Approaching Eldan’s and Lee & Vempala’s bounds for the KLS conjecture in a unified method

App oaching Eldan’s and Lee & Vempala’s bounds o he KLS conjec u e in a uni ied me hod Da id Alonso-Gu ié ez, Jesús Bas e o Depa amen o de Ma emá icas - I.U.M.A. Uni e sidad de Za agoza Campus Plaza San F ancisco 50009 Za agoza, Spain alonso[email p o ec ed], [email p o ec ed] Re . Real Academia de Ciencias. Za agoza. 75: 85–110, (2020). ISSN: 0370-3207 Resumen La p incipal idea de es e a ículo es e isa las p uebas de las mejo es es imaciones conocidas pa a la conje u a KLS de sal o espec al, demos adas po Eldan y Lee & Vempal, aplicando el esquema de localización de Eldan a dos sis emas de ecuaciones di e enciales es ocás icas di e en es. Damos una p ueba uni icada de es as dos aco a- ciones ob eniendo la es imación de Eldan desde el sis ema de ecuaciones di e enciales es ocás icas conside ado po Lee & Vempala. Abs ac The main idea o his pape is o e iew he p oo on he bes known es ima es o he KLS spec al gap conjec u e, gi en by Eldan and Lee & Vempala by applying El- dan’s localiza ion scheme o wo di e en sys ems o s ochas ic di e en ial equa ions. We gi e a uni ied p oo o hese wo bes bounds ob aining Eldan’s es ima e om he sys em o s ochas ic equa ions conside ed by Lee & Vempala. 1 In oduc ion The Kannan-Lo ász-Simono i s spec al gap conjec u e (KLS) is a majo p oblem in asymp o ic geome ic analysis. I s o igin comes om heo e ical compu e sciences as a p oblem a ising in he s udy o he complexi y o an sampling algo i hm and i is ela ed o 85 many o he b anches o ma hema ics like con ex geome y, p obabili y, PDE’s, Riemannian geome y and in o ma ion o lea ning heo y (see [AB1], [BGVV] and he e e ences he ein (o [AB2] o a p esen a ion o he conjec u e w i en in Spanish)). I conce ns log-conca e p obabili ies and can be s a ed in he ollowing way: Conjec u e 1.1 (KLS spec al gap conjec u e).The e exis s an absolu e cons an C > 0 such ha , o any log-conca e p obabili y µin Rn (1) µ+(A)≥C pkCo µkop min{µ(A), µ(Ac)}, o any Bo el se A⊂Rn whe e µ+(A) = lim in ε→0 µ(Aε)−µ(A) ε, being Aε={a+x:a∈A, |x|< ε}, and kCo µkop is he ope a o no m o he co a iance ma ix o µ This is a Cheege ype isope ime ic inequali y. This conjec u e was posed in [KLS], whe e he au ho s p o ed he Cheege ype isope ime ic inequali y wi h cons an C Eµ|x| (whe e Eµ|x|deno es he expec ed alue o he Euclidean no m wi h espec o he p oba- bili y µ) ins ead o C √kCo µkop . The KLS conjec u e has an equi alen exp ession as a co esponding Poinca é ype inequali y: he e exis s an absolu e cons an C > 0such ha (2) ZRn| −Eµ |2dµ ≤CkCo µkop ZRn|∇ |2dµ o any log-conca e p obabili y µin Rnand Lipschi z µ-in eg able unc ion . The ac o kCo µkop appea ing in bo h exp essions (1) and (2) is jus a no maliza ion ac o . Indeed, since he conjec u e in ol es e e y Bo el se A⊆Rn, o e e y Lipschi z µ-in eg able unc ion in i s equi alen o m, making a change