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....4, 2002 1 Introduction Since the origins of percolation theory in the 1950s, the determination of critical probabilities has been an important and challenging problem. To date, exact solutions have been found only for arbitrary trees [12] and a small number of periodic two dimensional graphs [8, 9, 17, 18]. For other graphs of interest, the problem has been approached by simulation and estimation, and through rigorous bounds. An additional goal of these lines of research is to understand the dependence of the critical probability upon the detailed structure of the underlying graph and possibly to ....

....of the bowtie lattice is the solution of 1 6p = 0 in the interval [0,1] while p c (D(B) 1 p c (B) producing the approximations given in Table 1. A key to the exact solutions for bond percolation thresholds mentioned above is the use of duality. An important theorem of Kesten [9] states that the bond percolation critical probabilities of a dual pair of periodic lattices sum to one. 3.2 Substitution Method Bounds The substitution method determines critical probability bounds using stochastic ordering and coupling methods. Recent advances in computational techniques for ....

Harry Kesten (1982) Percolation Theory for Mathematicians, Birkhauser.

....bonds in a cube Q l (w) and hence at most 9j jl in B(A) and at most 15j jl in B(A ) Therefore j j d B (x; y) 15l . Now must be an l connected subset of Z with z 2 . The number of possible such lattice animals with j j = n is at most 8 for all n, by the argument of ([21], p. 85. Thus provided is suciently small we have, again by a routine Peierls type argument, P B; x y) P B; j j P B; jf( j 4 d B (x;y) 60l which proves the theorem for the metric d B . Let B (x; r) denote the ball of radius r centered at x. Consider ....

Kesten, H., Percolation Theory for Mathematicians, Birkhauser, Boston (1982).

....we obtain. Moreover, since G(n; r(n) can be connected without D being entirely covered by n discs of radius r(n) this approach does not lead to any necessary conditions on r(n) for asymptotic connectivity in G(n; r(n) Yet another related problem considered is in continuum percolation theory ([4, 8]) Nodes are assumed to be distributed with Poisson intensity in , and two nodes are connected to each other if the distance between them is less than r. Then the problem considered is to nd a critical value of r such that the origin is connected to an in nite order component. We will, in ....

H. Kesten, Percolation Theory for Mathematicians, Birkhauser, Boston, MA, 1982.

....models for connectivity in wireless networks since edges are introduced independent of the distance between nodes. Thus such a graph may have a link from a node to a faraway node, without a link to a nearer node. Connectedness has also been considered in the eld of continuum percolation theory [13, 14]. There, the model for the points is a Poisson point process on the in nite plane, and the focus is on the existence of an in nite size connected component under di erent models of connections. Recently, 15, 16] have addressed wireless networks and covering algorithms by the methods of continuum ....

H. Kesten, Percolation Theory for Mathematicians. Boston, MA: Birkhauser, 1982.

....and upper bounds can be found for these critical values for several lattice graphs, usually by methods that require extensive computer calculations. In classical percolation theory, the exact bond model critical probabilities or site model critical probabilities are known for only a few graphs [14, 15, 25, 26], thus making it important to determine rigorous bounds for unsolved graphs [4, 27, 28, 29, 30, 31] Many simulation studies have estimated critical probabilities of various graphs, in particular the Archimedean lattices [24] In first passage percolation, other than its counterpart for infinite ....

Harry Kesten (1982) Percolation Theory for Mathematicians, Birkhauser.

....a better result is obtained by replacing fi(G) by the corresponding quantity that results when K is required to contain a fixed point o in the definition of fi(G) The same proof applies. To prove Proposition 4. 7, we shall use the following bound analogous to the wellknown bound of Kesten [33] on site connected clusters. Lemma 4.9. Let G be a graph with degrees bounded by d. For any fixed o 2 V (G) let b n be the number of connected subgraphs of G that contain o and have exactly n edges. Then lim sup n 1 b 1=n n e(d Gamma 1) Proof. Let b n; denote the number of connected ....

Kesten, H. (1982) Percolation Theory for Mathematicians. Birkhauser, Boston, Mass.

....proposition, a better result is obtained by replacing (G) by the corresponding quantity that results when K is required to contain a xed point o in the de nition of (G) The same proof applies. To prove Proposition 4. 7, we shall use the following bound analogous to the wellknown bound of Kesten [33] on site connected clusters. Lemma 4.9. Let G be a graph with degrees bounded by d. For any xed o 2 V (G) let b n be the number of connected subgraphs of G that contain o and have exactly n edges. Then lim sup n 1 b 1=n n e(d 1) Proof. Let b n; denote the number of connected subgraphs ....

