http://math.berkeley.edu/~ilc/talks/2019/ymcstara.pdf

Free Stein IrregularityYMC * A 2019 - University of Copenhagen

Ian Charlesworth ( UC Berkeley )Joint with Brent Nelson ( Michigan State University )

What is free probability ?

÷::÷:::*⇒⑤von Neumann algebras

with states

( Fln ,µ , E) ← a)

IR - valued bounded random variable self-adjoint xeM ,

Independence ← tensor product free product

Hit ii. d. IGXI -0µm,s ) lXijfi.d.fn.Ixi@olm.sl

← #Information theory

Shannon entropy Free entropyFisher 's Information measure Free Fisher Information

LT1R ' , D8 ) I ilpi ,d8m) ? ? ?

II. → In random variables-7 Law : {" nice " test functions} → ①

* s.in

onyR"

→ HI , ,→In)dµ

X , , . . . > XnE Msa

→ this . . . ,xn : ① LT , . . . ,TnY → ①"non-commutative law

"

p→ 4( ptxs.mn))

When n - I , µ×.lp ) = 91pA)) = ↳ It )dR¥trd measure )

Regularity properties

( " niceness " of µ , ,→×n) ⇒ (interesting properties of WH.int )

④ xi , .no , Xn) > I W*fx , , . . . ,xd has no( " x , ,.→xu are easy to

approximate ⇒ Cartan,is non -M , Lvoicuksa

by matrices"

) is prime kitFree skew field (

"

norational functions

"

)

x. , - → xn )=n ⇒ 64Th .→ 1¥

.

d)

( " 4. → xn become nice afterwith T , ↳ × ,

" iated " " " ""

small perturbations by free [ MaiSpeicher Yin]

Brownian motion ")

( classical ) Fisher InformationIf I know the distribution of X

- ELXI, how much do I

learn about ELXI from sampling × once ?

X - EG3 - no , D - not much X - ELX] - it IS , - lots

## -1.1The smoother the distribution of X , the less you

learn .

ILX ) = 118*711 ? where D: HRidpY→MdtdpDp → ( djpi )

is viewed as an unbounded operator

What is the non-commutative Jacobian ?

Jj : ① CT , , . . . . 7→ CUT 's . - ⇒ ④ CNT . . . - →TN3P

linear,

Leibniz rule,

with 7. Ti = of 1×01 .

( Note , if nil , and viewing ECT ,I ① Ls ,t3 ,

J.pt , ) = pls ) - pit)

It. )

Set I : ECT . . . .in#MnLehTs.-TnY0EhT....iTnY)pi-l2;Pi)

For A- EMILY MOD) , if JHELUM )"

so that

{ H , PIX " . . ,kD= LA , If ix. . . . . ,XnDV. Peek . .

thensay

Aedomtfx with J×*A=H .

The free Fisher information of X

IE4X ) : = {11051115if I :-( . . edomI¥otherwise

.

What if we only ask for It to be close to doing ?

The free Stein irregularityof X

8*1×1 := dist 11 , domtfx ) .

Also,free Stein dimension

→

JLX) : - n - ETLXT = dimmµo×n⇒dom8×

Example : X - * c- Mule )set p ;fIk-H]/lx - I ;)

set gdtt-2.IE . Then lkxa,

to

iffa-j-blgapl-j.EI.ITa) " " Also , pjlx )=o .Then if he doin ,

= ? } " "=fI dply.mil Hence has :D -0It follows that domJ*= face

,dp ) where adti.li ) ;

is supported on the off - diagonal

That is ,* 2

Ill - act ? = In , so EIX) ← th o idimmndom # I 1

- I

Hence • LX) =L .

Theorems : 2*1×5+5145 I [ * IX. Y ) with equality if X andY are free

TIX ) ± 89×1 with equality if a- I

TIXI is an algebra invariant

%xyE