o a iables, we can assume ha µis cen e ed and ha Co µ=In(iden i y ma ix), i.e. he new log-conca e measu e is iso opic and hen we can e o mula e bo h conjec u es in he ollowing way: he e exis s an absolu e cons an Csuch ha o any iso opic log-conca e p obabili y in Rn (3) µ+(A)≥Cmin{µ(A), µ(Ac)}, o any Bo el se A⊂Rn 86 o , equi alen ly, he e exis s an absolu e cons an C (4) ZRn| −Eµ |2dµ ≤CZRn|∇ |2dµ o any iso opic log-conca e p obabili y µin Rnand any Lipschi z µ-in eg able unc ion . This conjec u e emains open and he bes es ima es known up o now, which depend on he dimension, o he alue o he cons an in (1) and (2) ha e been ob ained in wo di e en pape s by Eldan ([E1], see also [E2] o ano he app oach) and Lee & Vempala ([LV1], see also [LV2] o a nice su ey on his conjec u e), espec i ely. The esul s whose p oo s we wan o uni y a e gi en by he ollowing wo heo ems: Theo em 1.1 (Eldan, [E1]).The e exis s an absolu e cons an C > 0such ha o any iso opic log-conca e p obabili y µin Rn (5) µ+(A)≥C σnlog nmin{µ(A), µ(Ac)} o any Bo el se A⊂Rn whe e σn=qsup Eµ|X|−√n 2and he sup uns o e all iso opic log-conca e andom ec o s Xin Rn. Theo em 1.2 (Lee & Vempala, [LV1]).The e exis s an absolu e cons an C > 0such ha o any iso opic log-conca e p obabili y µin Rn (6) µ+(A)≥C n1/4min{µ(A), µ(Ac)} o any Bo el se A⊂Rn. The pa ame e σnappea ing in Eldan’s esul is ela ed wi h a di e en conjec u e, which is he hin shell wid h conjec u e p oposed by Bobko -Koldobsky ([BK]): he e exis s an absolu e cons an C > 0such ha o any iso opic, log-conca e p obabili y in Rnwe ha e σµ=qEµ|x|−√n 2≤C. I his conjec u e we e ue i would imply ha he mass in he iso opic log-conca e p obabili ies is concen a ed in a hin shell a ound a dis ance √n om he o igin, Besides, he esul (5) would imply ha he KLS conjec u e is ue up o a log n ac o . As i is also e y well known ha he KLS conjec u e is s onge han he hin shell wid h conjec u e, he esul (6) implies he bes known es ima e o he he las conjec u e, i.e. σn≤Cn1/4. 87 P e ious es ima es o his pa ame e we e ound by Kla ag [K] and Guedon-Milman [GM]. Mo e in o ma ion on hese ela ions can be seen, o ins ance, in [BGVV] and [AB1]. The p oo o bo h esul s, Theo ems 1.1 and 1.2, ollow he o iginal idea de eloped by Eldan, he localiza ion scheme in oduced in [E1]: gi en an iso opic log-conca e p obabili y µin Rn, a s ochas ic sys em o di e en ial equa ions o igina es a s ochas ic p ocess o (no necessa ily iso opic) log-conca e p obabili ies (µ ) ≥0which a e an I ô p ocess. We can ge “good” in o ma ion om some µTand hen come back o he o iginal µ. Howe e he wo p oo s p opose di e en s ochas ic sys ems o di e en ial equa ions in o de o ge s ochas ic p ocess (µ ) ≥0 om which we can ob ain es ima es. The main pu pose o his pape is o uni y he wo app oaches and gi e a p oo o bo h esul s oge he , which will ollow om he same s ochas ic sys em o di e en ial equa ions. E en hough we a e no in