A New Upper Bound for the Critical Probability of the Frog Model on Homogeneous Trees

The square lattice site percolation model critical probability is shown to be at most .679492, improving the best previous mathematically rigorous upper bound. This bound is derived by extending the substitution method to apply to site percolation models.

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The critical probability for the frog model is not a monotonic function of the graph

Journal of Applied Probability ◽

10.1239/jap/1077134688 ◽

2004 ◽

Vol 41 (1) ◽

pp. 292-298 ◽

Cited By ~ 5

Author(s):

L. R. Fontes ◽

F. P. Machado ◽

A. Sarkar

Keyword(s):

Percolation Model ◽

Monotonic Function ◽

Critical Probability ◽

Frog Model

We show that the critical probability for the frog model on a graph is not a monotonic function of the graph. This answers a question of Alves, Machado and Popov. The nonmonotonicity is unexpected as the frog model is a percolation model.

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An Improved Upper Bound for Bootstrap Percolation in All Dimensions

Combinatorics Probability Computing ◽

10.1017/s0963548319000130 ◽

2019 ◽

Vol 28 (06) ◽

pp. 936-960

Author(s):

Andrew J. Uzzell

Keyword(s):

Upper Bound ◽

Bootstrap Percolation ◽

Exact Constant ◽

Critical Probability ◽

Time Step ◽

Natural Logarithm ◽

Vertex Set ◽

Image Position

AbstractIn r-neighbour bootstrap percolation on the vertex set of a graph G, a set A of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least r previously infected neighbours. When the elements of A are chosen independently with some probability p, it is natural to study the critical probability pc(G, r) at which it becomes likely that all of V(G) will eventually become infected. Improving a result of Balogh, Bollobás and Morris, we give a bound on the second term in the expansion of the critical probability when G = [n]d and d ⩾ r ⩾ 2. We show that for all d ⩾ r ⩾ 2 there exists a constant cd,r > 0 such that if n is sufficiently large, then $$p_c (\left[ n \right]^d ,{\rm{ }}r){\rm{\le }}\left( {\frac{{\lambda (d,r)}}{{\log _{(r - 1)} (n)}} - \frac{{c_{d,r} }}{{(\log _{(r - 1)} (n))^{3/2} }}} \right)^{d - r + 1} ,$$where λ(d, r) is an exact constant and log(k) (n) denotes the k-times iterated natural logarithm of n.

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An Improved Upper Bound for the Hexagonal Lattice Site Percolation Critical Probability

Combinatorics Probability Computing ◽

10.1017/s0963548302005345 ◽

2002 ◽

Vol 11 (6) ◽

pp. 629-643 ◽

Cited By ~ 5

Author(s):

JOHN C. WIERMAN

Keyword(s):

Upper Bound ◽

Lattice Site ◽

Hexagonal Lattice ◽

Critical Probability ◽

Site Percolation ◽

Substitution Method ◽

Site Model ◽

Bond Model

The hexagonal lattice site percolation critical probability is shown to be at most 0.79472, improving the best previous mathematically rigorous upper bound. The bound is derived by using the substitution method to compare the site model with the bond model, the latter of which is exactly solved. Shortcuts which eliminate a substantial amount of computation make the derivation of the bound possible.

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Upper bounds for the critical probability of oriented percolation in two dimensions

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1993.0012 ◽

1993 ◽

Vol 440 (1908) ◽

pp. 201-220 ◽

Cited By ~ 4

Keyword(s):

Upper Bound ◽

Upper Bounds ◽

Two Dimensions ◽

Oriented Percolation ◽

Bond Percolation ◽

Critical Probability ◽

Site Percolation

We give a method for obtaining upper bounds on the critical probability in oriented bond percolation in two dimensions. This method enables us to prove that the critical probability is at most 0.6863, greatly improving the best published upper bound, 0.84. We also prove that our method can be used to give arbitrarily good upper bounds. We also use a slight variant of our method to obtain an upper bound of 0.72599 for the critical probability in oriented site percolation.

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Upper bounds on the rate of low density stabilizer codes for the quantum erasure channel

Quantum Information and Computation ◽

10.26421/qic13.9-10-4 ◽

2013 ◽

Vol 13 (9&10) ◽

pp. 793-826

Author(s):

Nicolas Delfosse ◽

Gilles Zemor

Keyword(s):

Upper Bound ◽

Percolation Theory ◽

Hyperbolic Plane ◽

Upper Bounds ◽

Low Density ◽

Critical Probability ◽

Stabilizer Codes ◽

Erasure Channel ◽

Finite Quotients ◽

Quantum Erasure

Using combinatorial arguments, we determine an upper bound on achievable rates of stabilizer codes used over the quantum erasure channel. This allows us to recover the no-cloning bound on the capacity of the quantum erasure channel, $R \leq 1-2p$, for stabilizer codes: we also derive an improved upper bound of the form $R \leq 1-2p-D(p)$ with a function $D(p)$ that stays positive for $0<p<1/2$ and for any family of stabilizer codes whose generators have weights bounded from above by a constant -- low density stabilizer codes. We obtain an application to percolation theory for a family of self-dual tilings of the hyperbolic plane. We associate a family of low density stabilizer codes with appropriate finite quotients of these tilings. We then relate the probability of percolation to the probability of a decoding error for these codes on the quantum erasure channel. The application of our upper bound on achievable rates of low density stabilizer codes gives rise to an upper bound on the critical probability for these tilings.

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The critical probability for the frog model is not a monotonic function of the graph

Journal of Applied Probability ◽

10.1017/s0021900200014248 ◽

2004 ◽

Vol 41 (01) ◽

pp. 292-298 ◽

Cited By ~ 1

Author(s):

L. R. Fontes ◽

F. P. Machado ◽

A. Sarkar

Keyword(s):

Percolation Model ◽

Monotonic Function ◽

Critical Probability ◽

Frog Model

We show that the critical probability for the frog model on a graph is not a monotonic function of the graph. This answers a question of Alves, Machado and Popov. The nonmonotonicity is unexpected as the frog model is a percolation model.

Download Full-text