A convergence result for the Kuramoto model with all-to-all coupling

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1 A covergece reslt for the Kramoto model with all-to-all coplig Mark Verwoerd ad Oliver Maso The Hamilto Istitte, Natioal Uiversity of Irelad, Mayooth Abstract We prove a covergece reslt for the stadard Kramoto model with all-to-all coplig. Specifically, we show that the critical coplig stregth associated with the oset of completely phase-locked behavior coverges i probability as the mber of oscillators teds to ifiity. Key words: Sychroizatio, Copled Oscillators, Phase-Lockig, Kramoto model, Covergece PACS: Xt Itrodctio The Kramoto model of a system of weakly copled oliear oscillators o a complete graph K,, is give by [,,, 4]: θ i = x i + k si(θ θ i ), i =,...,, () where θ i ( ) R ad x i R deote the phase ad itrisic (or atral) freqecy of oscillator i respectively, ad k R 0 deotes the coplig coefficiet or coplig stregth. The behavior of the system () depeds strogly po the vale of the coplig stregth, as described i [] ad elsewhere. At low vales of k, the oscillators ted to oscillate more or less idepedetly, address: mark.verwoerd@im.ie (Mark Verwoerd ad Oliver Maso). The reaso we se x i rather tha the more commo otatio ω i to deote the itrisic freqecy is that, i keepig with stadard probabilistic otatio, the latter symbol is sed here to defie a probabilistic evet. Preprit sbmitted to Elsevier 4 Agst 00

2 ad little or o coheret behavior is observed. As k is gradally icreased, the oscillators cotie to oscillate icoheretly for a while, bt the slowly lock ito step as k exceeds the theshold vale defied by the Kramoto coplig [, 5, 6] (for a symmetric imodal distribtio of itrisic freqecies, the vale of the Kramoto coplig (i the limit as ) is give by k =, where g is the derlyig desity fctio ad µ is the first momet of g []). From this poit o, more ad more oscillators oi the clster of πg(µ) locked oscillators, til evetally a state is reached i which all the oscillators are (pairwise) phase-locked. The vale of the coplig that defies this fial trasitio to a global (or completely) phase-locked state is referred to here as the critical coplig, ad is deoted by k c. The preset paper is cocered with the behavior of k c i the limit as the mber of oscillators () teds to ifiity. Of the two trasitios metioed above, the first trasitio, from icoheret to partially coheret behavior, has geerally received most attetio, owig perhaps to its more immediate physical sigificace. Oe drawback that presets itself whe stdyig the first trasitio, is that it is hard to characterize mathematically. For this reaso, a approach based o limits sch as developed here for the secod trasitio wold fail whe applied to the first trasitio. The difficlty with the first trasitio is that there is o obvios critical pheomeo i the fiite-dimesioal Kramoto model that oe ca associate with the trasitio from icoheret to partially coheret behavior. For the secod trasitio, sch a critical pheomeo does exist, ad is easily defied. Ideed, for this case the critical pheomeo is precisely the emergece of a (completely) phase-locked soltio which defies the critical vale of the coplig coefficiet, k c. I the preset paper we show the radom variable k c, which geerally depeds po the particlar realizatio of itrisic freqecies, coverges i probability as the mber of oscillators teds to ifiity. To the best of or kowledge, this is the first covergece reslt for the classical Kramoto model. There are may papers that have dealt directly or idirectly with the secod trasitio. We metio some recet cotribtios most relevat to the preset work. Refereces [] ad [] provide a detailed accot of the spectral properties of the phase-locked state (k k c ) ad the partially phase-locked state (k < k c ) respectively. I referece [8], the athors of the preset paper describe a merical algorithm for comptig k c give ay fiite realizatio of itrisic freqecies [8] (see also Appedix A). A extesio of this reslt to the case of a complete bipartite graph is described i the follow-p paper [9]. The strctre of the paper is as follows. I Sectios ad we fix or otatio ad review relevat backgrod material. The mai reslt is preseted i Sectio 4, ad this is followed by a discssio of its applicatios i Sectio 5. Sectio 6 closes with coclsios.

3 Mathematical prelimiaries. Notatio Throghot the paper, R (C) deotes the field of real (complex) mbers, R 0 deotes the set of all o-egative real mbers, ad N deotes the set of positive itegers. For v R ad i, v i deotes the i th compoet of v. For ay real mber x, x deotes the absolte vale of x. For ay -tple x = (x, x,..., x ), the otatio x deotes the arithmetic mea of x, that is x := x.. Probability The termiology ad otatio adopted here is stadard. For backgrod o basic probability theory, the iterested reader shold coslt [0]. Let (Ω, F, P) be a probability space. A radom variable is a fctio X : Ω R with the property that {ω Ω : X(ω) x} F for each x R. We say that X is cotios if its distribtio F (x) = P(X x) ca be writte as F (x) = x f()d for some itegrable f : R [0, ). The fctio f is called the (probability) desity fctio of X. The expectatio of a cotios radom variable X with desity fctio f is give by EX = f()d wheever this itegral exists. Two radom variables X ad Y are idepedet if {X x} ad {Y y} are idepedet evets for all x, y R. If X ad Y are idepedet the for all a, b R we have that E(aX + by ) = aex + bey. Let X, X, X,... be radom variables o some probability space (Ω, F, P). a.s. We say that X X almost srely, writte X X, if {ω Ω : X (ω) X(ω) as } is a evet with probability. We say that X X i probability, writte X P X, if P( X X > ϵ) 0 as for all ϵ > 0. a.s. The followig implicatio holds: (X X) (X P X). We shall eed the followig reslts: Theorem (Hoeffdig s ieqality) Let X,..., X be idepedet radom variables. Sppose there exist a, b R sch that P(X i [a i, b i ]) =. Let

