No CrossRef data available.

Published online by Cambridge University Press:
**05 August 2019**

The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let ${m=d(n-1)}$ be the degree and $V = n^d$ be the number of vertices of H(d, n). Let $p_c^{(d)}$ be the critical point for bond percolation on H(d, n). We show that, for $d \in \mathbb{N}$ fixed and $n \to \infty$,

$$p_c^{(d)} = {1 \over m} + {{2{d^2} - 1} \over {2{{(d - 1)}^2}}}{1 \over {{m^2}}} + O({m^{ - 3}}) + O({m^{ - 1}}{V^{ - 1/3}}),$$

which extends the asymptotics found in [10] by one order. The term $O(m^{-1}V^{-1/3})$ is the width of the critical window. For $d=4,5,6$ we have $m^{-3} =
O(m^{-1}V^{-1/3})$, and so the above formula represents the full asymptotic expansion of $p_c^{(d)}$. In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation on H(d, n) for $d=2,3,4$. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdös–Rényi random graph.- Type
- Paper
- Information
- Copyright
- © Cambridge University Press 2019

Aizenman, M. and Barsky, D. J. (1987) Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 489–526.CrossRefGoogle Scholar

Aizenman, M. and Newman, C. M. (1984) Tree graph inequalities and critical behavior in percolation models. J. Statist. Phys. 36 107–143.CrossRefGoogle Scholar

Aldous, D. (1997) Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 812–854.CrossRefGoogle Scholar

van den Berg, J. and Kesten, H. (1985) Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556–569.CrossRefGoogle Scholar

Bhamidi, S., van der Hofstad, R. and van Leeuwaarden, J. S. H. (2010) Scaling limits for critical inhomogeneous random graphs with finite third moments. Electron. J. Probab. 15 1682–1703.CrossRefGoogle Scholar

Bhamidi, S., van der Hofstad, R. and van Leeuwaarden, J. S. H. (2012) Novel scaling limits for critical inhomogeneous random graphs. Ann. Probab. 40 2299–2361.CrossRefGoogle Scholar

Bollobás, B. (1984) The evolution of random graphs. Trans. Amer. Math. Soc. 286 257–274.CrossRefGoogle Scholar

Bollobás, B. (2001) Random Graphs, second edition, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.CrossRefGoogle Scholar

Borgs, C., Chayes, J., van der Hofstad, R., Slade, G. and Spencer, J. (2005) Random subgraphs of finite graphs, I: The scaling window under the triangle condition. Random Struct. Alg. 27 137–184.CrossRefGoogle Scholar

Borgs, C., Chayes, J., van der Hofstad, R., Slade, G. and Spencer, J. (2005) Random subgraphs of finite graphs, II: The lace expansion and the triangle condition. Ann. Probab. 33 1886–1944.CrossRefGoogle Scholar

Borgs, C., Chayes, J., van der Hofstad, R., Slade, G. and Spencer, J. (2006) Random subgraphs of finite graphs, III: The phase transition for the *n*-cube. Combinatorica 26 395–410.CrossRefGoogle Scholar

Brydges, D. and Spencer, T. (1985) Self-avoiding walk in 5 or more dimensions. Comm. Math. Phys. 97 125–148.CrossRefGoogle Scholar

Durrett, R. (2007) Random Graph Dynamics, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.Google Scholar

Erdős, P. and Spencer, J. (1979) Evolution of the *n*-cube. Comput. Math. Appl. 5 33–39.CrossRefGoogle Scholar

Federico, L., van der Hofstad, R., den Hollander, F. and Hulshof, T. (2019) The scaling limit for critical percolation on the Hamming graph. In preparation.Google Scholar

Federico, L., van der Hofstad, R. and Hulshof, T. (2016) Connectivity threshold for random subgraphs of the Hamming graph. Electron. Commun. Probab., 21 #27.CrossRefGoogle Scholar

Fitzner, R. and van der Hofstad, R. (2013) Non-backtracking random walk. J. Statist. Phys. 150 264–284.CrossRefGoogle Scholar

Grimmett, G. (2012) Percolation and disordered systems. In Percolation Theory at Saint-Flour, Springer, pp. 141–303.Google Scholar

Hara, T. and Slade, G. (1990) Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128 333–391.CrossRefGoogle Scholar

