Maximum No-Guess-solvable mine density in a finite $n^3$ 3D Minesweeper grid...
BackgroundI recently built a free browser game: a 3D Minesweeper where thegrid is a real cube of $n^3$ cells ($n = 4$ to $8$), and each inner cellhas up to $26$ neighbors instead of $8$ in classic 2D...
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Connecting Origin to the Boundary of a Box Implies Crossing Events
I'm working with a renormalization argument to prove a sub-exponential decay of the one-arm event (the origin is connected to the boundary $\partial \mathcal{B}_n$ of a box $\mathcal{B}_n$ with side...
View ArticleIntersection of Bernoulli Bond Percolation Infinite Cluster with an Arbitrary...
I found this problem through the internet:Consider bond percolation on $\mathbb{Z}^d$ and assume $p>p_c$. Let A be an infinite subset of vertices of $\mathbb{Z}^d$. Prove that$$P_p(\mbox{the...
View ArticleBond percolation: graph of trifurcation points $(T_L,\sim)$ is acyclic from...
Consider bond percolation graph in $\mathbb{Z}^d$, $p>0$. Let $B_L = \{-L,\dots, L\}^d$.We call $x$ a trifurcation point if it is in an infinite open cluster, there are exactly three edges with $x$...
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Identifying/characterizing a phase transition in a 2D growth model
This is an old question that's kicked around in my head every so often. We generate a semi-infinite array $\{a(j,k)\}_{j,k\geq 0}$ of bits as follows. First, set $a(0,j)=a(j,0)=1\text{ for all }j$. For...
View ArticleHow can I prove $q_{2n} \leq (c \cdot q_n)^2$ for $q_n = 1 -...
I am trying to solve the following exercise on Percolation theory:On $\mathbb{Z}^2$, consider the event$C_{2n,n} = \{\text{there exists an open path from left to right in } [0,2n]\times[0,n]\},$where...
View ArticleErdos random graph and percolation
For random graph, we chose the space $(\Omega, F, P) = ([0,1]^{\binom{n}{2}}, B^{\otimes}[0,1], Leb^{\otimes}{[0,1]})$ and we can define $U_{ij} = \pi_i$ (Projection maps, fix some labelleing). It is...
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Conditions for removing the multiplicative constant in exponential decay...
I'm studying percolation theory and I'm confused with some standard things on exponential decay. To get the things started, let $(x_n)$ a sequence of non-negative numbers. I will say that $(x_n)$ have...
View ArticleInequality instead of equality
Why in Introduction to Bernoulli Percolation does the author writeLet us start by showing that $\mathbb{P}_p[E_{<\infty} ∖ E_{≤1}] = 0$. By ergodicity, $E_{<\infty}$ and $E_{≤1}$ both...
View ArticleUniqueness of Infinite Cluster in Percolation Model Proof Question
In A. Gandolfi, G. Grimmett, and L. Russo's proof On the Uniqueness of the Infinite Cluster in the Percolation Model, they have on equation (8) that$$1-\mu_p(B_\Lambda)\leq |\Lambda|...
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The number of infinite clusters in Bernoulli percolation is a random variable...
In the proof of the Uniqueness of the infinite connected component (Theorem 3.1 of lecture note), it use the fact that {the number of infinite clusters is $K$} and {$K$ infinite clusters intersect the...
View ArticleIs the set of critical values for percolation dense in $(0,1)$?
Given a locally finite, countably infinite connected graph, define its critical probability as$$p_c(G):=\inf\left\{p\in (0,1]: (V(G),E(G)_p) \substack{\text{ contains an infinite connected}\\...
View ArticleThe Bethe lattice and the infinite dimensional hypercubic lattice
The Bethe lattice is a special lattice where each site has $z$ neighboring sites such that each branch gives rise to $z-1$ other branches. The Bethe lattice with $z=2$ corresponds to the...
View ArticleExercise 15 in Duminil-Copin's Percolation Notes
We are in the context of Bernoulli bond percolation in $\Bbb{Z}^{d}$.Exercise 15 in Hugo Duminil-Copin's notes on Percolation states that if for a finite set $S$ containing the origin, if...
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Site Percolation and S-clusters with $n\times m$ grid where $n\not =m$?
Consider a site percolation but change the dimension of the lattice from $n\times n$ to $n\times m$ where $n\not = m.$ S-clusters are defined for $n\times n$ lattice. The occupation of each site are...
View ArticleFind the Minimum Spanning Hyperedge
I am looking for an algorithm that finds a Tree that spans a set of Terminals in a hypergraph. Let's say we have (3) terminals {A, B, C}. I define the Tree as a set of hyperedges and edges:$ T \equiv...
View ArticleExponential Growth and Decay Model for Human Genealogy (Common Ancestor)
First time poster, and, as my post will intimate, not a mathematician, just someone searching for answers. My question has two parts:1) In a genealogical chart for a single individual (called an...
View ArticleSize of connected regions on a randomly-colored infinite chessboard
Consider an infinite chessboard where each square is colored white with probability $p$ and black with probability $1-p$. Suppose without loss of generality that the square at $(0,0)$ is white. We can...
View ArticleExpected largest size of the largest connected component on a randomly...
Consider a randomly colored $m \times n$ chessboard with each square having a probability $0 < p < 1$ of being colored white. We say that a maximal monochromatic region is a collection of squares...
View ArticleWhat does it mean to "determine" the time constant for first passage...
A fundamental question in FPP is to determine the time constant for some non-trivial distribution on $\mathbb{Z}^2$ (https://arxiv.org/pdf/1511.03262, question 1). We would all be very happy if there...
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