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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)