# Questions tagged [percolation]

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### The uniform odd and even subgraph of $\mathbb{Z}^2$

Given a (first finite and later infinite) graph $G =(V,E)$ the uniform even graph is the uniform probability measure on the set of spanning even subgraphs. That is subgraphs (V, E') with $E' \subset E$...
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### Multi-scale 3- and 5-arm exponents for critical planar percolation

Consider critical site percolation on the planar triangular lattice. Denote by $A_j(m,n)$ the event that there are $j$ arms (paths from the inner boundary to the outer boundary) of alternating colour ...
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### The fluctuations of a random path

Suppose I have a $n \times n$ square grid and for each square, I assign 1 with probability $\frac{1}{2}$ and 0 with probability $\frac{1}{2}$. On the boundary, I put 1s on the lower half and 0s on the ...
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### A disc contains many random points. Each point is connected to its nearest neighbor. What is the expectation of average cluster size?

A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points. For example, here are $20$ random ...
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### Can I explore the infinite cluster of Bernoulli percolation in $\mathbb{Z}^2$?

In Bernoulli percolation on the square lattice $\mathbb{Z}^2$ every edge is kept with a probability $q$ and erased with probability $1-q$. It is classical that whenever $q > \frac{1}{2}$ there is ...
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### How slowly can one be absorbed in the absorbing phase of the directed percolation universality class?

In absorbing state transitions on a lattice, one often considers "active" and "inactive" sites, with update rules describing how activity spreads or decays. When the lattice sites ...
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### Infinite clusters for loopless percolation

I feel like this is maybe an incredibly trivial problem, and I'm just missing something. I may also be describing a well-known model that I cannot find the name for, so any comment/suggestion is ...
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### Likelihood ratio of non-trivial cycles in an inhomogeneous random square lattice graph embedded on a toroidal surface

Consider a square lattice (random) graph $G$ embedded on a toroidal surface. Each edge $(i, j)$ of the graph has an associated likelihood probability $p_{ij}$. The probabilities $p_{ij}$ come from a ...
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### An elementary question in bond percolation

Consider a locally finite, connected graph and "bond (edge) percolation" on this graph. Each edge is open with probability $p.$ There is a parameter $\alpha$, $0<\alpha<1.$ The ...
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### Existence of a bigeodesic in last passage percolation is $0$-$1$ event

On the bottom of page two of This paper, the authors remark the following: '...by translation invariance and ergodicity, we know that existence of a bigeodesic is a $0−1$ event and hence it follows ...
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### Understanding the wrapping criterion in percolation theory

Context: When studying percolation in finite sized systems, there exist various definitions and criteria for determining when a given system is percolating, i.e., given a definition for connectivity, ...
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Let $G$ be an acyclic directed graph with $MN$ vertices arranged into $M$ generations of $N$ vertices each. We stipulate that edges may only go from generation $j$ to generation $j+1$, so there are (... 1 vote 0 answers 50 views ### Vertical and horizontal percolation on heterogeneous honeycomb lattice I have a regular honeycomb lattice where a bond in the unit cell aligns with(1,0)$; call this the horizontal direction. Each horizontal bond in the lattice is open with probability$p$and each "... 1 vote 0 answers 88 views ### Percolation and diameter of graph Is the critical probability in percolation and diameter of graph related. I guess larger the diameter higher the probability. Is there any result like this? By critical probability I mean the ... 0 votes 0 answers 113 views ### How to mathematically justify the "sampling" over only$100$random matrices to estimate percolation thresholds? As mentioned in the textbook "Introduction to Percolation Theory" (Chapter 4) by Stauffer et al., the variation of spanning cluster percolation probability$\Pi$in a finite$L < \infty$square ... 2 votes 1 answer 87 views ### Generalization: (The "number" of) smaller sized clusters in large random binary matrices follow a descending order. Why? This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix? In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ... 3 votes 1 answer 175 views ### Why is number of single cell clusters always greatest in a random matrix? Consider a large$N\times N$square lattice, where each cell has a probability$p$of being "occupied" (let's call denote them as "black") and a probability$1-p$of being empty (let's denote them as "... 6 votes 0 answers 116 views ### Length of optimal play in Hex as a function of size Consider Hex on an$n \times n$board without a swap rule, so that the first player wins. Assume the first player tries to minimize the length of the game, and the second player tries to maximize the ... • 1,678 1 vote 1 answer 112 views ### Probability for a group of stones to live on an infinite Go board Suppose on an infinite two dimensional Go board the tengen is occupied by a black stone, and every other grid point is occupied by a black stone, or a white stone, or nothing, with probability 1/3 ... • 5,109 1 vote 1 answer 153 views ### Does there exist any analogous result for site percolation? This is a sequel to the question: Proof and interpretation of the following percolation theory result for$n\times n$square grid In the paper: The Birth of the Infinite Cluster:Finite-Size Scaling ... 2 votes 1 answer 380 views ### Proof and interpretation of the following percolation theory result for$n\times n$square grid While I was discussing this question with @JamesMartin, he mentioned a result here that: In a$n\times n$finite square grid, if$p\geq p_c+\epsilon$, such that$\epsilon>0$and$p_c$is the ... 2 votes 1 answer 95 views ### References on the structure of bond percolation on the (finite) 2D-grid in the sub-critical regime (e.g p=1/10) Would appreciate references to the most up-to-date results for the structure of bond percolation on the (finite) 2D-grid in the sub-critical regime (e.g,$p=1/10$). Thank you. • 131 0 votes 0 answers 80 views ### How to calculate the exact probability$p\$ at which maxima occurs in the curves in an infinite system?

I'm writing with respect to this paper: Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice Here's a ...