$ visited nodes.

cond-mat.dis-nn↗

arXiv2017

The distribution of first hitting times of random walks on Erdős-Rényi networks

Analytical results for the distribution of first hitting times of random walks on Erdős-Rényi networks are presented. Starting from a random initial node, a random walker hops between adjacent nodes until it hits a node which it has already visited before. At this point, the path terminates. The path length, namely the number of steps, $d$, pursued by the random walker from the initial node up to its termination is called the first hitting time or the first intersection length. Using recursion equations, we obtain analytical results for the tail distribution of the path lengths, $P(d > \ell)$. The results are found to be in excellent agreement with numerical simulations. It is found %turns out that the distribution $P(d > \ell)$ follows a product of an exponential distribution and a Rayleigh distribution. The mean, median and standard deviation of this distribution are also calculated, in terms of the network size and its mean degree. The termination of an RW path may take place either by backtracking to the previous node or by retracing of its path, namely stepping into a node which has been visited two or more time steps earlier. We obtain analytical results for the probabilities, $p_b$ and $p_r$, that the cause of termination will be backtracking or retracing, respectively. It is shown that in dilute networks the dominant termination scenario is backtracking while in dense networks most paths terminate by retracing. We also obtain expressions for the conditional distributions $P(d=\ell | b)$ and $P(d=\ell | r)$, for those paths which are terminated by backtracking or by retracing, respectively. These results provide useful insight into the general problem of survival analysis and the statistics of mortality rates when two or more termination scenarios coexist.

physics.soc-ph↗

arXiv2016

Yield statistics of interpolated superoscillations

Yield Optimized Interpolated Superoscillations (YOIS) have been recently introduced as a means for possibly making the use of the phenomenon of superoscillation practical. In this paper we study how good is a superoscillation that is not optimal. Namely, by how much is the yield decreased when the signal departs from the optimal one. We consider two situations. One is the case where the signal strictly obeys the interpolation requirement and the other is when that requirement is relaxed. In the latter case the yield can be increased at the expense of deterioration of signal quality. An important conclusion is that optimizing superoscillations may be challenging in terms of the precision needed, however, storing and using them is not at all that sensitive. This is of great importance in any physical system where noise and error are inevitable.

cond-mat.stat-mech↗

arXiv2016

The universal coefficient of the exact correlator of a large-$N$ matrix field theory

Exact expressions have been proposed for correlation functions of the large-$N$ (planar) limit of the $(1+1)$-dimensional ${\rm SU}(N)\times {\rm SU}(N)$ principal chiral sigma model. These were obtained with the form-factor bootstrap. The short-distance form of the two-point function of the scaling field $Φ(x)$, was found to be $N^{-1}\langle {\rm Tr}\,Φ(0)^{\dagger} Φ(x)\rangle=C_{2}\ln^{2}mx$, where $m$ is the mass gap, in agreement with the perturbative renormalization group. Here we point out that the universal coefficient $C_{2}$, is proportional to the mean first-passage time of a Lévy flight in one dimension. This observation enables us to calculate $C_{2}=1/16π$.

hep-th↗

arXiv2016

The distribution of path lengths of self avoiding walks on Erdős-Rényi networks

We present an analytical and numerical study of the paths of self avoiding walks (SAWs) on random networks. Since these walks do not retrace their paths, they effectively delete the nodes they visit, together with their links, thus pruning the network. The walkers hop between neighboring nodes, until they reach a dead-end node from which they cannot proceed. Focusing on Erdős-Rényi networks we show that the pruned networks maintain a Poisson degree distribution, $p_t(k)$, with an average degree, $\langle k \rangle_t$, that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, $n_T(\ell)$, increases dramatically as a function of $\ell$. We also obtain analytical results for the path-length distribution, $P(\ell)$, of the SAW paths which are actually pursued, starting from a random initial node. It turns out that $P(\ell)$ follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.

cond-mat.dis-nn↗

arXiv2016

Distance distribution in configuration model networks

We present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks. The results, which are based on recursion equations, are shown to be in good agreement with numerical simulations for networks with degenerate, binomial and power-law degree distributions. The mean, mode and variance of the distribution of shortest path lengths are also evaluated. These results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution, illuminating the connection between the two distributions.

