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Sebastian Müller

Publications and source records attributed to Sebastian Müller.

At least 73 records · Page 4Linked to original sources
arXiv2015

Interacting growth processes and invariant percolation

The aim of this paper is to underline the relation between reversible growth processes and invariant percolation. We present two models of interacting branching random walks (BRWs), truncated BRWs and competing BRWs, where survival of the growth process can be formulated as the existence of an infinite cluster in an invariant percolation on a tree. Our approach is fairly conceptual and allows generalizations to a wider set of "reversible" growth processes.

math.PR↗
arXiv2014

Particle Diagrams and Statistics of Many-Body Random Potentials

We present a method using Feynman-like diagrams to calculate the statistical properties of random many-body potentials. This method provides a promising alternative to existing techniques typically applied to this class of problems, such as the method of supersymmetry and the eigenvector expansion technique pioneered in [1]. We use it here to calculate the fourth, sixth and eighth moments of the average level density for systems with $m$ bosons or fermions that interact through a random $k$-body Hermitian potential ($k \le m$); the ensemble of such potentials with a Gaussian weight is known as the embedded Gaussian Unitary Ensemble (eGUE) [2]. Our results apply in the limit where the number $l$ of available single-particle states is taken to infinity. A key advantage of the method is that it provides an efficient way to identify only those expressions which will stay relevant in this limit. It also provides a general argument for why these terms have to be the same for bosons and fermions. The moments are obtained as sums over ratios of binomial expressions, with a transition from moments associated to a semi-circular level density for $m < 2k$ to Gaussian moments in the dilute limit $k \ll m \ll l$. Regarding the form of this transition, we see that as $m$ is increased, more and more diagrams become relevant, with new contributions starting from each of the points $m = 2k, 3k, \ldots, nk$ for the $2n$-th moment.

cond-mat.other↗
arXiv2014

Particle diagrams and embedded many-body random matrix theory

We present a new method which uses Feynman-like diagrams to calculate the statistical quantities of embedded many-body random matrix problems. The method provides a promising alternative to existing techniques and offers many important simplifications. We use it here to find the fourth, sixth and eighth moments of the level density for k fermions or bosons interacting through a random hermitian potential in the limit where the number of possible single-particle states is taken to infinity. All share the same transition, starting immediately after 2k = m, from moments arising from a semi-circular level density to gaussian moments. The results also reveal a striking feature; the domain of the 2n'th moment is naturally divided into n subdomains specified by the points 2k = m, 3k = m, ..., nk = m.

cond-mat.stat-mech↗
arXiv2013

Cookie branching random walks

We consider a branching random walk on $\Z$, where the particles behave differently in visited and unvisited sites. Informally, each site on the positive half-line contains initially a cookie. On the first visit of a site its cookie is removed and particles at positions with a cookie reproduce and move differently from particles on sites without cookies. Therefore, the movement and the reproduction of the particles depend on the previous behaviour of the population of particles. We study the question if the process is recurrent or transient, i.e., whether infinitely many particles visit the origin or not.

math.PR↗
arXiv2013

Polylogarithmic Cuts in Models of V^0

We study initial cuts of models of weak two-sorted Bounded Arithmetics with respect to the strength of their theories and show that these theories are stronger than the original one. More explicitly we will see that polylogarithmic cuts of models of $\mathbf{V}^0$ are models of $\mathbf{VNC}^1$ by formalizing a proof of Nepomnjascij's Theorem in such cuts. This is a strengthening of a result by Paris and Wilkie. We can then exploit our result in Proof Complexity to observe that Frege proof systems can be sub exponentially simulated by bounded depth Frege proof systems. This result has recently been obtained by Filmus, Pitassi and Santhanam in a direct proof. As an interesting observation we also obtain an average case separation of Resolution from AC0-Frege by applying a recent result with Tzameret.

cs.LO↗
arXiv2013

GSE statistics without spin

Energy levels statistics following the Gaussian Symplectic Ensemble (GSE) of Random Matrix Theory have been predicted theoretically and observed numerically in numerous quantum chaotic systems. However in all these systems there has been one unifying feature: the combination of half-integer spin and time-reversal invariance. Here we provide an alternative mechanism for obtaining GSE statistics that is based on geometric symmetries of a quantum system which alleviates the need for spin. As an example, we construct a quantum graph with a particular discrete symmetry given by the quaternion group Q8. GSE statistics is then observed within one of its subspectra.