oducing any uly new ideas in his pape , a he han ca e ully mixing and gluing he a gumen s om he a o emen ioned au ho s, i is ou desi e o cla i y and shed ligh on he a gumen s o his beau i ul and in e es ing heo y wha has mo ed us o w i e his wo k and b ing i close o he in e es ed people e en i hey a e less expe in he ield. The heo em we a e going o p o e in his wo k is he ollowing, which collec s bo h Theo ems 1.1 and 1.2. Theo em 1.3. The e exis s an absolu e cons an C > 0such ha o any iso opic log- conca e p obabili y µin Rn he ollowing isope ime ic inequali y holds µ+(A)≥C min{σnlog n, n1/4}min{µ(A), µ(Ac)} o any Bo el se A⊆Rn. The pape is o ganized in he ollowing way. In Sec ion 2 we will in oduce no a ion, some de ini ions and some p e ious esul s we a e going o use in o de o de elop ou p oo . In Sec ion 3 we will in oduce Eldan’s localiza ion scheme, p esen ing he sys em o s ochas ic di e en ial equa ions we will conside in his wo k, which will de ine he a o emen ioned s ochas ic p ocess o log-conca e p obabili ies (µ ) ≥0. In Sec ion 4 we will gi e an o e iew o he s a egy we ollow in o de o s ess ou he e en whose p obabili y is needed so ha he es ima es o he KLS cons an can be ob ained. The ace o he co a iance ma ix o he p obabili ies (µ ) ≥0will be needed o bound he p obabili y o 88 such e en om below. They will be p o ed o be small enough wi h some p obabili y in Sec ion 5. Finally, in Sec ion 6 we will pu all he inequali ies oge he o comple e he p oo o Theo em 1.3. 2 No a ion and de ini ions In his sec ion we will in oduce some no a ion and de ini ions which a e common in his amewo k. Some well-known esul s will also be explained ei he by gi ing hei p oo o a e e ence o i . We will deno e by |·| he Euclidean no m in Rnand also he absolu e alue on R.Sn−1 will deno e he Euclidean uni sphe e. A p obabili y measu e µon Rnis called log-conca e i o any compac subse s A, B ⊆Rnand o any 0≤λ≤1 µ((1 −λ)A+λB)≥µ(A)1−λµ(B)λ. The ollowing heo em by Bo ell [B], cha ac e izes his kind o p obabili ies: Le µbe a non degene a e log-conca e p obabili y measu e on Rn, (i.e. no concen a ed in any hype plane). Then, µis log-conca e i and only i µis absolu ely con inuous wi h espec o he Lebesgue measu e and i s densi y is log-conca e, i.e. dµ(x) = (x)dx =e−V(x)dx, whe e he unc ion V:Rn→(−∞,∞]is con ex. In he sequel we will use he p obabilis ic no a ion Eµg:= RRng(x)dµ(x)and Va µg:= Eµ(g−Eµg)2 o any µ-in eg able unc ion g. Fi s educ ions: We say ha µis iso opic i i s ba ycen e bµ:= Eµx= 0 and i s co a iance ma ix Co µ=Aµ:= Eµ(x−bµ)⊗(x−bµ) = In, whe e Inis he iden i y ma ix. E e y non degene a e log-conca e p obabili y dµ(x) = (x)dx admi s an a ine ans o ma ion such ha dν(y) = |de (Aµ)|1/2 (bµ+A1/2 µy)dy is an iso opic log-conca e p obabili y. In pa icula wi h his change o a iables, i is easy o p o e ha i an iso opic p obabili y µsa is ies Poinca é’s inequali y (2) wi h some cons an C, hen o any