Heydenreich, M. and van der Hofstad, R. (2017) Progress in High-Dimensional Percolation and Random Graphs, CMR Short Courses, Springer.CrossRefGoogle Scholar

van der Hofstad, R. (2017) Random Graphs and Complex Networks, Vol. 1, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.CrossRefGoogle Scholar

van der Hofstad, R. and Luczak, M. J. (2010) Random subgraphs of the 2D Hamming graph: The supercritical phase. Probab. Theory Related Fields 147 1–41.CrossRefGoogle Scholar

van der Hofstad, R., Luczak, M. J. and Spencer, J. (2010) The second largest component in the supercritical 2D Hamming graph. Random Struct. Alg. 36 80–89.CrossRefGoogle Scholar

van der Hofstad, R. and Nachmias, A. (2014) Unlacing hypercube percolation: A survey. Metrika 77 23–50.CrossRefGoogle Scholar

van der Hofstad, R. and Nachmias, A. (2017) Hypercube percolation. J. Eur. Math. Soc. 19 725–814.CrossRefGoogle Scholar

van der Hofstad, R. and Slade, G. (2005) Asymptotic expansions in $n^{-1}$ for percolation critical values on the *n*-cube and ${\mathbb Z}^n$. Random Struct. Alg. 27 331–357.CrossRefGoogle Scholar

van der Hofstad, R. and Slade, G. (2006) Expansion in $n^{-1}$ for percolation critical values on the *n*-cube and ${\mathbb Z}^n$: The first three terms. Combin. Probab. Comput. 15 695–713.CrossRefGoogle Scholar

Janson, S., Knuth, D., Łuczak, T. and Pittel, B. (1993) The birth of the giant component. Random Struct. Alg. 4 231–358.CrossRefGoogle Scholar

Janson, S. and Warnke, L. (2018) On the critical probability in percolation. Electron. J. Probab. 23 #1.CrossRefGoogle Scholar

Joseph, A. (2014) The component sizes of a critical random graph with given degree sequence. Ann. Appl. Probab. 24 2560–2594.CrossRefGoogle Scholar

Łuczak, T., Pittel, B. and Wierman, J. C. (1994) The structure of a random graph at the point of the phase transition. Trans. Amer. Math. Soc. 341 721–748.CrossRefGoogle Scholar

Menshikov, M. V. (1986) Coincidence of critical points in percolation problems. Soviet Math. Doklady 33 856–859.Google Scholar

Miłoś, P. and Şengül, B. (2019) Existence of a phase transition of the interchange process on the Hamming graph. Electronic J. Probab. 24 64.CrossRefGoogle Scholar

Nachmias, A. and Peres, Y. (2007) Component sizes of the random graph outside the scaling window. ALEA Lat. Am. J. Probab. Math. Stat. 3 133–142.Google Scholar

Nachmias, A. and Peres, Y. (2008) Critical random graphs: Diameter and mixing time. Ann. Probab. 36 1267–1286.CrossRefGoogle Scholar

Nachmias, A. and Peres, Y. (2010) Critical percolation on random regular graphs. Random Struct. Alg. 36 111–148.CrossRefGoogle Scholar

Pakes, A. G. (1971) Some limit theorems for the total progeny of a branching process. Adv. Appl. Probab. 3 176–192.CrossRefGoogle Scholar

Pittel, B. (2001) On the largest component of the random graph at a nearcritical stage. J. Combin. Theory Ser. B 82 237–269.CrossRefGoogle Scholar

Ráth, B. (2018) A moment-generating formula for Erdős–Rényi component sizes. Electron. Commun. Probab. 23 #24.CrossRefGoogle Scholar

Russo, L. (1981) On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56 229–237.CrossRefGoogle Scholar

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0

Total number of PDF views: 59 *

View data table for this chart

* Views captured on Cambridge Core between 05th August 2019 - 21st April 2021. This data will be updated every 24 hours.

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Expansion of Percolation Critical Points for Hamming Graphs

- Volume 29, Issue 1

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Expansion of Percolation Critical Points for Hamming Graphs

- Volume 29, Issue 1

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Expansion of Percolation Critical Points for Hamming Graphs

- Volume 29, Issue 1

×
####
Submit a response