in the binomial random graph Tel Aviv University Wojciech Samotij - PDF document

Deviations of triangle counts in the binomial random graph Tel Aviv University Wojciech Samotij joint work with Matan Harel Frank Mousset SC MS online seminar

Our setting Gmp the binomial random graph with lo n's and edge prob p Vx set X EEN 131ps triangles in Gmp Problem For every 8 0 determine the asymptotic of log PIX Ht d EIN logarithmicupper tail probability

Theorem Chatterjee DeMarco Kahn 20121 login In then for every d O If p log PIXHto HEINI Od n'p tog Holt Question What is the implicit constant Definition For d O minded EIX I G Gnp a HOHEN yo Eo EXT

Proposition If 4181 then 00 t t 401 z p't PIX 24 81 EEN Proof Pol I P f IGE Gmp Ge kn with YldtE edges Pick a small E O and a st Eo EN z Hot E ELI Notethat lil Pluto Is PIGE Gmp PLUTO pH IR UTd.CiilEoIXIePolUTo4 lltdHEMtlPdUTd lb E EEN 1 5 Ep potto Potato Finally Iiit O

If n'Epal Theorem l then o Wo PIX HOMELY ept 9 42 Chatterjee Dembo 12014 a _if Eldan 120161 2 3 Cook Dembo 12018 L Augen12018 a

Theorem Hard Mousset S 2019 tf pal n login then for every d 0 401 PIX HSIEH pl Theorem Lubetzky Zhao 20141 For every8 0 if n park 842 41811 n'p n'keep 1 if min 8442,8 3 To p g 3np hub clique Mfdp ggBTnp.is

Proof Fix a small E O and a large C Definition A graph GE kn a seed if is E Crip log 01 n'p4 d G 41 Hp he Hd E EEN a Eo IN Lemma t Pl UT E HoH IP Gmp contains a seed Union bound over all seeds Too naive a seed with m Not E edges 2M e G Go Go u G is a seed 2 man Ibm psms nI Eli seeds in Gmp Rfp

core if Definition A graph is a s Crip log 14 41 e E p a Eo EX Hd 2E IE EX IE until off gap IE o EXT Have Fact Every seed contains a core Proof Iteratively remove from a seed all edges failing 3 Lemma 2 For every M e p Em cores with m edges

Assuming both lemmas Ll EP Gmp contains a seed P UT F IP Gmp contains a core EL cores in Gnp Em EI cores with m edges in Gmp e Hd 2E L2 e pl Em m 8 Ems Wo ee p p 418 2E E edges in a core

Proof of Lemma 1 based on Janson Oleszkiewicz Rucinski D Gmp does not contain a seed Let 2 As ZELO lb E PIZ O PIX 2 Hor EEN P1 Xs Hot EIN 4 pm EP Xs HOHEN Enough to show 1EUR14 He I Markov's inequality HOY EEN ELIX 2 Ye Ho E EEN Claim 31 Crip log Hp

1EUR19 ECHO EYELID Claim 31 Crip'logHp Proof a Guys a 2 PIT EUR14 1 u Te u 2 PIT u ute a Gmp E Ti Te u Te not a seed Tu Hd E EN Et ua TX eltu.ve e3eeCrip4ogl4pl E 2 PIT u ute.ie Gmp PITeeGn.pl Tu uTeieGn.p T Te u Te not a Tu seed Htc E EEN e o

Foof of Lemma 2 Suppose G core graph with medges Earth Eo w X aw aura Ours EEEX 840 I 4p Clip'loglyp log Owe nlp f du.ir t p du idvllp p4 Knt U V UN Yu u either du idrzg.nl ogHplordu.vZjnp logHpl

Cor Have dutdrzjnllogllplordu.vs.jp logHpl Define A well dw 28h1 pH 1210g'll we VIE dw 2 jnpllogHp B Obs 2M 246 1 Eno dw Z Max At 271g 1131 Engin's 8 1 2m10 4M 98 IBI e b I Ate a maxi y n p ma has 2 endpts in B edge every or Z l endpt in A A B Ibm imam le e p Imax cores w medges E lanna D

Problem Calculate xt.tl loglplx.zdtdHEL where Xp copies of H in Gmp State of the art H Kr Haret Morisset S H connected Basak Basu HMS regular Hi irregular Cook Dembo