The Self-Avoiding Walk

Madras, Neal; Slade, Gordon

doi:10.1007/978-1-4612-4132-4

Cited by 279 publications

(514 citation statements)

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“…The smallest of such transition "temperatures" corresponds to the case of nearest neighbors, Eq. (18). Finally, we note that, under the same assumptions that we used for the ohmic bath, superohmic environments are local and therefore will always yield a finite transition "temperature".…”

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confidence: 87%

Surface Code Threshold in the Presence of Correlated Errors

Novais

Mucciolo

2013

Phys. Rev. Lett.

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We study the fidelity of the surface code in the presence of correlated errors induced by the coupling of physical qubits to a bosonic environment. By mapping the time evolution of the system after one quantum error correction cycle onto a statistical spin model, we show that the existence of an error threshold is related to the appearance of an order-disorder phase transition in the statistical model in the thermodynamic limit. This allows us to relate the error threshold to bath parameters and to the spatial range of the correlated errors.Comment: 5 pages, 2 figure

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confidence: 87%

Surface Code Threshold in the Presence of Correlated Errors

Novais

Mucciolo

2013

Phys. Rev. Lett.

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“…For the two dimensional proofs we will make extensive use of the following theorem due to Madras and reported in [44], Theorem 9.7.2, pag. 356:…”

Section: Definitionmentioning

confidence: 99%

“…Therefore, an additional factor of N should be added for the kink-kink bilocal algorithm: in this case, we expect τ ∼ > N 3 . It is interesting to notice that the kink-kink 9 We refer the reader interested in more precise and rigorous statements to [44,48]. 10 In [49] the following rigorous lower bound was obtained:…”

Section: Dynamic Critical Behaviourmentioning

confidence: 99%

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Caracciolo

Causo

Ferraro

et al. 2000

Journal of Statistical Physics

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“…This is a random walk which does not visit the same node more than once [25]. At each step, the walker chooses its next move randomly from the neighbors of its present node, excluding nodes that were already visited.…”

Section: Introductionmentioning

confidence: 99%

The distribution of path lengths of self avoiding walks on Erdős–Rényi networks

Tishby¹,

Biham²,

Katzav³

2016

J. Phys. A: Math. Theor.

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Abstract. We present an analytical and numerical study of the paths of self avoiding walks (SAWs) on random networks. Since these walks do not retrace their paths, they effectively delete the nodes they visit, together with their links, thus pruning the network. The walkers hop between neighboring nodes, until they reach a deadend node from which they cannot proceed. Focusing on Erdős-Rényi networks we show that the pruned networks maintain a Poisson degree distribution, p t (k), with an average degree, k t , that decreases linearly in time. We enumerate the SAW paths of any given length and find that the number of paths, n T (ℓ), increases dramatically as a function of ℓ. We also obtain analytical results for the path-length distribution, P (ℓ), of the SAW paths which are actually pursued, starting from a random initial node. It turns out that P (ℓ) follows the Gompertz distribution, which means that the termination probability of an SAW path increases with its length.

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The Self-Avoiding Walk

Cited by 279 publications

References 0 publications

Surface Code Threshold in the Presence of Correlated Errors

Surface Code Threshold in the Presence of Correlated Errors

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The distribution of path lengths of self avoiding walks on Erdős–Rényi networks

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