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We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e the behavior of the "ant in the labyrinth". It is natural to conjecture (see [16] and [8]) that the scaling limit for random walks on…
We study families of dependent site percolation models on the triangular lattice ${\mathbb T}$ and hexagonal lattice ${\mathbb H}$ that arise by applying certain cellular automata to independent percolation configurations. We analyze the…
We obtain infinitely many boundary operators in the Brownian loop soup in the subcritical phase by analyzing the conformal block expansion of the two-point function that computes the probability of having two marked points on the upper…
We identify the local scaling limit of the Uniform Infinite Planar Quadrangulation (UIPQ) and of critical Boltzmann quadrangulations, when one simultaneously rescales the distances and reroot them far away from the root of a distinguished…
It is well-known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalization, to the local time of a standard Brownian excursion. In this work, we study the…
We study matricial approximations of master fields we constructed in a previous work. These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in $\mathbb{R},…
A global picture of a random particle movement is given by the convex hull of the visited points. We obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of…
We derive theorems which outline explicit mechanisms by which anomalous scaling for the probability density function of the sum of many correlated random variables asymptotically prevails. The results characterize general anomalous scaling…
We introduce a model of tree-rooted planar maps weighted by their number of $2$-connected blocks. We study its enumerative properties and prove that it undergoes a phase transition. We give the distribution of the size of the largest…
Reflecting boundary conditions cause two one-dimensional random walks to synchronize if a common direction is chosen in each step. The mean synchronization time and its standard deviation are calculated analytically. Both quantities are…
Motivated by automated junction recognition in tracking data, we study a problem of placing a square or disc of fixed size in an arrangement of lines or line segments in the plane. We let distances among the intersection points of the lines…
We prove generalised concentration inequalities for a class of scaled self-bounding functions of independent random variables, referred to as ${(M,a,b)}$ self-bounding. The scaling refers to the fact that the component-wise difference is…
Distance covariance is a popular measure of dependence between random variables. It has some robustness properties, but not all. We prove that the influence function of the usual distance covariance is bounded, but that its breakdown value…
In this paper we investigate the scaling limit of the range (the set of visited vertices) for a class of critical lattice models, starting from a single initial particle at the origin. We give conditions on the random sets and an associated…
We look at geometric limits of large random non-uniform permutations. We mainly consider two theories for limits of permutations: permuton limits, introduced by Hoppen, Kohayakawa, Moreira, Rath, and Sampaio to define a notion of scaling…
We study interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. Our main objective has been to study the long range behavior of the…
Irregular boundary lines can be characterized by fractal dimension, which provides important information for spatial analysis of complex geographical phenomena such as cities. However, it is difficult to calculate fractal dimension of…
Quantum many-body systems realise many different phases of matter characterised by their exotic emergent phenomena. While some simple versions of these properties can occur in systems of free fermions, their occurrence generally implies…
The relative distance between eigenvalues of the compression of a not necessarily semibounded self-adjoint operator to a closed subspace and some of the eigenvalues of the original operator in a gap of the essential spectrum is considered.…
Quantum Cram\'er--Rao theory is intrinsically local: it bounds precision near a specified parameter value, and its saturating measurement generally depends on that value. Barankin-type bounds use finite parameter displacements, but remain…