A general law of large permanent

JÓzsef Balogh; Hoi Nguyen

doi:10.3934/dcds.2017229

Article Contents

2017, Volume 37, Issue 10: 5285-5297. Doi: 10.3934/dcds.2017229

This issue Previous Article A locally integrable multi-dimensional billiard system Next Article Nodal Bubble-Tower Solutions for a semilinear elliptic problem with competing powers

A general law of large permanent

JÓzsef Balogh^1, and
Hoi Nguyen^2,

1.
Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
2.
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA

Received: February 2017

Revised: April 2017

Published: October 2017

J. Balogh is partially supported by NSF grant DMS-1500121, Arnold O. Beckman Research Award (UIUC Campus Research Board 15006) and Langan Professional Scholarship. H. Nguyen is partially supported by NSF grant DMS-1600782.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

In this short note we establish a law of large permanent for matrices with entries from an $\mathbf{N}^2$-indexed stochastic process. This answers a question by Bochi, Iommi and Ponce in [4].

Keywords:

Permanent,
law of large number.

Mathematics Subject Classification: 15A15, 26E60, 60B20.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	S. Aaronson and H. Nguyen , Near invariance of the hypercube, Israel Journal of Mathematics, 212 (2016) , 385-417. doi: 10.1007/s11856-016-1291-z.
	N. Alon and S. Friedland, The maximum number of perfect matchings in graphs with a given degree sequence, Electron. J. Combin., 15 (2008), Note 13, 2 pp.
	A. Barvinok , Polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor, Random Structures and Algorithms, 14 (1999) , 29-61. doi: 10.1002/(SICI)1098-2418(1999010)14:1<29::AID-RSA2>3.0.CO;2-X.
	J. Bochi , G. Iommi and M. Ponce , The scaling mean and a law of large permanents, Adv. Math., 292 (2016) , 374-409. doi: 10.1016/j.aim.2016.01.013.
	J. Bochi, G. Iommi and M. Ponce, Perfect matching in inhomogeneous random bipartite graphs in random environment, submitted, https://arxiv.org/pdf/1605.06137v2.pdf.
	L. M. Bregman , Some properties of non-negative matrices and their permanents, Soviet Math. Dokl., 14 (1973) , 945-949.
	R. Brualdi , S. Parter and H. Schneider , The diagonal equivalence of a non-negative matrix to a stochastic matrix, J. Math. Anal. Appl., 16 (1966) , 31-50. doi: 10.1016/0022-247X(66)90184-3.
	K. P. Costello and V. Vu , Concentration of random determinants and permanent estimators, SIAM J. Discrete Math., 23 (2009) , 1356-1371. doi: 10.1137/080733784.
	G. Egorychev , The solution of van der Waerden's problem for permanents, Adv. in Math., 42 (1981) , 299-305. doi: 10.1016/0001-8708(81)90044-X.
	D. Falikman , Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix, Mat. Zametki, 29 (1981) , 931-938, 957.
	S. Friedland , Positive diagonal scaling of a nonnegative tensor to one with prescribed slice sums, Linear Algebra Appl., 434 (2011) , 1615-1619. doi: 10.1016/j.laa.2010.02.007.
	S. Friedland and S. Karlin , Some inequalities for the spectral radius of non-negative matrices and applications, Duke Math. J., 42 (1975) , 459-490.
	S. Friedland , B. Rider and O. Zeitouni , Concentration of permanent estimators for certain large matrices, Annals Appl. Prob, 14 (2004) , 1559-1576. doi: 10.1214/105051604000000396.
	S. Friedland , S. Low and C. Tan , Nonnegative matrix inequalities and their application to non-convex power control optimization, SIAM J. Matrix Anal. Appl., 32 (2011) , 1030-1055. doi: 10.1137/090757137.
	C. D. Godsil and I. Gutman, On the matching polynomial of a graph, in Algebraic Methods in Graph Theory I-II (L. Lovász and V. T. Sós, eds. ), North-Holland, Amsterdam, 25 (1981), 241-249.
	N. R. Goodman , Distribution of the determinant of a complex Wishart distributed matrix, Annals Stat., 34 (1963) , 178-180. doi: 10.1214/aoms/1177704251.
	A. Guionnet and O. Zeitouni , Concentration of the spectral measure for large matrices, Elec. Comm. Probab., 5 (2000) , 119-136. doi: 10.1214/ECP.v5-1026.
	G. Halász and G. Székely , On the elementary symmetric polynomials of independent random variables, Acta Math. Acad. Sci. Hungar., 28 (1976) , 397-400. doi: 10.1007/BF01896806.
	G. Keller, Equilibrium States in Ergodic Theory London Math. Soc. Stud. Texts, vol. 42, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781107359987.
	N. Linial , A. Samorodnitsky and A. Wigderson , A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents, Combinatorica, 20 (2000) , 545-568. doi: 10.1007/s004930070007.
	A. Marshall and I. Olkin , Scaling of matrices to achieve specified row and column sums, Numer. Math., 12 (1968) , 83-90. doi: 10.1007/BF02170999.
	M. Menon , Matrix links, an extremization problem, and the reduction of a non-negative matrix to one with prescribed row and column sums, Canad. J. Math., 20 (1968) , 225-232. doi: 10.4153/CJM-1968-021-9.
	H. Minc , Upper bounds for permanents of $(0, 1)$-matrices, Bull. Amer. Math. Soc., 69 (1963) , 789-791. doi: 10.1090/S0002-9904-1963-11031-9.
	G. Rempala and J. Wesołowski, Symmetric Functionals on Random Matrices and Random Matchings Problems, Springer, New York, NY, 2008.
	M. Rudelson and O. Zeitouni , Singular values of Gaussian matrices and permanent estimators, Random Structures and Algorithms, 48 (2016) , 183-212. doi: 10.1002/rsa.20564.
	R. Sinkhorn , Continuous dependence on $A$ in the $D_1AD_2$ theorems, Proc. Amer. Math. Soc., 32 (1972) , 395-398. doi: 10.2307/2037825.
	R. Sinkhorn and P. Knopp , Concerning nonnegative matrices and doubly stochastic matrices, Pacific J. Math., 21 (1967) , 343-348. doi: 10.2140/pjm.1967.21.343.
	G. Soules , New permanental upper bounds for nonnegative matrices, Linear Multilinear Algebra, 51 (2003) , 319-337. doi: 10.1080/0308108031000098450.
	B. van der Waerden , Aufgabe 45, Jber. Deutsch. Math. Verein., 35 (1926) , p117.
	A. Widgerson, Matrix and Operator scaling and their many applications, http://www.math.ias.edu/avi/talks.