MATH 182 : HIDDEN DATAIN RANDOM MATRICES

Todd Kemp ,APM 5202

www.math.ucsd.edu/ntkemp/182

TODAY : Histograms and linear statisticsNEXT : The Trace and the Large - N Limit

HW 2 :

Lab z :} Regrade Window wed Feb 128am → Fri Feb 148pm

HistogramsofEmpni-aawsh.eeA histogram hi, of a pent set µI encodes the histogram statistics

heap, lls)-

- t # fj : 3- tlgbl)http.

is an" approximate density "

"shout"

Ex

for the empirical random variable" " t:* ::*

.

¥a b

( if spp . --XD

lPlE± e- A) = In #Sj : Ajc. A} ie. MEI tlgblfhca

,bill

suppose E± has a "density" fi gab f , "c⇒dx£

Histogramsv.s.pe/ynomial4h-earSatsticshcgbgHI--d-#fJ:Xje-Lgbt =L,

Diabelli)-

f- §,

baht Example ofa linearstatstio→

:&! Eaite¥.

b. Mend a' weight function

"

Hard to compute with weight function * team .

More familiar examples :N

Eg . In = f- It; Lucas ⇒c)J"- A *n

,A-AT 03 . di - - - H

Eg.

Mads ) -- k¥45 THA) = bit - - - tha .

Eg .

w is a polynomial . woe ) - botbpetbzxlx - - - tbnxn

Approximatingwithpdynomialsttwill turn out to be possible to really compute linearstatistics of eigenvalues, with a polynomial weight function w .

we'd really like to do it with W -- Ic,by . Fortunately ,

we can

approximate Icap, with polynomials .theorem ( Bernstein

,1920 s)

Let La, pi be a closed interval , and let w : Lgpl→ IR

.

There is a sequence Bn of polynomials (deg Pn -- n ) suchthat

feng Bnl a) = w la) for all at Lgplnon-cent.pe ly s t

.W is continuous @ x-

• Can't do it on all of IR bk Ica )→ to as a→ Ies.

proofofBernstein-therem.aepopes on

Steph : reduce to Lgpl - cell . GilW squirmsBn e

# ⇒y. Bn-tnt-B.ktp.at

# T ta

po ' alseapdynenia)

x i- =p of degree n .

c- p - a

step 2 : the " Probabilistic Method ""9-4+4×9

Fix pefgll . Let Xylh,-→Xn it:D. Berlp) rt 's .

Sn, # xp

- -- tin Bmlnsp)

tbsp ⇒ p GLLN )

i. If wi56nkmars@pc-ie_unespeutslmibsaep.wftnSypj-swcp2.i. E(wltnsnp ) )→ wept .

Established that157. ⑤ ( wltnsn.pl ) = wop )

Snp ~ Binh,p) Snp Els 's . .>n }

Msn,p=k)= IL ) pka -pin- k

Ef wltnsn.pl) = # lglsn.pl) glxtwltnx)= # glkjplsn.ph )= E. wtf) lnujpkci - pink .

' : Bnlp )

%I Bncpl - wept s ¥3 - (Bernstein

,b 's)