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October  2017, 37(10): 5285-5297. doi: 10.3934/dcds.2017229

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  June 2017

Fund Project: 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.

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: JÓzsef Balogh, Hoi Nguyen. A general law of large permanent. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5285-5297. doi: 10.3934/dcds.2017229
References:
[1]

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.

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

L. M. Bregman, Some properties of non-negative matrices and their permanents, Soviet Math. Dokl., 14 (1973), 945-949. 

[7]

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.

[8]

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.

[9]

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.

[10]

D. Falikman, Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix, Mat. Zametki, 29 (1981), 931-938, 957. 

[11]

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.

[12]

S. Friedland and S. Karlin, Some inequalities for the spectral radius of non-negative matrices and applications, Duke Math. J., 42 (1975), 459-490. 

[13]

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.

[14]

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.

[15]

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.

[16]

N. R. Goodman, Distribution of the determinant of a complex Wishart distributed matrix, Annals Stat., 34 (1963), 178-180.  doi: 10.1214/aoms/1177704251.

[17]

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.

[18]

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.

[19]

G. Keller, Equilibrium States in Ergodic Theory London Math. Soc. Stud. Texts, vol. 42, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781107359987.

[20]

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.

[21]

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.

[22]

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.

[23]

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.

[24] G. Rempala and J. Wesołowski, Symmetric Functionals on Random Matrices and Random Matchings Problems, Springer, New York, NY, 2008. 
[25]

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.

[26]

R. Sinkhorn, Continuous dependence on $A$ in the $D_1AD_2$ theorems, Proc. Amer. Math. Soc., 32 (1972), 395-398.  doi: 10.2307/2037825.

[27]

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.

[28]

G. Soules, New permanental upper bounds for nonnegative matrices, Linear Multilinear Algebra, 51 (2003), 319-337.  doi: 10.1080/0308108031000098450.

[29]

B. van der Waerden, Aufgabe 45, Jber. Deutsch. Math. Verein., 35 (1926), p117. 

[30]

A. Widgerson, Matrix and Operator scaling and their many applications, http://www.math.ias.edu/avi/talks.

show all references

References:
[1]

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.

[2]

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.

[3]

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.

[4]

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.

[5]

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.

[6]

L. M. Bregman, Some properties of non-negative matrices and their permanents, Soviet Math. Dokl., 14 (1973), 945-949. 

[7]

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.

[8]

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.

[9]

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.

[10]

D. Falikman, Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix, Mat. Zametki, 29 (1981), 931-938, 957. 

[11]

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.

[12]

S. Friedland and S. Karlin, Some inequalities for the spectral radius of non-negative matrices and applications, Duke Math. J., 42 (1975), 459-490. 

[13]

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.

[14]

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.

[15]

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.

[16]

N. R. Goodman, Distribution of the determinant of a complex Wishart distributed matrix, Annals Stat., 34 (1963), 178-180.  doi: 10.1214/aoms/1177704251.

[17]

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.

[18]

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.

[19]

G. Keller, Equilibrium States in Ergodic Theory London Math. Soc. Stud. Texts, vol. 42, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781107359987.

[20]

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.

[21]

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.

[22]

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.

[23]

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.

[24] G. Rempala and J. Wesołowski, Symmetric Functionals on Random Matrices and Random Matchings Problems, Springer, New York, NY, 2008. 
[25]

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.

[26]

R. Sinkhorn, Continuous dependence on $A$ in the $D_1AD_2$ theorems, Proc. Amer. Math. Soc., 32 (1972), 395-398.  doi: 10.2307/2037825.

[27]

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.

[28]

G. Soules, New permanental upper bounds for nonnegative matrices, Linear Multilinear Algebra, 51 (2003), 319-337.  doi: 10.1080/0308108031000098450.

[29]

B. van der Waerden, Aufgabe 45, Jber. Deutsch. Math. Verein., 35 (1926), p117. 

[30]

A. Widgerson, Matrix and Operator scaling and their many applications, http://www.math.ias.edu/avi/talks.

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