non-degene a e linea map T he log-conca e p obabili y measu e 89 µ◦T, gi en by (µ◦T)(A) = µ(T(A)) o any Bo el se A, sa is ies (2) wi h he same cons an C. The e o e, i he e exis s a cons an Cnsuch ha e e y iso opic log-conca e p obabili y in Rnsa is ies (2) wi h cons an Cn, hen e e y log-conca e p obabili y in Rn sa is ies (2) wi h he same cons an Cnand i he e exis s a cons an ˜ Cnsuch ha e e y iso opic log-conca e p obabili y in Rnwi h compac suppo sa is ies (2) wi h cons an ˜ Cn, hen e e y log-conca e p obabili y in Rnwi h compac suppo sa is ies (2) wi h he same cons an ˜ Cn. Fu he mo e, i any log-conca e p obabili y in Rnwi h compac suppo sa is ies (2) wi h some cons an Cn>2√2(which we can always assume), hen any iso opic log- conca e p obabili y sa is ies (2) wi h cons an 5Cnand so any log-conca e (non-necessa ily iso opic) p obabili y in Rnsa is ies (2) wi h cons an 5Cn. Indeed, le µbe an iso opic log-conca e p obabili y, dµ =e−V(x)dx, wi h V:Rn→(−∞,∞]con ex and le be any Lipschi z µin eg able unc ion . I we ake Ka con ex body such ha •RKe−V(x)dx ≥1 2 •RK( (x)−Eµ (x))2dµ(x)≥1 2RRn( (x)−Eµ (x))2dµ(x), •(EµK −Eµ )2≤Eµ|∇ |2 deno ing by µK he p obabili y suppo ed on Kwi h densi y dµK(x) = e−V(x)dx RKe−V(x)dx, and aking in o accoun ha o any log-conca e p obabili y νone has ha he ope a o no m o i s co a iance ma ix e i ies kCo νkop = sup θ∈Sn−1Eνhx, θi2−(Eνhx, θi)2, we ob ain Va µ ≤2√2Va µK + 2√2(EµK −Eµ )2≤2√2Va µK + 2√2Eµ|∇ |2 ≤CnkCo µKkopEµK|∇ |2+ 2√2Eµ|∇ |2 =Cnsup θ∈Sn−1EµKhx, θi2−(EµKhx, θi)2EµK|∇ |2+ 2√2Eµ|∇ |2 ≤Cnsup θ∈Sn−1 EµKhx, θi2EµK|∇ |2+ 2√2Eµ|∇ |2 ≤4Cnsup θ∈Sn−1 Eµhx, θi2Eµ|∇ |2+ 2√2Eµ|∇ |2 = (4Cn+ 2√2)Eµ|∇ |2≤5CnEµ|∇ |2. 90 The e o e, one can conside only compac ly suppo ed iso opic log-conca e p obabili ies in Rnin o de o p o e (2). By using a nice esul by E. Milman, [EM], in o de o p o e (2) i is enough o gi e an uppe bound o he a iance o by an absolu e cons an imes k∇ k2 ∞ o any Lipschi z µin eg able unc ion. Besides, i is a 1-Lipschi z µ-in eg able unc ion one has Va µ ≤Eµ| − (0)|2≤Eµ|x|2=n. As a consequence one ob ains ha o e e y ixed n∈N, he alue o he cons an such ha (2) holds o e e y log-conca e p obabili y µin Rnand Lipschi z µ-in eg able unc ion is bounded by a cons an Cn, depending on N, The e o e, i is enough o p o e Theo em 1.3 o e e y n∈Nla ge han some ixed n0, since, changing he alue o he cons an C, one can immedia ely ob ain he esul o e e y dimension n∈N. In conclusion, one can conside only compac ly suppo ed log-conca e iso opic p oba- bili ies in Rn o n≥n0 o some n0∈Nin o de o p o e (2). We will include some p elimina y ac s o esul s we a e going o use. Lemma 2.1. Le µbe any p obabili y on Rnand z∈Rn, hen Eµhx−bµ, zi2=hAµz, zi. P oo . Simply expand bo h exp essions. P oposi ion 2.2 (Re e se Hölde ’s inequali y).The e exis s an absolu e cons an C > 0 such ha o e e y log-conca e p obabili y µon Rn, any semino m g:Rn→Rand 1≤ p≤qwe ha e (Eµgp)1/p ≤(Eµgq)1/q ≤Cq p(Eµgp)1/p . P oo . See [BGVV, Theo em 2.4.6.], The nex esul says ha we only need o ake in o accoun Bo el se s wi h p obabili y 1/2. P oposi ion 2.3. Le µbe an iso opic log-conca e p obabili y on Rn. Assume ha he e exis wo posi i e numbe s Θ, C > 0such ha µ(EΘ E)≥C 91 o any Bo el se E∈Rnsuch ha µ(E) = 1 2, whe e EΘis he Θ-dila ion o E, i.e. EΘ={e+x∈Rn:e∈E, |x|<Θ}.Then µ+(A)≥C Θmin{µ(A), µ(Ac)} o any Bo el se A⊂Rn. P oo . See [EM2]. In o de o con ol he p obabili y o dila ions o Bo el se s, he ollowing concen a ion esul s o mo e con ex han Gaussian p obabili ies can be applied P oposi ion 2.4. Le φbe a con ex unc ion φ:Rn→Rand le > 0. Assume ha dµ(x) = e−φ(x)− 2|x|2dx, is a cen e ed p obabili y on Rn. Then o e e y Bo el se A⊂Rnsuch ha 1 10 ≤µ(A)≤9 10 we ha e µAD √ ≥95 100, whe e D > 0is a sui ably chosen absolu e cons an independen o e e y o he pa ame e and AD/√ is he D/√ -dila ion o A. The p oo o his ac ollows om [BGVV, Theo em 14.6.6] (see also [AB1, Theo em 3.8]). Nex we a e going o desc ibe some esul s on I ô p ocesses we a e going o use. (see o ins ance, [O], [Kle]). Le (Ω,F,P)a p obabili y space and (F ) ∈[0,T ]a il a ion in Ω, i.e., a amily o sub-σ- algeb as on Ωsuch ha F 1⊆ F 2⊆ F, whene e 0≤ 1≤ 2≤T. A one-dimensional I ô p ocess (X( )) ∈[0,T ]on Ωis a eal s ochas ic p ocess ha ing he o m X( ) = X(0) + Z 0 U(s)ds +Z 0 V(s)dW(s),0≤ ≤T, whe e X(0) is F0-measu able and he p ocesses U( )and V( )a e F -adap ed and such ha EPRT 0|U( )|d < ∞,EPRT 0V2( )d < ∞, and (W( )) ≥0is a Wiene p ocess (o B ownian 92 mo ion). I is said ha he p ocess (X( )) ∈[0,T ]has he s ochas ic di e en ial on [0, T] dX( ) = U( )d +V( )dW( ),0≤ ≤T. The p ocess (U( )) ∈[0,T ]is called he d i and (V( )) ∈[0,T]is called he di usion o (X( )) ∈[0,T]. No e ha he p ocesses (U( )) ∈[0,T ]and (V( )) ∈[0,T ]may (and o en do) depend on (X( )) ∈[0,T ]o he Wiene p ocess (W( )) ≥0as well. In he case ha he p ocesses (U( )) ∈[0,T ]is Rn- alued, (V( )) ∈[0,T]is an (n×n)ma ix and (W( )) ≥0is an n-dimensional Wiene p ocess, we say ha Xis an n-dimensional I ô p ocess. Le (X1( )) ∈[0,T],(X2( )) ∈[0,T ]be wo 1-dimensional I ô p ocesses. The quad a ic co- a ia ion o [X1, X2] is de ined by [X1, X2] = lim kPk→0 N X k=0 X1(τk+1)−X1(τk)X2(τk+1)−X2(τk) whe e P={0 = τ0≤τ1≤ ··· ≤ τN≤T}is a s ochas ic pa i ion o he non-nega i e eal numbe s, kPk= max(τn−τn−1)is called he mesh o Pand he limi is de ined using con e gence in p obabili y. I X2=X1we will deno e [X1] := [X1, X1] o e e y ∈[0, T]. In he case whe e dXi( ) = Ui( )d +hVi( ), dW( )i, o i= 1,2, whe e (Ui( )) ∈[0,T]and (Vi( )) ∈[0,T](i= 1,2), a e n-dimensional adap ed s ochas ic p ocesses and (W( )) ≥0is an n-dimensional Wiene p ocess [X1, X2] is also an I ô p ocess wi hou di usion and (7) d[X1, X2] =hV1( ), V2( )id . P oposi ion 2.5 (I ô’s o mula).Le (X( )) ∈[0,T ]be an n-dimensional I ô p ocess gi en by dX( ) = U( )d +V( )dW( ), whe e U( )∈Rn,V( )is an n×nma ix and W( )is a n-dimensional Wiene p ocess. Le g:Rn→Rbe a unc ion wi h g∈ C2)(Rn). Then he s ochas ic p ocess (Y( )) ∈[0,T ]gi en by Y( ) = g(X( )) e i ies dY ( ) = dg(X1( ), . . . , Xn( )) = n X i=1 ∂ ∂xi g(X1( ), . . . , Xn( ))dXi( ) +1 2 n X i,j=1 ∂2 ∂xi∂xj g(X1( ), . . . , Xn( ))d[Xi, Xj] . 