Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-20T19:04:42.444Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant probability measures and dynamics of exponential linear type maps

Published online by Cambridge University Press:  01 October 2008

DAVID GAMARNIK ,
TOMASZ NOWICKI  and
GRZEGORZ ŚWIRSZCZ
DAVID GAMARNIK
Affiliation:
MIT Sloan School of Management, Cambridge, MA 02139, USA IBM Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY 10598, USA (email: gamarnik@mit.edu, tnowicki@us.ibm.com, swirszcz@us.ibm.com)
TOMASZ NOWICKI
Affiliation:
MIT Sloan School of Management, Cambridge, MA 02139, USA IBM Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY 10598, USA (email: gamarnik@mit.edu, tnowicki@us.ibm.com, swirszcz@us.ibm.com)
GRZEGORZ ŚWIRSZCZ
Affiliation:
MIT Sloan School of Management, Cambridge, MA 02139, USA IBM Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, NY 10598, USA (email: gamarnik@mit.edu, tnowicki@us.ibm.com, swirszcz@us.ibm.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of the asymptotic size of the random maximum-weight matching of a sparse random graph, which we translate into dynamics of the operator in the space of distribution functions. A tight condition for the uniqueness of the globally attracting fixed point is provided, which extends the result of Karp and Sipser [Maximum matchings in sparse random graphs. 22nd Ann. Symp. on Foundations of Computer Science (Nashville, TN, 28–30 October, 1981). IEEE, New York, 1981, pp. 364–375] from deterministic weight distributions (Dirac measures μ) to general ones. Given a probability measure μ which corresponds to the weight distribution of a link of a random graph, we form a positive linear operator Φμ (convolution) on distribution functions and then analyze a family of its exponents, with parameter λ, which corresponds to the connectivity of a sparse random graph. The operator 𝕋 relates the distribution F on the subtrees to the distribution 𝕋F on the node of the tree by 𝕋F=exp (−λΦμF). We prove that for every probability measure μ and every λ<e, there exists a unique globally attracting fixed point of the operator; the probability measure corresponding to this fixed point can then be used to compute the expected maximum-weight matching on a sparse random graph. This result is called the e-cutoff phenomenon. For deterministic distributions and λ>e, there is no fixed point attractor. We further establish that the uniqueness of the invariant measure of the underlying operator is not a monotone property of the average connectivity; this parallels similar non-monotonicity results in the statistical physics context.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008 , pp. 1479 - 1495
Copyright
Copyright © 2008 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aldous, D.. Asymptotics in the random assignment problem. Probab. Theory Related Fields 93(4) (1992), 507–534.CrossRefGoogle Scholar
[2]Aldous, D.. The ζ(2) limit in the random assignment problem. Random Structures Algorithms 18(4) (2001), 381–418.CrossRefGoogle Scholar
[3]Aldous, D. and Steele, J. M.. The objective method: Probabilistic combinatorial optimization and local weak convergence. Discrete Combinatorial Probability. Ed. H. Kesten. Springer, Berlin, 2003.Google Scholar
[4]Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, Singapore, 2000.CrossRefGoogle Scholar
[5]van der Berg, J. and Steif, J.. Percolation and hard-core lattice gas model. Stochastic Process. Appl. 49 (1994), 179–197.CrossRefGoogle Scholar
[6]Berger, N., Kenyon, C., Mossel, E. and Peres, Y.. Glauber dynamics on trees and hyperbolic graphs. Proc. 42nd IEEE Symp. on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA, 2001.Google Scholar
[7]Bollobas, B.. Random Graphs. Academic Press, New York, 1985.Google Scholar
[8]Brightwell, G. R., Häggström, O. and Winkler, P.. Nonmonotonic behavior in hard-core and Widom-Rowlinson models. J. Stat. Phys. 94(3–4) (1999), 415–435.CrossRefGoogle Scholar
[9]Brightwell, G. R. and Winkler, P.. Gibbs extremality for the hard-core model on a Bethe lattice. Preprint, 2003.Google Scholar
[10]Gamarnik, D.. Linear phase transition in random linear constraint satisfaction problems. Probab. Theory Related Fields 29(3) (2004), 410–440.CrossRefGoogle Scholar
[11]Gamarnik, D., Nowicki, T. and Świrszcz, G.. Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method. Approx–Random 2004 (Lecture Notes in Computer Science, 3122). Springer, Berlin, 2004, pp. 357–368.Google Scholar
[12]Janson, S., Luczak, T. and Rucinski, A.. Random Graphs. John Wiley, New York, 2000.CrossRefGoogle Scholar
[13]Karp, R. M. and Sipser, M.. Maximum matchings in sparse random graphs. 22nd Ann. Symp. on Foundations of Computer Science (Nashville, TN, 28–30 October, 1981). IEEE, New York, 1981, pp. 364–375.Google Scholar
[14]Martin, J. B.. Reconstruction thresholds on regular trees. Discrete Random Walks, DRW’03 (Discrete Mathematics and Theoretical Computer Science Proceedings AC). Eds. C. Banderier and C. Krattenthaler. Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2003, pp. 191–204.Google Scholar
[15]Mossel, E.. Survey: information flow on trees. Graphs, Morphisms and Statistical Physiscs. Eds. J. Nesetril and P. Winkler. American Mathematical Society, Providence, RI, 2004, pp. 155–170.CrossRefGoogle Scholar
[16]Martinelli, F., Sinclair, A. and Weitz, D.. The Ising model on trees: boundary conditions and mixing time. Proc. 44th IEEE Symp. on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA, 2003.Google Scholar