Kesten, H. (1982) Percolation Theory for Mathematicians. Birkhauser, Boston, Mass.

....physics There is an extensive literature in mathematical physics concerned with asymptotic enumeration, especially in Ising models of statistical mechanics and percolation methods. Many of the methods are related to combinatorial enumeration. For an introduction to them, see Chapter or the books [30, 226]. 16.4. Classical applied mathematics There are many techniques, such as the ray method and the WKB method, that have been developed for solving di#erential and integral equations in what we might call classical applied mathematics. An introduction to them can be found in [31] They are powerful, ....

H. Kesten, Percolation Theory for Mathematicians, Birkhauser, 1982.

....about an incipient infinite cluster at p c . This brings us to our third motivation. The Incipient Infinite Cluster At p c , it is believed that with probability one there is no infinite cluster. On the other hand, the expected size of the cluster of the origin is infinite at p c , see [Ham57] [Kes82], Cor. 5.1, and [AN84] This suggests that from the perspective of an observer at the origin, all clusters are finite, with larger and larger clusters appearing as one considers larger and larger length scales. Physicists have called the emerging object the incipient infinite cluster. In the ....

....in general dimension, much less is know about L 0 (p) or A 0 (p) above p c . In particular, below p c , it is easy to see that L 0 (p) is monotone increasing, left continuous and piecewise constant. Moreover, p c , 2.27) because R L,3L (p c ) is bounded away from 0 (e.g. by Theorem 5. 1 in [Kes82]) Furthermore, the jumps in L 0 (p) are uniformly bounded on a logarithmic scale. In particular, by the methods of [ACCFR83] CC86] CCF85] and [Kes87] we have R 2L,6L a(d) R L,3L , 2.28) which in turn implies L 0 (p #) L 0 (p) # 2 , 2.29) provided p p c and # a(d) By ....

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H. Kesten, Percolation Theory for Mathematicians, Birkhauser, Boston, 1982.

....property in Proposition 1 which allows us to let # go to 0 afterwards. This continuity property needs proof because T may be infinite. In finite networks continuity of the e#ective resistance between two vertices, as a function of the resistances of the individual edges, is comparatively easy (see Kesten (1982) Ch. 11) 2. Preliminaries Some standard ways to combine resistances were already discussed in Section 2 of Grimmett and Kesten (1983) We need some (known) extensions of these rules, especially for the case where individual edges may have zero resistance. Let G be a finite connected graph and A ....

....over all edges e of G with one endpoint in the class corresponding to # v and the other endpoint outside this class; the class of the endpoint of e outside # v is denoted by # w(e) Note that any R(e) appearing in (2. 1) is strictly positive by our choice of the classes # v and # w (see Kesten (1982) Ch. 11 for more details) We shall frequently appeal to the following probabilistic interpretation of V ( see Doyle and Snell (1982) Gri#eath and Liggett (1982) Consider a Markov chain X # on # G with transition probability (2.2) P (#v, # w) # # v 1 R(e) # 1 # v,w 1 ....

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Kesten, H., Percolation Theory for Mathematicians, Birkhauser, Boston, 1982.

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H. Kesten, Percolation Theory for Mathematicians, Birkhauser 1982.

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Kesten, H. (1982). Percolation Theory for Mathematicians. Birkhauser.

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H. Kesten. Percolation Theory for Mathematicians, volume 2. Birkhauser, Boston, 1982.

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H. Kesten. Percolation Theory for Mathematicians, volume 2. Birkhauser, Boston, 1982.

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Kesten, H. (1982) Percolation theory for mathematicians. Boston MA, Birkhauser.

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Kesten, H., Percolation theory for mathematicians, Birkhauser, (1982) New York.

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Kesten, H. (1982) Percolation Theory for Mathematicians, Birkhauser, Boston.

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H. KESTEN. Percolation Theory for Mathematicians, volume 2 of Progress in Probability. Birkh#user, Boston, 1982.

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H. KESTEN. Percolation Theory for Mathematicians, volume 2 of Progress in Probability. Birkh#user, Boston, 1982.

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H. Kesten, Percolation Theory for Mathematicians. Birkhauser, 1982.

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Kesten, H. (1982) Percolation Theory for Mathematicians. Birkhauser, Boston, Mass.

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Kesten, H. (1982). Percolation theory for mathematicians (Birkhauser, Boston).

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H. Kesten (1980). Percolation Theory for Mathematicians, Birkhauser.

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H. Kesten (1980). Percolation Theory for Mathematicians, Birkhauser.

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H. Kesten. Percolation theory for mathematicians. Birkhauser Boston, Mass., 1982.

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