93 whe e δ is an adap ed, wi h bounded a ia ion p ocess, such ha δ ≤(C p2σ2 nlog nT (Ap )1+ 1 p,i p≥3 CT (A2 )3/2,i p= 2 and | | ≤ CpT (Ap )1+ 1 2p∀p≥2, whe e C > 0is an absolu e cons an and σ2 n= sup E|X|−√n 2and he sup uns o e all iso opic log-conca e andom ec o s in Rn. P oo . We ollow Eldan’s me hod o compu e d(T (Ap )). Howe e , o p= 2 we will use he idea gi en by Lee-Vempala. In o de o do ha we will exp ess A in e ms o a special o hono mal basis. Le 0≥0be a ixed ime. Le ( i)n i=1 be an o hono mal basis composed by he eigen- ec o s o A 0and (αii( 0))n i=1 he co esponding eigen alues. Assume ha he o hono mal basis ( i)n i=1 is o de ed in such way ha α11( 0)≥α22( 0)≥ ··· ≥ αnn( 0). Le also, o any ≥0,αi,j =αi,j( ) := hA i, ji. We can exp ess, o any ≥0, A = n X i,j=1 αij i⊗ j. I is no di icul o see ha o any na u al numbe p≥2and o any ≥0, T (Ap ) = Xαi1i2αi2i3. . . αipi1, whe e he sum uns o e all indices i1,...ip∈ {1, . . . , n}. No ice ha i = 0 hen αij( 0) = hA 0 i, ji=δij, he K onecke del a. The e o e, di e en ia ing a = 0, d(T (Ap ))| = 0=Xd(αi1i2αi2i3. . . αipi1) = 0 = n X i=1 d(αp ii)| = 0+X i6=j k1+k2+k3=p−2 d(αk1 ii αijαk2 jj αjiαk3 ii ) = 0 = n X i=1 d(αp ii)| = 0+X i6=j 0≤k≤p−2 d(αk iiαp−k−2 jj α2 ij) = 0 100 ( he es o he e ms a e 0by I ô’s o mula). Acco ding o he exp ession o d(A )we ha e ha d(αij)| = 0=hd(A )| = 0 i, ji =hEµ 0hx−b 0, iihx−b 0, ji(x−b 0), dW i−hA2 0 i, jid =hξij, dW i−hA 0 i, A 0 jid =hξij, dW i−αiiαjjδijd , whe e ξij a e he ec o s ξij =ξi,j( 0) = Eµ 0hx−b 0, iihx−b 0, ji(x−b 0)∈Rn By I ô’s o mula we ob ain he ollowing es ima es d(αp ii)| = 0=pαp−1 ii dαii| = 0+1 2p(p−1)αp−2 ii d[αii] 0 =1 2p(p−1)αp ii |ξii|2 α2 ii −pαp+1 ii d +pαp ii ξii αii , dW  and o i<jand 0≤k≤p−2, since αij = 0 and αii ≥αjj, d((αii)k(αjj)p−k−2(αij)2) = (αii)k(αjj)p−k−2d[αij] = (αii)k+1(αjj)p−k−1|ξij|2 αiiαjj d ≤(αii)p|ξij|2 αiiαjj d . The e o e, T (Ap )is an I ô p ocess wi h d(T (Ap )) = δ d +h , dW i, whe e o any = 0 δ 0=1 2p(p−1) n X i=1 (αii)p|ξii|2 (αii)2−p n X i=1 (αii)p+1 +X i6=j 0≤k≤p−2 (αii)k+1(αjj)p−k−1|ξij|2 αiiαjj and 0=p n X i=1 αp ii ξii αii . I is now enough o bound o m abo e δ and | |a each pa icula = 0. Fi s o all 101 we es ima e | |. By using Cauchy-Schwa z and Bo ell’s e e se Hölde inequali ies (P oposi ion 2.2), he e exis s an absolu e cons an C > 0such ha o e e y 1≤i≤n |ξii|=ξii,ξii |ξii|=Eµ hx−b , ii2x−b ,ξii |ξii| ≤qEµ hx−b , ii4sEµ x−b ,ξii |ξii|2 ≤CEµ hx−b , ii2sEµ x−b ,ξii |ξii|2 Taking in o accoun ha , by Lemma 2.1, Eµ hx−b , zi2=hA z, zi ∀z∈Rn we ob ain ha o e e y 1≤i≤n |ξii| ≤ ChA i, iiA ξii |ξii|,ξii |ξii|1/2 ≤CαiikA k1/2 op . Hence | | ≤ CpkA k1/2 op T (Ap )≤Cp(T (Ap ))1+1/(2p), o some absolu e cons an C > 0. Nex we will es ima e δ i) Case p= 2 No e ha his necessa ily implies k= 0, and so we ha e a simple exp ession o δ , δ = n X i,j=1 |ξij|2−2 n X i=1 (αii)3. 