Exact calculations of first-passage quantities on recursive networks

We present general methods to exactly calculate mean-first passage quantities on self-similar networks defined recursively. In particular, we calculate the mean first-passage time and the splitting probabilities associated to a source and one or several targets; averaged quantities over a given set of sources (e.g., same-connectivity nodes) are also derived. The exact estimate of such quantities highlights the dependency of first-passage processes with respect to the source-target distance, which has recently revealed to be a key parameter to characterize transport in complex media. We explicitly perform calculations for different classes of recursive networks (finitely ramified fractals, scale-free (trans)fractals, non-fractals, mixtures between fractals and non-fractals, non-decimable hierarchical graphs) of arbitrary size. Our approach unifies and significantly extends the available results in the field.Comment: 16 pages, 10 figure

Random Words, Toeplitz Determinants and Integrable Systems. I

It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject to an overall constraint that the eigenvalues lie in a hyperplane.Comment: 15 pages, no figure

Random multi-index matching problems

The multi-index matching problem (MIMP) generalizes the well known matching problem by going from pairs to d-uplets. We use the cavity method from statistical physics to analyze its properties when the costs of the d-uplets are random. At low temperatures we find for d>2 a frozen glassy phase with vanishing entropy. We also investigate some properties of small samples by enumerating the lowest cost matchings to compare with our theoretical predictions.Comment: 22 pages, 16 figure

Random walks pertaining to a class of deterministic weighted graphs

In this note, we try to analyze and clarify the intriguing interplay between some counting problems related to specific thermalized weighted graphs and random walks consistent with such graphs

Stationary Kolmogorov Solutions of the Smoluchowski Aggregation Equation with a Source Term

In this paper we show how the method of Zakharov transformations may be used to analyze the stationary solutions of the Smoluchowski aggregation equation for arbitrary homogeneous kernel. The resulting massdistributions are of Kolmogorov type in the sense that they carry a constant flux of mass from small masses to large. We derive a ``locality criterion'', expressed in terms of the asymptotic properties of the kernel, that must be satisfied in order for the Kolmogorov spectrum to be an admissiblesolution. Whether a given kernel leads to a gelation transition or not can be determined by computing the mass capacity of the Kolmogorov spectrum. As an example, we compute the exact stationary state for the family of kernels,$K_\zeta(m_1,m_2)=(m_1m_2)^{\zeta/2}$ which includes both gelling and non-gelling cases, reproducing the known solution in the case $\zeta=0$. Surprisingly, the Kolmogorov constant is the same for all kernels in this family.Comment: This article is an expanded version of a talk given at IHP workshop "Dynamics, Growth and Singularities of Continuous Media", Paris July 2003. Updated 01/04/04. Revised version with additional discussion, references added, several typographical errors corrected. Revised version accepted for publication by Phys. Rev.

Expected length of the longest common subsequence for large alphabets

We consider the length L of the longest common subsequence of two randomly uniformly and independently chosen n character words over a k-ary alphabet. Subadditivity arguments yield that the expected value of L, when normalized by n, converges to a constant C_k. We prove a conjecture of Sankoff and Mainville from the early 80's claiming that C_k\sqrt{k} goes to 2 as k goes to infinity.Comment: 14 pages, 1 figure, LaTe

Rapid Mixing for Lattice Colorings with Fewer Colors

We provide an optimally mixing Markov chain for 6-colorings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colorings of the square lattice.Comment: Appeared in Proc. LATIN 2004, to appear in JSTA

Abrupt Convergence and Escape Behavior for Birth and Death Chains

We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the convergence times for the latter are exponentially distributed. We compare and study both phenomena for discrete-time birth-and-death chains on Z with drift towards zero. In particular, this includes energy-driven evolutions with energy functions in the form of a single well. Under suitable drift hypotheses, we show that there is both an abrupt convergence towards zero and escape behavior in the other direction. Furthermore, as the evolutions are reversible, the law of the final escape trajectory coincides with the time reverse of the law of cut-off paths. Thus, for evolutions defined by one-dimensional energy wells with sufficiently steep walls, cut-off and escape behavior are related by time inversion.Comment: 2 figure

Heteroepitaxial growth of ferromagnetic MnSb(0001) films on Ge/Si(111) virtual substrates

Molecular beam epitaxial growth of ferromagnetic MnSb(0001) has been achieved on high quality, fully relaxed Ge(111)/Si(111) virtual substrates grown by reduced pressure chemical vapor deposition. The epilayers were characterized using reflection high energy electron diffraction, synchrotron hard X-ray diffraction, X-ray photoemission spectroscopy, and magnetometry. The surface reconstructions, magnetic properties, crystalline quality, and strain relaxation behavior of the MnSb films are similar to those of MnSb grown on GaAs(111). In contrast to GaAs substrates, segregation of substrate atoms through the MnSb film does not occur, and alternative polymorphs of MnSb are absent

Flip dynamics in octagonal rhombus tiling sets

We investigate the properties of classical single flip dynamics in sets of two-dimensional random rhombus tilings. Single flips are local moves involving 3 tiles which sample the tiling sets {\em via} Monte Carlo Markov chains. We determine the ergodic times of these dynamical systems (at infinite temperature): they grow with the system size $N_T$ like $Cst. N_T^2 \ln N_T$; these dynamics are rapidly mixing. We use an inherent symmetry of tiling sets and a powerful tool from probability theory, the coupling technique. We also point out the interesting occurrence of Gumbel distributions.Comment: 5 Revtex pages, 4 figures; definitive versio