Let $L_n^{X}(x)$ denote the number of visits to $x \in {\bf Z}^2$ of the simple planar random walk $X$, up till step $n$. Let $X'$ be another simple planar random walk independent of $X$. We show that for any $0<b<1/(2 π)$, there are $n^{1-2πb+o(1)}$ points $x \in {\bf Z}^2$ for which $L_n^{X}(x)L_n^{X'}(x)\geq b^2 (\log n)^4$. This is the discrete counterpart of our main result, that for any $a<1$, the Hausdorff dimension of the set of {\it thick intersection points} $x$ for which $\limsup_{r \to 0} {\mathcal I}(x,r)/(r^2|\log r|^4)=a^2$, is almost surely $2-2a$. Here ${\mathcal I}(x,r)$ is the projected intersection local time measure of the disc of radius $r$ centered at $x$ for two independent planar Brownian motions run till time 1. The proofs rely on a `multi-scale refinement' of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius $r$ centered at $x$ by $x+rK$ for general sets $K$.
It is shown that if a planar graph admits no non-constant bounded harmonic functions then the trajectories of two independent simple random walks intersect almost surely.
Let B_n be the number of self-intersections of a symmetric random walk with finite second moments in the integer planar lattice. We obtain moderate deviation estimates for B_n - E B_n and E B_n- B_n, which are given in terms of the best constant of a certain Gagliardo-Nirenberg inequality. We also prove the corresponding laws of the iterated logarithm.
The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-i…
Fourier-Wiener transform of the formal expression for multiple self-intersection local time is described in terms of the integral, which is divergent on the diagonals. The method of regularization we use in this work related to regularization of functions with non-integrable singularities. The strong local nondeterminism property, which is more restrictive than the property of local nondeterminism introduced by S.Berman is considered. Its geometrical meaning in the construction of the regularization is investigated. As the example the problem of regularization is solved for the compact perturb…
In this article we calculate the third and fourth moment of the renormalized intersection local time of a planar Brownian motion. The third moment is calculated anlaytically, the fourth moment numerically. For the closed planar random walk the third moment of the distribution of the multiple point range is also calculated in leading order.
We prove that the set of $γ$-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all $γ\neq 0$. Our proof relies on the coupling between a GFF and the nested CLE$_4$. In particular, we show that the thick points of the GFF are the same as those of the weighted CLE$_4$ nesting field and establish the almost sure total disconnectedness of the complement of a nested CLE$_κ$, $κ\in (8/3,4]$. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.
In this article we study the distribution of the number of points of a simple random walk, visited a given number of times (the k-multiple point range). In a previous article we had developed a graph theoretical approach which is now extended from the closed to the the non restricted walk. Based on a study of the analytic properties of the first hit determinant a general method to define and calculate the generating functions of the moments of the distribution as analytical functions and express them as an absolutely converging series of graph contributions is given. So a method to calculate t…
We define and study a family of generalized non-intersection exponents for planar Brownian motions that is indexed by subsets of the complex plane: For each $A\subset\CC$, we define an exponent $ξ(A)$ that describes the decay of certain non-intersection probabilities. To each of these exponents, we associate a conformally invariant subset of the planar Brownian path, of Hausdorff dimension $2-ξ(A)$. A consequence of this and continuity of $ξ(A)$ as a function of $A$ is the almost sure existence of pivoting points of any sufficiently small angle on a planar Brownian path.
