Related papers: Tree structure of the percolating Universe
The morphological nature of structures that form under gravitational instability has been of central interest to cosmology for over two decades. A remarkable feature of large scale structures in the Universe is that they occupy a relatively…
The fractal structure and scaling properties of a 2d slice of the 3d Ising model is studied using Monte Carlo techniques. The percolation transition of geometric spin (GS) clusters is found to occur at the Curie point, reflecting the…
The stochastic addition of either vertices or connections in a network leads to the observation of the percolation transition, a structural change with the appearance of a connected component encompassing a finite fraction of the system.…
We present in this paper a quantitative method for defining void size in large-scale structure based on percolation threshold density. Beginning with two-dimensional gravitational clustering simulations smoothed to the threshold of…
The study of Gaussian free field level sets on supercritical Galton-Watson trees has been initiated by Ab\"acherli and Sznitman in Ann. Inst. Henri Poincar\'{e} Probab. Stat., 54(1):173--201, 2018. By means of entirely different tools, we…
Tree models for rigidity percolation are introduced and solved. A probability vector describes the propagation of rigidity outward from a rigid border. All components of this ``vector order parameter'' are singular at the same rigidity…
We prove non-universality results for first-passage percolation on the configuration model with i.i.d. degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of…
We consider $Z^d$, with d bigger or equal to three. We investigate the vacant set of random interlacements in the strongly percolative regime, the vacant set of the simple random walk, and the excursion set above a given level of the…
We analyze the critical connectivity of systems of penetrable $d$-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers…
We investigate the non-gaussian properties of cosmic-string-seeded linear density perturbations with cold and hot dark matter backgrounds, using high-resolution numerical simulations. We compute the one-point probability density function of…
Percolation problems appear in a large variety of different contexts ranging from the design of composite materials to vaccination strategies on community networks. The key observable for many applications is the percolation threshold.…
We consider a Bernoulli bond percolation on a random recursive tree of size $n\gg 1$, with supercritical parameter $p_n=1-c/\ln n$ for some $c>0$ fixed. It is known that with high probability, there exists then a unique giant cluster of…
We study the morphology of magnetic domain growth in disordered three dimensional magnets. The disordered magnetic material is described within the random-field Ising model with a Gaussian distribution of local fields with width $\Delta$.…
The rigidity transition occurs when, as the density of microscopic components is increased, a disordered medium becomes able to transmit and ensure macroscopic mechanical stability, owing to the appearance of a space-spanning rigid…
In the standard picture of cosmological structure formation, the Universe we see today is evolved under the gravitational instability from tiny random fluctuations. In this talk I discuss the onset of non-linearity in the large scale…
The large-scale structure of the Universe, as traced by the distribution of galaxies, is now being revealed by large-volume cosmological surveys. The structure is characterized by galaxies distributed along filaments, the filaments…
We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches…
A set of discrete individual points located in an embedding continuum space can be seen as percolating or non-percolating, depending on the radius of the discs/spheres associated with each of them. This problem is relevant in theoretical…
We consider geometrical clusters (i.e. domains of parallel spins) in the square lattice random field Ising model by varying the strength of the Gaussian random field, $\Delta$. In agreement with the conclusion of previous investigation…
A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in…