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Christoforos Panagiotis

Publications and source records attributed to Christoforos Panagiotis.

26 records · Page 2Linked to original sources
arXiv2021

On the exponential growth rates of lattice animals and interfaces

We introduce a formula for translating any upper bound on the percolation threshold of a lattice \g into a lower bound on the exponential growth rate of lattice animals $a(G)$ and vice-versa. We exploit this to improve on the best known asymptotic bounds on $a(\mathbb{Z}^d)$ as $d\to \infty$. Our formula remains valid if instead of lattice animals we enumerate certain sub-species called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation threshold. Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of $p\in (0,1)$.

math.PR↗
arXiv2021

On the exponential growth rates of lattice animals and interfaces II: new asymptotic bounds

We introduce a method for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate $a(G)$ of lattice animals and vice-versa. We exploit this in both directions. We improve on the best known asymptotic lower and upper bounds on $a(\mathbb{Z}^d)$ as $d\to \infty$. We use percolation as a tool to obtain the latter, and conversely we use the former to obtain lower bounds on $p_c(\mathbb{Z}^d)$. We obtain the rigorous lower bound $\dot{p}_c(\mathbb{Z}^3)>0.2522$ for 3-dimensional site percolation.

math.PR↗
arXiv2020

Convergence of square tilings to the Riemann map

A well-known theorem of Rodin \& Sullivan, previously conjectured by Thurston, states that the circle packing of the intersection of a lattice with a simply connected planar domain $Ω$ into the unit disc $\mathbb{D}$ converges to a Riemann map from $Ω$ to $\mathbb{D}$ when the mesh size converges to 0. We prove the analogous statement when circle packings are replaced by the square tilings of Brooks et al.

math.PR↗
arXiv2020

Site percolation and isoperimetric inequalities for plane graphs

We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least $7$ have site percolation threshold bounded away from $1/2$, which was conjectured by Benjamini and Schramm, and make progress on a conjecture of Angel, Benjamini and Horesh that the critical probability is at most $1/2$ for plane triangulations of minimum degree $6$. We prove additional bounds for stronger minimum degree conditions, and for graphs without triangular faces.

math.PR↗
arXiv2017

Universal partial sums of Taylor series as functions of the centre of expansion

V. Nestoridis conjectured that if $Ω$ is a simply connected subset of $\mathbb{C}$ that does not contain $0$ and $S(Ω)$ is the set of all functions $f\in \mathcal{H}(Ω)$ with the property that the set $\left\{T_N(f)(z)\coloneqq\sum_{n=0}^N\dfrac{f^{(n)}(z)}{n!} (-z)^n : N = 0,1,2,\dots \right\}$ is dense in $\mathcal{H}(Ω)$, then $S(Ω)$ is a dense $G_δ$ set in $\mathcal{H}(Ω)$. We answer the conjecture in the affirmative in the special case where $Ω$ is an open disc $D(z_0,r)$ that does not contain $0$.

math.CV↗