Sharp Bounds for Sums Associated to Graphs of Matrices. at the WaldenFest. Roland Speicher Universität des Saarlandes Saarbrücken, Germany

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1 Sharp Bounds for Sums Assocated to Graphs of Matrces at the WaldenFest Roland Specher Unverstät des Saarlandes Saarbrücken, Germany

2 Academc famly tree of Wlhelm von Waldenfels Schürmann Köng Wese Köng Specher Waldenfels Skede

3 Academc famly tree of Wlhelm von Waldenfels Schürmann Köng Wese Köng Specher and sons Waldenfels Skede

4 Hedelberg... at the good old tmes... last century

5 Some thngs we learned... thnkng s hard work...

6 Some thngs we learned... thnkng s hard work...

7 Some thngs we learned... look for a (hdden) conceptual structure behnd your complcated concrete problem...

8 Queston What s the asyptotc behavour n N of 1,..., 2m some constrants on equalty of ndces 1 2 t (2) 3 4 t (m) 2m 1 2m wth gven matrces T k = ( t (k) ) N,=1

9 Examples t,=1 T =1 t T,,k=1 t(2) k t(3) k T 1 T 2 k,,k,l=1 t(2) k t(3) l T 2 T 1 k l

10 That s easy? Okay, so look on ths: 1,..., 8 =1 1 2 t (2) 3 2 t (3) 3 4 t (4) 4 4 t (5) 5 3 t (6) 2 5 t (7) 6 5 t (8) 6 5 t (9) 6 6 t (10) 7 5 t (11) 8 7 t (12) T T 2 T 2 1 T 5 T 6 T 7 T 6 11 T 10 T 8 8 T 9 T

11 Problem For a gven drected graph G (multple edges and loops allowed), wth matrces attached to the edges, denote the correspondng sum by S G (N) Queston: What s the optmal bound r(g) n S G (N) N r(g) m k=1 T k

12 Optmal asymptotcs n N? 4 T T 2 T 2 1 T 5 T 6 T 7 T 6 11 T 10 T 8 8 T 9 T t (2) 2 3 t (3) 2 3 t (4) 4 4 t (5) 4 5 t (6) 3 2 t (7) 5 6 t (8) 5 6 t (9) 5 6 t (10) 6 7 t (11) 5 8 t (12) 7 8 N??? 7 12 =1 T

13 Motvaton such sums appear and have to be asymptotcally controlled n calculatons of moments of products of random (Wgner) matrces and determnstc matrces Yn + Krshnaah, 1983 Ba (+ Slversten), 1999 (2006) Mngo + Specher, 2012 JFA (asymptotc freeness of Wgner and determnstc matrces) Male, 2012 traffcs

14 Example =1 t T Then or =1 t =1 T = N T =1 t = Tr(T ) = N tr(t ) N T

15 Example =1 t T Then or =1 t =1 T = N T =1 t = Tr(T ) = N tr(t ) N T

16 Example,,k=1 t(2) k t(3) k T 1 T 2 k,,k=1 t(2) k t(3) k = Tr(T 1T 2 ) = N tr(t 1 T 2 ) N T 1 T 2 N T 1 T 2

17 Example t,=1 T trval estmate:,=1 t,=1 T = N 2 T But we can do better...

18 t,=1 T... wth usng the vectors we can wrte e := (0,..., 1,..., 0), e := (1, 1,..., 1) and thus,=1 t =,=1 e, T e = e, T e snce,=1 t e, T e e 2 T = N T e = N

19 t,=1 T... wth usng the vectors we can wrte e := (0,..., 1,..., 0), e := (1, 1,..., 1) and thus,=1 t =,=1 e, T e = e, T e snce,=1 t e, T e e 2 T = N T e = N

20 ,,k,l=1 Example t(2) k t(3) l T 2 T 1 k l t(2) k t(3) l =,,k,l=1,,k,l e, T 1 e e, T 2 e k e, e l }{{} e e,t 2 e k e l = e, T 1 e e e }{{} V :C N C N C N T 2, e e = e, T 1 V (T 2 ) e e

21 ,,k,l=1 Example t(2) k t(3) l T 2 T 1 k l,,k,l=1 t(2) k t(3) l =,,k,l e, T 1 e e, T 2 e k e, e l }{{} e e,t 2 e k e l = e, T 1 e e e }{{} V :C N C N C N T 2, e e = e, T 1 V (T 2 ) e e

