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

Optimal spatial transportation networks where link-costs are sublinear in link-capacity

Consider designing a transportation network on $n$ vertices in the plane, with traffic demand uniform over all source-destination pairs. Suppose the cost of a link of length $\ell$ and capacity $c$ scales as $\ell c^\beta$ for fixed $0<\beta<1$. Under appropriate standardization, the cost of the minimum cost Gilbert network grows essentially as $n^{\alpha(\beta)}$, where $\alpha(\beta) = 1 - \frac{\beta}{2}$ on $0 < \beta \leq {1/2}$ and $\alpha(\beta) = {1/2} + \frac{\beta}{2}$ on ${1/2} \leq \beta < 1$. This quantity is an upper bound in the worst case (of vertex positions), and a lower bound under mild regularity assumptions. Essentially the same bounds hold if we constrain the network to be efficient in the sense that average route-length is only $1 + o(1)$ times average straight line length. The transition at $\beta = {1/2}$ corresponds to the dominant cost contribution changing from short links to long links. The upper bounds arise in the following type of hierarchical networks, which are therefore optimal in an order of magnitude sense. On the large scale, use a sparse Poisson line process to provide long-range links. On the medium scale, use hierachical routing on the square lattice. On the small scale, link vertices directly to medium-grid points. We discuss one of many possible variant models, in which links also have a designed maximum speed $s$ and the cost becomes $\ell c^\beta s^\gamma$.Comment: 13 page

Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices

We develop a scaling theory for KPZ growth in one dimension by a detailed study of the polynuclear growth (PNG) model. In particular, we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape. These distribution functions are computed using the partition function of Gaussian random matrices in a cosine potential.Comment: 4 pages, 3 figures, 1 table, RevTeX, revised version, accepted for publication in PR

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

Condensation of the roots of real random polynomials on the real axis

We introduce a family of real random polynomials of degree n whose coefficients a_k are symmetric independent Gaussian variables with variance = e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly the mean number of real roots for large n. As \alpha is varied, one finds three different phases. First, for 0 \leq \alpha \sim (\frac{2}{\pi}) \log{n}. For 1 < \alpha < 2, there is an intermediate phase where grows algebraically with a continuously varying exponent, \sim \frac{2}{\pi} \sqrt{\frac{\alpha-1}{\alpha}} n^{\alpha/2}. And finally for \alpha > 2, one finds a third phase where \sim n. This family of real random polynomials thus exhibits a condensation of their roots on the real line in the sense that, for large n, a finite fraction of their roots /n are real. This condensation occurs via a localization of the real roots around the values \pm \exp{[\frac{\alpha}{2}(k+{1/2})^{\alpha-1} ]}, 1 \ll k \leq n.Comment: 13 pages, 2 figure

Random tree growth by vertex splitting

We study a model of growing planar tree graphs where in each time step we separate the tree into two components by splitting a vertex and then connect the two pieces by inserting a new link between the daughter vertices. This model generalises the preferential attachment model and Ford's $\alpha$-model for phylogenetic trees. We develop a mean field theory for the vertex degree distribution, prove that the mean field theory is exact in some special cases and check that it agrees with numerical simulations in general. We calculate various correlation functions and show that the intrinsic Hausdorff dimension can vary from one to infinity, depending on the parameters of the model.Comment: 47 page

Particle Systems with Stochastic Passing

We study a system of particles moving on a line in the same direction. Passing is allowed and when a fast particle overtakes a slow particle, it acquires a new velocity drawn from a distribution P_0(v), while the slow particle remains unaffected. We show that the system reaches a steady state if P_0(v) vanishes at its lower cutoff; otherwise, the system evolves indefinitely.Comment: 5 pages, 5 figure

The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion

For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^{-3/2}, plus an error term of order n^{-3/2-\epsilon} with \epsilon >0. This is a strong statement that the critical exponent delta is given by delta =2.Comment: 56 pages, 3 Postscript figures, in AMS-LaTeX, with graphicx, epic, and xr package

Real roots of Random Polynomials: Universality close to accumulation points

We identify the scaling region of a width O(n^{-1}) in the vicinity of the accumulation points $t=\pm 1$ of the real roots of a random Kac-like polynomial of large degree n. We argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent, identically distributed coefficients c_i, as long as the second moment \sigma=E(c_i^2) is finite. In particular, we reveal a gradual (in contrast to the previously reported abrupt) and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value \mu_n = E(c_i) scaled as \mu_n\sim n^{-1/2}.Comment: Some minor mistakes that crept through into publication have been removed. 10 pages, 12 eps figures. This version contains all updates, clearer pictures and some more thorough explanation

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