math-ph↗
arXiv2012

No anti-correlation between cyclotron line energy and X-ray flux in 4U 0115+634

We report on an outburst of the high mass X-ray binary 4U 0115+63 with a pulse period of 3.6s in 2008 March/April as observed with RXTE and INTEGRAL. During the outburst the neutron star's luminosity varied by a factor of 10 in the 3--50\,keV band. In agreement with earlier work we find evidence for five cyclotron resonance scattering features at ~10.7, 21.8, 35.5, 46.7, and 59.7keV. Previous work had found an anti-correlation between the fundamental cyclotron line energy and the X-ray flux. We show that this apparent anti-correlation is probably due to the unphysical interplay of parameters of the cyclotron line with the continuum models used previously, e.g., the negative and positive exponent power law (NPEX). For this model, we show that cyclotron line modeling erroneously leads to describing part of the exponential cutoff and the continuum variability, and not the cyclotron lines. When the X-ray continuum is modeled with a simple exponentially cutoff power law modified by a Gaussian emission feature around 10keV, the correlation between the line energy and the flux vanishes and the line parameters remain virtually constant over the outburst. We therefore conclude that the previously reported anti-correlation is an artifact of the assumptions adopted in the modeling of the continuum.

astro-ph.HE↗
arXiv2012

The reawakening of the sleeping X-ray pulsar XTE J1946+274

We report on a series of outbursts of the high mass X-ray binary XTE J1946+274 in 2010/2011 as observed with INTEGRAL, RXTE, and Swift. We discuss possible mechanisms resulting in the extraordinary outburst behavior of this source. The X-ray spectra can be described by standard phenomenological models, enhanced by an absorption feature of unknown origin at about 10 keV and a narrow iron K alpha fluorescence line at 6.4 keV, which are variable in flux and pulse phase. We find possible evidence for the presence of a cyclotron resonance scattering feature at about 25 keV at the 93% level. The presence of a strong cyclotron line at 35 keV seen in data from the source's 1998 outburst and confirmed by a reanalysis of these data can be excluded. This result indicates that the cyclotron line feature in XTE J1946+274 is variable between individual outbursts.

astro-ph.HE↗
arXiv2012

Semiclassical approach to discrete symmetries in quantum chaos

We use semiclassical methods to evaluate the spectral two-point correlation function of quantum chaotic systems with discrete geometrical symmetries. The energy spectra of these systems can be divided into subspectra that are associated to irreducible representations of the corresponding symmetry group. We show that for (spinless) time reversal invariant systems the statistics inside these subspectra depend on the type of irreducible representation. For real representations the spectral statistics agree with those of the Gaussian Orthogonal Ensemble (GOE) of Random Matrix Theory (RMT), whereas complex representations correspond to the Gaussian Unitary Ensemble (GUE). For systems without time reversal invariance all subspectra show GUE statistics. There are no correlations between non-degenerate subspectra. Our techniques generalize recent developments in the semiclassical approach to quantum chaos allowing one to obtain full agreement with the two-point correlation function predicted by RMT, including oscillatory contributions.

nlin.CD↗
arXiv2011

Random walks on Galton-Watson trees with random conductances

We consider the random conductance model, where the underlying graph is an infinite supercritical Galton--Watson tree, the conductances are independent but their distribution may depend on the degree of the incident vertices. We prove that, if the mean conductance is finite, there is a deterministic, strictly positive speed $v$ such that $\lim_{n\to\infty} \frac{|X_n|}{n}= v$ a.s.\ (here, $|\cdot|$ stands for the distance from the root). We give a formula for $v$ in terms of the laws of certain effective conductances and show that, if the conductances share the same expected value, the speed is not larger than the speed of simple random walk on Galton--Watson trees. The proof relies on finding a reversible measure for the environment observed by the particle.

math.PR↗
arXiv2011

Branching Random Walks on Free Products of Groups

We study certain phase transitions of branching random walks (BRW) on Cayley graphs of free products. The aim of this paper is to compare the size and structural properties of the trace, i.e., the subgraph that consists of all edges and vertices that were visited by some particle, with those of the original Cayley graph. We investigate the phase when the growth parameter $λ$ is small enough such that the process survives but the trace is not the original graph. A first result is that the box-counting dimension of the boundary of the trace exists, is almost surely constant and equals the Hausdorff dimension which we denote by $Φ(λ)$. The main result states that the function $Φ(λ)$ has only one point of discontinuity which is at $λ_{c}=R$ where $R$ is the radius of convergence of the Green function of the underlying random walk. Furthermore, $Φ(R)$ is bounded by one half the Hausdorff dimension of the boundary of the original Cayley graph and the behaviour of $Φ(R)-Φ(λ)$ as $λ\uparrow R$ is classified. In the case of free products of infinite groups the end-boundary can be decomposed into words of finite and words of infinite length. We prove the existence of a phase transition such that if $λ\leq \tildeλ_{c}$ the end boundary of the trace consists only of infinite words and if $λ>\tildeλ_{c}$ it also contains finite words. In the last case, the Hausdorff dimension of the set of ends (of the trace and the original graph) induced by finite words is strictly smaller than the one of the ends induced by infinite words.

