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Register a new userA Lemma for Color Switching on the Square Lattice
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We consider 2d critical Bernoulli percolation on the square lattice. We prove an approximate color-switching lemma comparing k arm probabilities for different polychromatic color sequences. This result is well-known for site percolation on the triangular lattice in [Nolin08]. To handle the complications arising from the dual lattice, we introduce a shifting transformation to convert arms between the primal and the dual lattices.
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On the square lattice raindrops fall on an edge with midpoint $x$ at rate $|x|_infty^{-alpha}$. The edge becomes open when the first drop falls on it. Let $rho(x,t)$ be the probability that the edge with midpoint $x=(x_1,x_2)$ is open at time $t$ and let $n(p,t)$ be the distance at which edges are open with probability $p$ at time $t$. We show that with probability tending to 1 as $t to infty$: (i) the cluster containing the origin $mathbb C_0(t)$ is contained in the square of radius $n(p_c-epsilon,t)$, and (ii) the cluster fills the square of radius $n(p_c+epsilon,t)$ with the density of points near $x$ being close to $theta(rho(x,t))$ where $theta(p)$ is the percolation probability when bonds are open with probability $p$ on $mathbb Z^2$. Results of Nolin suggest that if $N=n(p_c,t)$ then the boundary fluctuations of $mathbb C_0(t)$ are of size $N^{4/7}$.
In Poisson percolation each edge becomes open after an independent exponentially distributed time with rate that decreases in the distance from the origin. As a sequel to our work on the square lattice, we describe the limiting shape of the component containing the origin in the oriented case. We show that the density of occupied sites at height $y$ in the cluster is close to the percolation probability in the corresponding homogeneous percolation process, and we study the fluctuations of the boundary.
In this paper we consider independent site percolation in a triangulation of $mathbb{R}^2$ given by adding $sqrt{2}$-long diagonals to the usual graph $mathbb{Z}^2$. We conjecture that $p_c=frac{1}{2}$ for any such graph, and prove it for almost every such graph.
Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least $theta$ occupied neighbors. The initially occupied set is random, given by a uniform product measure with a low density $p$. Our main focus is on this process on the product graph $mathbb{Z}^2times K_n^2$, where $K_n$ is a complete graph. We investigate how $p$ scales with $n$ so that a typical site is eventually occupied. Under critical scaling, the dynamics with even $theta$ exhibits a sharp phase transition, while odd $theta$ yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on $mathbb{Z}^2times K_n$. The main tool is heterogeneous bootstrap percolation on $mathbb{Z}^2$.
Quantitative formulations of Feffermans counterexample for the ball multiplier are naturally linked to square function and vector-valued estimates for directional singular integrals. The latter are usually referred to as Meyer-type lemmas and are traditionally attacked by combining weighted inequalities with sharp estimates for maximal directional averaging operators. This classical approach fails to give sharp bounds. In this article we develop a novel framework for square function estimates, based on directional Carleson embedding theorems and multi-parameter time-frequency analysis, which overcomes the limitations of weighted theory. In particular we prove the sharp form of Meyers lemma, namely a sharp operator norm bound for vector-valued directional singular integrals, in both one and two parameters, in terms of the cardinality of the given set of directions. Our sharp Meyer lemma implies an improved quantification of the reverse square function estimate for tangential $deltatimes delta^2$-caps on $mathbb S^1$. We also prove sharp square function estimates for conical and radial multipliers. A suitable combination of these estimates yields a new and currently best known bound for the Fourier restriction to a $N$-gon, improving on previous results of A. Cordoba.
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