cond-mat.dis-nn↗

arXiv2016

Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation

Using the weak-noise theory, we evaluate the probability distribution $\mathcal{P}(H,t)$ of large deviations of height $H$ of the evolving surface height $h(x,t)$ in the Kardar-Parisi-Zhang (KPZ) equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height $H$ at time $t$. We argue that the tails of $\mathcal{P}$ behave, at arbitrary time $t>0$, and in a proper moving frame, as $-\ln \mathcal{P}\sim |H|^{5/2}$ and $\sim |H|^{3/2}$. The $3/2$ tail coincides with the asymptotic of the Gaussian orthogonal ensemble Tracy-Widom distribution, previously observed at long times.

cond-mat.stat-mech↗

arXiv2015

Steady-state propagation speed of rupture fronts along one-dimensional frictional interfaces

The rupture of dry frictional interfaces occurs through the propagation of fronts breaking the contacts at the interface. Recent experiments have shown that the velocities of these rupture fronts range from quasi-static velocities proportional to the external loading rate to velocities larger than the shear wave speed. The way system parameters influence front speed is still poorly understood. Here we study steady-state rupture propagation in a one-dimensional (1D) spring-block model of an extended frictional interface, for various friction laws. With the classical Amontons--Coulomb friction law, we derive a closed-form expression for the steady-state rupture velocity as a function of the interfacial shear stress just prior to rupture. We then consider an additional shear stiffness of the interface and show that the softer the interface, the slower the rupture fronts. We provide an approximate closed form expression for this effect. We finally show that adding a bulk viscosity on the relative motion of blocks accelerates steady-state rupture fronts and we give an approximate expression for this effect. We demonstrate that the 1D results are qualitatively valid in 2D. Our results provide insights into the qualitative role of various key parameters of a frictional interface on its rupture dynamics. They will be useful to better understand the many systems in which spring-block models have proved adequate, from friction to granular matter and earthquake dynamics.

physics.class-ph↗

arXiv2015

Analytical results for the distribution of shortest path lengths in random networks

We present two complementary analytical approaches for calculating the distribution of shortest path lengths in Erdos-Rényi networks, based on recursion equations for the shells around a reference node and for the paths originating from it. The results are in agreement with numerical simulations for a broad range of network sizes and connectivities. The average and standard deviation of the distribution are also obtained. In the case that the mean degree scales as $N^α$ with the network size, the distribution becomes extremely narrow in the asymptotic limit, namely almost all pairs of nodes are equidistant, at distance $d=\lfloor 1/α\rfloor$ from each other. The distribution of shortest path lengths between nodes of degree $m$ and the rest of the network is calculated. Its average is shown to be a monotonically decreasing function of $m$, providing an interesting relation between a local property and a global property of the network. The methodology presented here can be applied to more general classes of networks.

cond-mat.dis-nn↗

arXiv2014

Phase transitions in the condition number distribution of Gaussian random matrices

We study the statistics of the condition number $κ=λ_{\mathrm{max}}/λ_{\mathrm{min}}$ (the ratio between largest and smallest squared singular values) of $N\times M$ Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large $N$ the cumulative $\mathcal{P}[κ x]$ distributions of $κ$. We find that these distributions decay as $\mathcal{P}[κ x]\approx\exp\left(-βN Φ_{+}(x)\right)$, where $β$ is the Dyson index of the ensemble. The left and right rate functions $Φ_{\pm}(x)$ are independent of $β$ and calculated exactly for any choice of the rectangularity parameter $α=M/N-1>0$. Interestingly, they show a weak non-analytic behavior at their minimum $\langleκ\rangle$ (corresponding to the average condition number), a direct consequence of a phase transition in the associated Coulomb fluid problem. Matching the behavior of the rate functions around $\langleκ\rangle$, we determine exactly the scale of typical fluctuations $\sim\mathcal{O}(N^{-2/3})$ and the tails of the limiting distribution of $κ$. The analytical results are in excellent agreement with numerical simulations.

cond-mat.stat-mech↗

arXiv2013

Yield--Optimized Superoscillations

Superoscillating signals are band--limited signals that oscillate in some region faster their largest Fourier component. While such signals have many scientific and technological applications, their actual use is hampered by the fact that an overwhelming proportion of the energy goes into that part of the signal, which is not superoscillating. In the present article we consider the problem of optimization of such signals. The optimization that we describe here is that of the superoscillation yield, the ratio of the energy in the superoscillations to the total energy of the signal, given the range and frequency of the superoscillations. The constrained optimization leads to a generalized eigenvalue problem, which is solved numerically. It is noteworthy that it is possible to increase further the superoscillation yield at the cost of slightly deforming the oscillatory part of the signal, while keeping the average frequency. We show, how this can be done gradually, which enables a trade-off between the distortion and the yield. We show how to apply this approach to non-trivial domains, and explain how to generalize this to higher dimensions.