math.PR↗
arXiv2011

Short Propositional Refutations for Dense Random 3CNF Formulas

Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems. Lower bounds for random 3CNF refutations in many propositional proof systems are known. Most notably are the exponential-size resolution refutation lower bounds for random 3CNF formulas with $Ω(n^{1.5-ε}) $ clauses [Chvatal and Szemeredi (1988), Ben-Sasson and Wigderson (2001)]. On the other hand, the only known non-trivial upper bound on the size of random 3CNF refutations in a non-abstract propositional proof system is for resolution with $Ω(n^{2}/\log n) $ clauses, shown by Beame et al. (2002). In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable ratio. Specifically, we demonstrate polynomial-size propositional refutations whose lines are $TC^0$ formulas (i.e., $TC^0$-Frege proofs) for random 3CNF formulas with $ n $ variables and $ Ω(n^{1.4}) $ clauses. The idea is based on demonstrating efficient propositional correctness proofs of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek (2006). Since the soundness of these witnesses is verified using spectral techniques, we develop an appropriate way to reason about eigenvectors in propositional systems. To carry out the full argument we work inside weak formal systems of arithmetic and use a general translation scheme to propositional proofs.

cs.CC↗
arXiv2010

On the trace of branching random walks

We study branching random walks on Cayley graphs. A first result is that the trace of a transient branching random walk on a Cayley graph is a.s. transient for the simple random walk. In addition, it has a.s. critical percolation probability less than one and exponential volume growth. The proofs rely on the fact that the trace induces an invariant percolation on the family tree of the branching random walk. Furthermore, we prove that the trace is a.s. strongly recurrent for any (non-trivial) branching random walk. This follows from the observation that the trace, after appropriate biasing of the root, defines a unimodular measure. All results are stated in the more general context of branching random walks on unimodular random graphs.

math.PR↗
arXiv2010

A note on spider walks

Spider walks are systems of interacting particles. The particles move independently as long as their movement do not violate some given rules describing the relative position of the particles; moves that violate the rules are not realized. The goal of this paper is to study qualitative properties, as recurrence, transience, ergodicity, and positive rate of escape of these Markov processes.

math.PR↗
arXiv2009

Dynamical sensitivity of recurrence and transience of branching random walks

Consider a sequence of i.i.d. random variables $X_n$ where each random variable is refreshed independently according to a Poisson clock. At any fixed time $t$ the law of the sequence is the same as for the sequence at time 0 but at random times almost sure properties of the sequence may be violated. If there are such \emph{exceptional times} we say that the property is \emph{dynamically sensitive}, otherwise we call it \emph{dynamically stable}. In this note we consider branching random walks on Cayley graphs and prove that recurrence and transience are dynamically stable in the sub-and supercritical regime. While the critical case is left open in general we prove dynamical stability for a specific class of Cayley graphs. Our proof combines techniques from the theory of ranching random walks with those of dynamical percolation.

math.PR↗
arXiv2009

The Debris Disk of Vega: A Steady-State Collisional Cascade, Naturally

It has been argued that the photometric data and images of the archetypical debris disk around Vega may be in contradiction with the standard, steady-state collisional scenario of the disk evolution. Here we perform physical modeling of the Vega disk "from the sources". We assume that dust is maintained by a "Kuiper belt" of parent planetesimals at ~ 100 AU and employ our collisional and radiative transfer codes to consistently model the size and radial distribution of the disk material and then thermal emission of dust. In doing so, we vary a broad set of parameters, including the stellar properties, the exact location, extension, and dynamical excitation of the planetesimal belt, chemical composition of solids, and the collisional prescription. We are able to reproduce the spectral energy distribution in the entire wavelength range from the near-infrared to millimeter, as well as the mid-IR and sub-millimeter radial brightness profiles of the Vega disk. Thus our results suggest that the Vega disk observations are compatible with a steady-state collisional dust production, and we put important constraints on the disk parameters and physical processes that sustain it. The total disk mass in < 100 km-sized bodies is estimated to be ~ 10 Earth masses. Provided that collisional cascade has been operating over much of the Vega age of ~ 350 Myr, the disk must have lost a few Earth masses of solids during that time. We also demonstrate that using an intermediate luminosity of the star between the pole and the equator, as derived from its fast rotation, is required to reproduce the debris disk observations. Finally, we show that including cratering collisions into the model is mandatory.

astro-ph.EP↗