22 ,,k,l=1 Example t(2) k t(3) l T 2 T 1 k l,,k,l=1 t(2) k t(3) l =,,k,l e, T 1 e e, T 2 e k e, e l }{{} e e,t 2 e k e l = e, T 1 e e e }{{} V :C N C N C N T 2, e e = e, T 1 V (T 2 ) e e

23 ,,k,l=1 Example t(2) k t(3) l T 2 T 1 k l,,k,l=1 t(2) k t(3) l =,,k,l e, T 1 e e, T 2 e k e, e l }{{} e e,t 2 e k e l = e, T 1 e e e }{{} V :C N C N C N T 2, e e = e, T 1 V (T 2 ) e e

24 ,,k,l=1 t(2) k t(3) l T 2 T 1 k l,,k,l=1 t(2) k t(3) l = e, T 1 V (T 2 ) e e e e e }{{} e e T 1 V T }{{} 2 =1 = N 3/2 T 1 T 2 snce V : C N C N C N, e e s an sometry, and thus V = 1. e, f = 0, f

25 General structure In ths example, our sum S(N) s gven as nner product where S(N) ˆ= output, stuff nput each nput and each output vertex contrbutes factor N 1/2 nternal vertces do not contrbute, summaton over them corresponds to matrx multplcaton or, more general, partal sometres

26 Note that one can also modfy,,k=1 t(2) k t(3) k T 1 T 2 k to an nput-output form: T 1 T 2 t k

27 General strategy Modfy gven graph to equvalent nput-output graph then each nput and output vertex gves factor N 1/2 Man problems: Can we always do such a modfcaton? If yes, how can we recognze how many nput/output vertces we need?

28 Asymptotcs determned by structure of F(G) F(G) = forrest of two-edge-connected components 4 T T 2 T 2 1 T 5 2 = 3 = 5 = 6 5 T 6 T 7 T 6 11 T 10 T 8 8 T 9 T = 8

29 Theorem [Mngo, Specher, JFA 2012] We have the followng optmal estmate: :V [N] e E t (e) t(e), s(e) N 1 2 #leaves of F(G) e E T e (Trval leaves,.e., only vertces of trval trees, count twce!)

30 Optmal asymptotcs n N! 4 T T 2 T 2 1 T 5 2 = 3 = 5 = 6 5 T 6 T 7 T 6 11 T 10 T 8 8 T 7 12 T9 1 7 = t (2) 2 3 t (3) 2 3 t (4) 4 4 t (5) 4 5 t (6) 3 2 t (7) 5 6 t (8) 5 6 t (9) 5 6 t (10) 6 7 t (11) 5 8 t (12) 7 8 N 3/ =1 T

31 Replace by equvalent nput-output problem 4 T 4 4 T T 2 T 2 1 T 5 2 T t 2 T 5 5 T 6 T 7 6 T 6 T 7 T 6 11 T 10 T 8 8 T 9 T T 1 T 10 T8 t T 9 T 11 T

32 Addng addtonal vertces 4 T 4 T T t 2 T 5 5 T 5 2 T 6 T 7 6 T t 2 T 7 I T N 6 6 T 9 T 9 T 1 T 10 T8 t 5 7 T 11 T 12 T 10 T t 8 T 1 T 11 T

33 H n V n,2 : H n H H T 4 T 4 V 2,1 : H H H T 5 V 1,2 : H H H T 5 T t 2 T 7 I T N 6 V 1,1 V 1,3 : H H) H (H H H) T t 2 T 6 T 7 V 2,1 V 1,1 V 1,2 : (H H) H H H H (H H) T 9 T 9 V 1,1 V 1,1 V 2,1 : H H (H H) H H H T 1 T8 t T 10 T 11 T 12 T t 8 V 1,1 V 2,1 : H (H H) H H T 10 V 1,1 V 1,2 : H H H (H H) T 1 T 11 T 12 H out H out V 1,out V 2,out : H (H H) H out H out

34 Theorem [Mngo, Specher, JFA 2012] We have the followng optmal estmate: :V [N] e E t (e) t(e), s(e) N 1 2 #leaves of F(G) e E T e (Trval leaves,.e., only vertces of trval trees, count twce!)

35 Happy Brthday, Wlhelm!!!