102 Thus, using again Bo ell’s e e se Hölde ’s inequali y wice δ ≤ n X i,j=1 |ξij|2= n X i,j=1 Eµ hx−b , iihx−b , ji(x−b ) 2 = n X i,j,k=1 |Eµ hx−b , iihx−b , jihx−b , ki|2 =Eµ ,x⊗µ ,y hx−b , y −b i3≤CEµ ,x Eµy, hx−b , y −b i23/2 =CEµ ,x hA (x−b ), x −b i3/2=CEµ ,x |A1/2 (x−b )|3 ≤C2Eµ ,x |A1/2 (x−b )|23/2≤C2Eµ ,x hA (x−b ), x −b i3/2 =C2(T (A2 ))3/2. ii) Case p≥3. Now δ ≤1 2p(p−1) n X i=1 (αii)p|ξij|2 (αii)2+p(p−1) X 1≤i<j≤n (αii)p|ξij|2 αiiαjj ≤p(p−1) n X i=1 (αii)p n X j=1 |ξij|2 αiiαjj . Le us ix 1≤i≤n. Then n X j=1 |ξij|2 αiiαjj = n X j=1  Eµ x−b , i √αii x−b , j √αjj (x−b ) 2 = (?) We pe o m in he in eg al de ining he expec a ion he change o a iables x−b =A1/2 y. The in eg al wi h espec o he new a iable ycan be ega ded as an expec a ion wi h espec o a p obabili y ν , which is iso opic and, since he ec o s (ηi)n i=1 wi h ηi= 103 A1/2 i/√αii o m an o hono mal basis, we ha e (?) = n X j=1  Eν A1/2 y, i √αii A1/2 y, j √αjj A1/2 y 2 = n X j=1 A1/2 (Eν hy, ηiihy, ηjiy) 2 ≤ n X j=1 kA1/2 k2 op |Eν hy, ηiihy, ηjiy|2 ≤ kA kop sup θ∈Sn−1 n X j=1 |Eν hy, θihy, ηjiy|2 =kA kop sup θ∈Sn−1kEν y⊗yhy, θik2 HS Thus δ ≤C p2kA kopT (Ap ) sup θ∈Sn−1kEν y⊗yhy, θik2 HS Eldan p o ed in [E1, Lemma 1.6] ha he exp ession be o e is bounded om abo e by sup θ∈Sn−1kEν y⊗yhy, θik2 HS ≤Cσ2 nlog n which gi es us he co esponding es ima e. P oposi ion 5.2. The e exis n0∈Nand C > 0such ha o any n≥n0we ha e P{kA kop ≤4,∀ ∈[0, T]}>0.9, o T=1 4C(log n)2σ2 n and P(kA kop ≤√51 7√n, ∀ ∈[0, T])≥0.9, o T=1 256C√n P oo . Le p≥2, ixed. Conside he unc ion Φ( ) = −(n+T (Ap ))−1/p. Then Φ( )is an 104 I ô P ocess and d(Φ( )) = 1 p d(T (Ap )) (n+T (Ap ))1+1/p −1 2p1 + 1 pd[T (Ap )] (n+T (Ap ))2+1/p =1 p δ (n+T (Ap ))1+1/p −1 + p 2p2| |2 (n+T (Ap ))2+1/p d + p(n+T (Ap ))1+1/p , dW =α d +dZ , whe e α is an adap ed p ocess o bounded a ia ion and Z a ma ingale e m wi h Z0= 0. By he p eceding P oposi ion δ ≤Lp(T (Ap ))1+1/p whe e Lpis a di e en exp ession depending on whe he p= 2 o p≥3. The e o e, α ≤Lp(T (Ap ))1+1/p p(n+T (Ap ))1+1/p ≤Lp p=(C p σ2 nlog n, i p≥3 C, i p= 2.. The quad a ic a ia ion o Z is d[Z] =| |2 p2(n+T (Ap ))2+2/p d ≤C(T (Ap ))2+1/p (n+T (Ap ))2+2/p d ≤C n1/p d . Then Φ( )−Φ(0) = Z 0 αsds +Z ∀ ≥0. We ix T > 0, hen max 0≤ ≤TΦ( ) + (2n)−1/p ≤Lp pT+ max 0≤ ≤TZ . By he Dambis and Dubins-Schwa z heo em (see P oposi ion 2.7) we know ha Z is equal in law o a B ownian mo ion ˜ W[Z] , so o any γ > 0we ha e, by he e lec ion p inciple (see P oposi ion 2.6), ha Pmax 0≤ ≤TΦ( ) + (2n)−1/p −Lp pT > γ≤Pmax 0≤ ≤TZ > γ =Pmax 0≤ ≤T ˜ W[Z] > γ≤Pmax 0≤s≤CT n−1/p ˜ Ws> γ = 2Pn˜ WCT n−1/p > γo≤2 exp −γ2 2CTn−1/p  105 We ake γ=1 4n1/p and T=p 256Lpn1/p and we achie e Pmax 0≤ ≤TΦ( )> n−1/p 1 4+1 256 −2−1/p≤2 exp −8Lp pC  Since p≥2, we ha