The connective constant of the honeycomb lattice equals sqrt(2+sqrt 2)

Duminil-Copin, Hugo; Smirnov, Stanislav

doi:10.4007/annals.2012.175.3.14

Cited by 167 publications

(260 citation statements)

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“…The limit is known as SLE 6 , which has an independent definition (to be given below) that does not depend on the lattice. We remark that Smirnov's proof only applies to the hexagonal lattice, although it is widely believed that Theorem 1.0.1 holds for several other lattices, such as the square lattice.…”

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confidence: 99%

Smoothness of Loewner slits

Wong

2013

Trans. Amer. Math. Soc.

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confidence: 99%

Smoothness of Loewner slits

Wong

2013

Trans. Amer. Math. Soc.

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“…Despite decades of efforts, these questions remain unsolved, especially in low dimensions. It has recently been shown [6] ≈ 2.64). For two-dimensional lattices, it is conjectured that γ = 43/32 and that, for the uniform distribution, ν = 3/4.…”

Section: Introductionmentioning

confidence: 88%

“…Despite decades of efforts, these questions remain unsolved, especially in low dimensions. It has recently been shown [6] that the connective constant for the hexagonal lattice on plane is 2 + √ 2. Conversely, for the square lattice on plane, it is conjectured [11] that the connective constant is the unique positive root of 13x…”

Section: Introductionmentioning

confidence: 99%

An optimal algorithm to generate extendable self-avoiding walks in arbitrary dimension

Préa

Rouault

Brucker

2017

Electronic Notes in Discrete Mathematics

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A self-avoiding walk (SAW) is extendable [10,13] if it can be extended into an infinite SAW. We give a simple proof that, for every lattice, extendable SAWs admit the same connective constant that the general SAWs and we give an optimal linear algorithm to generate random extendable SAWs. Our algorithm can generate every extendable SAW in dimension 2. For dimension d > 2, it generates only a subset of the extendable SAWs. We conjecture that this subset is "large" and has the same connective constant that the extendable SAWs. Our algorithm produces a kinetic distribution of the extendable SAWs, for which the critical exponent ν ≈ .57 for d = 2, .51 for d = 3 and .50 for d = 4, 5, 6. Keywords: Self Avoiding Walks, Connective Constant, Critical Exponent ν, Random Generation. IntroductionA walk on a lattice is self-avoiding if it never passes twice through the same vertex (see Figure 1). Self-avoiding walks (SAWs) appeared as a model for polymers [7]. They also have applications in statistical physics [8] and in probability theory [15]. Formally speaking, for instance on the square grid Z × Z on plane, a walk is a sequence W = (w 0 , w 1 , . . . w n ) of vertices such that, for all i < n, w i and w i+1 are neighbors (i.e. if w i = (x i , y i ) and w i+1 = (x i+1 , y i+1 ), then (x i+1 = x i and y i+1 = y i ± 1) or (y i+1 = y i and x i+1 = x i ± 1)). The walk W is self-avoiding if i = j ⇒ w i = w j .Despite a very simple and natural definition, and although they are very closed to random walks (which are standard objects, very well studied and known) self-avoiding walks ask many problems, both theoretical and practical.Given a lattice (for instance, the hexagonal lattice on plane or the cubic one Z × Z × Z in space), the two main theoretical questions concerning SAWs on that lattice are:• What is the number c n of SAWs of length n?• Given a distribution on the SAWs (for instance the uniform distribution on the SAWs of length n), what is the average Euclidean distance d n between 1 Part of this work was done while one of the authors (P.P.) was visiting the University of Cape Town. It is strongly believed that the critical exponents γ and ν depend only on the dimension (and not on the lattice) while the connective constant µ also depends on the lattice. It is known that the connective constant (which is equal to lim n→∞ c 1/n n ) exists for every lattice. Despite decades of efforts, these questions remain unsolved, especially in low dimensions. It has recently been shown [6] ≈ 2.64). For two-dimensional lattices, it is conjectured that γ = 43/32 and that, for the uniform distribution, ν = 3/4.From a practical point of view, it is very interesting to generate random self-avoiding walks since these walks arise as models for various physical phenomena. In addition, generating random self avoiding walks yields an approximation to the critical exponents and the connective constant. There are many algorithms to generate random self-avoiding walks; among them the pivot algorithm [14,12], the Berretti-Sokal algorithm ...

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“…For example, if we take ℱ N to be the set of all SAWs, we can split a walk w ∈ ℱ N into a unique initial part of length m and a final part of length N − m , which establishes (2). In general, determining the value of the connective constant requires significant ingenuity (see, e.g., [10], which establishes the value for SAWs on the two-dimensional hexagonal lattice).…”

Section: Introductionmentioning

confidence: 99%

A Study of the Boltzmann Sequence-Structure Channel

Magner

Szpankowski²

2017

Proc. IEEE

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We rigorously study a channel that maps sequences from a finite alphabet to self-avoiding walks in the two-dimensional grid, inspired by a model of protein folding from statistical physics and studied empirically by biophysicists. This channel, which we call the Boltzmann sequence-structure channel, is characterized by a Boltzmann/Gibbs distribution with a free parameter corresponding to temperature. In our previous work, we verified empirically that the channel capacity appears to have a phase transition for small temperature and decays to zero for high temperature. In this paper, we make some progress toward theoretically explaining these phenomena. We first estimate the conditional entropy between the input sequence and the output fold, giving an upper bound which exhibits a phase transition with respect to temperature. Next, we formulate a class of parameter settings under which the dependence between walk energies is governed by their number of shared contacts. In this setting, we derive a lower bound on the conditional entropy. This lower bound allows us to conclude that the mutual information tends to zero in a nontrivial regime of high temperature, giving some support to the empirical fact regarding capacity. Finally, we construct an example setting of the parameters of the model for which the conditional entropy is exactly calculable and which does not exhibit a phase transition.

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The connective constant of the honeycomb lattice equals sqrt(2+sqrt 2)

Cited by 167 publications

References 13 publications

Smoothness of Loewner slits

Smoothness of Loewner slits

An optimal algorithm to generate extendable self-avoiding walks in arbitrary dimension

A Study of the Boltzmann Sequence-Structure Channel

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