quant-ph↗

arXiv2013

A comparative study of crumpling and folding of thin sheets

Crumpling and folding of paper are at rst sight very di erent ways of con ning thin sheets in a small volume: the former one is random and stochastic whereas the latest one is regular and deterministic. Nevertheless, certain similarities exist. Crumpling is surprisingly ine cient: a typical crumpled paper ball in a waste-bin consists of as much as 80% air. Similarly, if one folds a sheet of paper repeatedly in two, the necessary force becomes so large that it is impossible to fold it more than 6 or 7 times. Here we show that the sti ness that builds up in the two processes is of the same nature, and therefore simple folding models allow to capture also the main features of crumpling. An original geometrical approach shows that crumpling is hierarchical, just as the repeated folding. For both processes the number of layers increases with the degree of compaction. We nd that for both processes the crumpling force increases as a power law with the number of folded layers, and that the dimensionality of the compaction process (crumpling or folding) controls the exponent of the scaling law between the force and the compaction ratio.

cond-mat.soft↗

arXiv2012

Spectral Properties of the Jacobi Ensembles via the Coulomb Gas approach

Using the Coulomb gas method and standard methods of statistical physics, we compute analytically the joint cumulative probability distribution of the extreme eigenvalues of the Jacobi-MANOVA ensemble of random matrices, in the limit of large matrices. This allows us to derive the rate functions for the large fluctuations to the left and the right of the expected values of the smallest and largest eigenvalues analytically. Our findings are compared with some available known exact results as well as with numerical simulations finding good agreement.

cond-mat.stat-mech↗

arXiv2012

Dynamic stability of crack fronts: Out-of-plane corrugations

The dynamics and stability of brittle cracks are not yet fully understood. Here we use the Willis-Movchan 3D linear perturbation formalism [J. Mech. Phys. Solids {\bf 45}, 591 (1997)] to study the out-of-plane stability of planar crack fronts in the framework of linear elastic fracture mechanics. We discuss a minimal scenario in which linearly unstable crack front corrugations might emerge above a critical front propagation speed. We calculate this speed as a function of Poisson's ratio and show that corrugations propagate along the crack front at nearly the Rayleigh wave-speed. Finally, we hypothesize about a possible relation between such corrugations and the long-standing problem of crack branching.

cond-mat.mtrl-sci↗

arXiv2011

Exponent Inequalities in Dynamical Systems

In this letter we derive exponent inequalities relating the dynamic exponent $z$ to the steady state exponent $Γ$ for a general class of stochastically driven dynamical systems. We begin by deriving a general exact inequality, relating the response function and the correlation function, from which the various exponent inequalities emanate. We then distinguish between two classes of dynamical systems and obtain different and complementary inequalities relating $z$ and $Γ$. The consequences of those inequalities for a wide set of dynamical problems, including critical dynamics and Kardar-Parisi-Zhang-like problems are discussed.

cond-mat.stat-mech↗

arXiv2011

Dynamical Inequality in Growth Models

A recent exponent inequality is applied to a number of dynamical growth models. Many of the known exponents for models such as the Kardar-Parisi-Zhang (KPZ) equation are shown to be consistent with the inequality. In some cases, such as the Molecular Beam Equation, the situation is more interesting, where the exponents saturate the inequality. As the acid test for the relative strength of four popular approximation schemes we apply the inequality to the exponents obtained for two Non Local KPZ systems. We find that all methods but one, the Self Consistent Expansion, violate the inequality in some regions of parameter space. To further demonstrate the usefulness of the inequality, we apply it to a specific model, which belongs to a family of models in which the inequality becomes an equality. We thus show that the inequality can easily yield results, which otherwise have to rely either on approximations or general beliefs.

cond-mat.stat-mech↗

arXiv2010

Large Deviations of the Smallest Eigenvalue of the Wishart-Laguerre Ensemble

We consider the large deviations of the smallest eigenvalue of the Wishart-Laguerre Ensemble. Using the Coulomb gas picture we obtain rate functions for the large fluctuations to the left and the right of the hard edge. Our findings are compared with known exact results for $β=1$ finding good agreement. We also consider the case of almost square matrices finding new universal rate functions describing large fluctuations.

cond-mat.dis-nn↗