e 1 4<2−1/p −1 4−1 256 <7 10 and hen Pmax 0≤ ≤TΦ( )>−7 10n−1/p≤2 exp −8Lp pC . We ema k ha max 0≤ ≤TΦ( )>−7 10n−1/p ⇐⇒ max [0,T]T (Ap )≥10 7p −1n Hence we ob ain ha Pmax [0,T]T (Ap )≥10 7p −1n≤2 exp −8Lp pC  E en ually we will conside wo alues o pin o de o ge ou esul . On he one hand, i we choose p= 2 we ha e T=1 256C√nand Pmax 0≤ ≤TT (A2 )>51 49 n≤2 exp (−8) and P(max 0≤ ≤TkA kop >√51 7√n)≤2 exp (−8) On he o he hand, i we choose p= log n. Then T=1 256Cσ2 n(log n)2and P(max 0≤ ≤TT (Alog n )> 10 7log n −1!n)≤2 exp −8σ2 n(log n)2 106 No e ha , conside ing he Gaussian dis ibu ion, we ob ain σ2 n≥1/2and hen P(max 0≤ ≤TT (Alog n )>10 7log n n)≤2 exp −2(log n)2 and Pmax 0≤ ≤TkA kop >10e 7≤2 exp −2(log n)2 6 Gluing he es ima es P oposi ion 6.1. The e exis s n0∈Nsuch ha i n≥n0,µis an iso opic log-conca e p obabili y measu e on Rnand Eis a Bo el se µ(E) = 1 2, gi en he sys em o s ochas ic di e en ial equa ions (8),µ be he measu e de ined by (9),g( ) = µ (E), and T=1 4C σ2 n(log n)2o T =1 256C√n, hen we ha e ha P g(T)−1 2 >1 4≤0.2. P oo . We know ha g(T)−1 2=g(T)−g(0) = ZT 0 dg( ) = ZT 0hη , dW i whe e η =ZE (x)(x−b )dx, being he densi y o he p obabili y measu e µ . The unc ion g( )is a ma ingale and so, by Dambis, Dubins-Schwa z heo em, P opo- si ion 2.7, we ha e ha in dis ibu ion g(T)−g(0) = ¯ W[g]T, ≥0 whe e ¯ Wsis a Wiene p ocess and [g]Tis he quad a ic a ia ion o g, which is, [g]T=ZT 0|η |2d . 107 Hence, o any M > 0, P{|g(T)−1/2|>1/4}=P{| ¯ W[g]T|>1/4} ≤ P{[g]T> M}+Pmax 0≤ ≤M|¯ W |>1 4. We will bound bo h summands om abo e. Taking in o accoun ha o e e y ≥0 |η |=η ,η |η |=ZE (x)(x−b ),η |η |dx ≤sEµ (x−b ),η |η |2 =sA η |η |,η |η |≤qkA kop we ha e ha [g]T≤ZT 0kA kop d ≤Tmax 0≤ ≤TkA kop. and hen (12) P{[g]T> M} ≤ Pmax 0≤ ≤TkA kop >M T On he o he hand, −¯ W  ≥0is also a B ownian mo ion and hen we ha e Pmax 0≤ ≤M|¯ W |>1 4≤Pmax 0≤ ≤M ¯ W >1 4+Pmax 0≤ ≤M−¯ W >1 4 = 4P¯ WM>1 4≤4 exp −1 32M. (13) We conside now wo cases: In he case T=1 4Cσ2 n(log n)2we choose M= 4Tand hen, by (12) and P oposi ion 5.2 P{[g]T> M} ≤ Pmax 0≤ ≤TkA kop >1 4≤0.1. and by (13) Pmax 0≤ ≤M|¯ W |>1 4≤4 exp −C2σ2 n(log n)2 32 ≤0.1 o nla ge enough. 108 In he case T=1 256C√nwe choose M=1 128C. Then by (12) and P oposi ion 5.2 P{[g]T> M} ≤ Pmax 0≤ ≤TkA kop >2√n≤0.1 and by (13) Pmax 0≤ ≤M|¯ W | ≤ 4≤exp (−4C)≤0.1, assuming ha C > 2, which we can assume wi hou loss o gene ali y. The la e esul , oge he wi h he discussion in Sec ion 4, gi e he p oo o Theo em 1.3. Acknowledgmen s. This wo k is pa ially suppo ed by MINECO/MICINN p ojec s MTM2016-77710-P, PID2019-105979GB-I00 and DGA p ojec E48_20R. Re e ences [AB1] D. Alonso-Gu ié ez, J. Bas e o App oaching he Kannan-Lo ász-Simono i s and a iance conjec u es. Lec u e No es in Ma hema ics, Sp inge . Monog aph. 2131, (2015). [AB2] D. Alonso-Gu ié ez, J. Bas e o Sob e la conje u a de sal o espec al de Kannan, Lo ász y Simono i s. 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