There is a close connection between intersections of Brownian motion paths and percolation on trees. Recently, ideas from probability on trees were an important component of the multifractal analysis of Brownian occupation measure, in joint work with A. Dembo, J. Rosen and O. Zeitouni. As a consequence, we proved two conjectures about simple random walk in two dimensions: The first, due to Erdős and Taylor (1960), involves the number of visits to the most visited lattice site in the first $n$ steps of the walk. The second, due to Aldous (1989), concerns the number of steps it takes a simple ra…
We consider a periodic planar Lorentz process with strictly convex obstacles and finite horizon. This process describes the displacement of a particle moving in the plane with unit speed and with elastic reflection on the obstacles. We call number of self-intersections of this Lorentz process the number V(n) of couples of integers (k,m) smaller than n such that the particle hits a same obstacle both at the k-th and at the m-th collision times. The aim of this paper is to prove that the variance of V(n) is equivalent to cn^2 (such a result has recently been proved for simple planar random walks…
For a symmetric random walk in $Z^2$ with $2+δ$ moments, we represent $|\mathcal{R}(n)|$, the cardinality of the range, in terms of an expansion involving the renormalized intersection local times of a Brownian motion. We show that for each $k\geq 1$ \[ (\log n)^k [ \frac{1}{n} |\mathcal{R}(n)| +\sum_{j=1}^k (-1)^j (\textstyle{\frac1{2π}}\log n +c_X)^{-j} γ_{j,n}]\to 0, \qquad a.s. \] where $W_t$ is a Brownian motion, $W^{(n)}_t=W_{nt}/\sqrt n$, $γ_{j,n}$ is the renormalized intersection local time at time 1 for $W^{(n)}$, and $c_X$ is a constant depending on the distribution of the random wal…
We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant of the desired thickness level and show a simple relation between the two objects. Our results extend those of Bass, Burdzy and Khoshnevisan and in particular cover the entire $L^1$-phase or subcritical regime. These results allow us to obtain a nondegenerate limit for the appropriately rescaled size of thick points, thereb…
We show that the intersection exponents for planar Brownian motions are analytic. More precisely, let $B$ and $B'$ be independent planar Brownian motions started from distinct points, and define the exponent $ξ(1, λ)$ by $$ E[P[B[0,t] \cap B'[0,t] = \emptyset | B[0,t]]^λ] \approx t^{-ξ(1, λ)/2}, t \to \infty. $$ Then the mapping $λ\mapsto ξ(1, λ)$ is real analytic in $(0,\infty)$. The same result is proved for the exponents $ξ(k, λ)$ where $k$ is a positive integer. In combination with the determination of $ξ(k, λ)$ for integer $k \ge 1$ and real $λ\ge 1$ in our previous papers, this gives the…
Let β_k(n) be the number of self-intersections of order k, appropriately renormalized, for a mean zero random walk X_n in Z^2 with 2+δmoments. On a suitable probability space we can construct X_n and a planar Brownian motion W_t such that for each k\geq 2, |β_k(n)-γ_k(n)|=O(n^{-a}), a.s. for some a>0 where γ_k(n) is the renormalized self-intersection local time of order k at time 1 for the Brownian motion W_{nt}/\sqrt n.
Let \ell be the projected intersection local time of two independent Brownian paths in R^d for d=2,3. We determine the lower tail of the random variable \ell(U), where U is the unit ball. The answer is given in terms of intersection exponents, which are explicitly known in the case of planar Brownian motion. We use this result to obtain the multifractal spectrum, or spectrum of thin points, for the intersection local times.
If β_t is renormalized self-intersection local time for planar Brownian motion, we characterize when Ee^{γβ_1} is finite or infinite in terms of the best constant of a Gagliardo-Nirenberg inequality. We prove large deviation estimates for β_1 and -β_1. We establish lim sup and lim inf laws of the iterated logarithm for β_t as t\to\infty.
We study some asymptotic properties of cylinder processes in the plane defined as union sets of dilated straight lines (appearing as mutually overlapping infinitely long strips) derived from a stationary independently marked point process on the real line, where the marks describe thickness and orientation of individual cylinders. Such cylinder processes form an important class of (in general non-stationary) planar random sets. We observe the cylinder process in an unboundedly growing domain $ρK$ when $ρ\to \infty\,$, where the set $K$ is compact and star-shaped w.r.t. the origin ${\bf o}$ bei…
A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In this paper we present conjectures for the scaling limit of the uniform measures on these objects. The conjectures are based on recent results on the stochastic Loewner evolution and non-disconnecting Brownian motions. New heuristic derivations are given for the critical exponents for SAWs and SAPs.
We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard three-dimensional flat torus with a fixed smooth reference curve, which has nowhere vanishing curvature. The expected intersection number is universally proportional to the length of the reference curve, times the wavenumber, independent of the geometry. Our main result gives a bound for the variance, if either the torsion of the curve is nowhere zero or if the curve is planar.
We construct the natural diffusion in the random geometry of planar Liouville quantum gravity. Formally, this is the Brownian motion in a domain $D$ of the complex plane for which the Riemannian metric tensor at a point $z \in D$ is given by $\exp (γh(z) - \frac12 γ^2 \E (h(z)^2))$. Here $h$ is an instance of the Gaussian Free Field on $D$ and $γ\in (0,2)$ is a parameter. We show that the process is almost surely continuous and enjoys certain conformal invariance properties. We also estimate the Hausdorff dimension of times that the diffusion spends in the thick points of the Gaussian Free Fie…
We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue $E>0$, both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the high-energy limit ($E\to \infty$). Our main result is that both the nodal length (real case) and the number of nodal intersections (complex case) verify a Central Limit Theorem, which is in sharp contrast with the non-Gaussian behaviour observed for real and complex arithmetic random waves on the flat $2$-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our fi…