4 S := X. The P(S ES t) exp ( t (b i a i ) ), t > 0. Theorem (Strog law of large mbers) Let X, X,..., X be idepedet idetically distribted radom variables. The X µ almost srely, as, for some costat µ, if ad oly if EX <. I this case, µ = EX. I the paper, we shall se a weaker versio of the strog law of large mbers, which may be stated as follows: Corollary Let X, X,..., X be idepedet idetically distribted radom variables ad sppose EX <. The X EX i probability, as. Backgrod ad Problem Statemet. Backgrod The asymptotic aalysis preseted i this paper is based o recet reslts o global phase-lockig obtaied by the athors [8, 9] (see [, ] for related work). Before we proceed with statig the problem, we review some relevat reslts from the aforemetioed work. To start with, we defie the otio of critical coplig, as follows: Defiitio (Critical coplig) Let N,, ad let x R be give. Cosider the Kramoto model of copled oscillators o the complete graph K (). For this model, we defie the critical coplig, k c, as follows: k c := mi k 0 : ϕ R s.t. k si(ϕ ϕ i ) = x x i for all i. I other words, k c is the smallest oegative vale of the coplig stregth for which the system () admits a (global) phase-locked soltio. 4

5 We have the followig reslt [8]: Theorem Let x R be give ad sppose x i x for some (i, ). The the eqatio ( x x ) = ( x x ). () has a iqe soltio (max x x, max x x ], ad we have that k c = ( ) () x x. If x i = x for all (i, ), we defie := 0 ad k c := 0. Eqatio () expresses the critical coplig k c i terms of, the soltio to Eqatio (). The latter ca be evalated to withi ser-defied precisio with the algorithm described i Appedix A.. Problem statemet The proof of or mai reslt, which we preset i Sectio 4., reqires several techical steps ad is qite ivolved, bt the mai idea is easy to follow. Ideed, the idea is to ivoke the law of large mbers o the expressios i Eqs. () ad () which allows oe i the limit to pass from sample meas to expected vales (from smmatios to itegrals). The mai techical difficlty with this approach is that i order to ivoke the law of large mbers oe mst reqire the terms der the smmatio to be idepedet, which i or case they are ot. What saves the argmet is the fact that i the limit, we ca replace x with µ, which gives s the idepedece we eed. Ufortately, the ecessary proofs reqire a rather otatio-heavy techical apparats. The prpose of the preset sectio is to set p this apparats ad to state the problem formally. Let X, X,..., X be idepedet ad idetically distribted radom variables with desity fctio p : R R 0 ad expected vale µ := EX = R x p(x)dx. Let S p := {x R : p(x) > 0} deote the spport of p. We assme the followig: Assmptio The desity fctio p is (piecewise) cotios ad has boded spport. Moreover, S p ( ) (x µ) (x µ) p(x)dx < (4) 5

6 for all sp x Sp x µ. Remark Assmptio is a techical assmptio that will facilitate the forthcomig aalysis. The class of distribtios satisfyig this assmptio is sfficietly rich for or preset prposes. Note i particlar that Assmptio is satisfied by all piecewise cotios p sch that sp x Sp x < (boded spport) ad sp x Sp p(x) < (boded rage). Moreover, sice the itegrad is a mootoe decreasig fctio of, if follows that if (4) is satisfied for = sp x Sp x µ, the it is satisfied for all sp x Sp x µ. Defie D := {(x,, a) R R R : > max x a > 0}, ad let f : D R ad f : [sp x Sp x µ, ) R be give by f (x,, a) := ( x a ) ( x a ), (5) ( ) f() := S p (x µ) (x µ) p(x)dx. (6) We ext defie the fctio : R R R. For all (x, a) R R sch that max x a > 0, (x, a) is the iqe soltio of f (x, (x, a), a) = 0. I the case that max x a = 0, we defie (x, a) = 0. Let X deote the -tple of iid radom variables (X,..., X ) ad let X := X. The radom variables (X, X ), (X, a) (for a R) are defied i the sal maer, as is the radom variable k c (X), k c (X) := (X, X ) ( ). X X (X, X ) The prpose of this ote is to show that, der sitable techical coditios, where (X, X ) P (X, µ) P, f (0) if f(sp := x Sp x µ ) > 0, sp x Sp x µ otherwise, It the follows that there exists k 0 sch that k c (X) P k. 6

7 4 A covergece reslt 4. Key techical reslts We shall ow preset some techical reslts that will be helpfl i provig or mai reslt. Proofs ca be fod i Appedix B. We adopt the followig otatio: ad f (x,, a) := f (x, ξ, a) ξ, a f (x,, a) := f (x,, ξ), ξ= ξ ξ=a aa f (x,, a) := f (x,, ξ) ξ. ξ=a Or first lemma establishes a few sefl facts abot f ad. Lemma The followig statemets are tre for all : (i) For all (x,, a) D, we have that a f (x,, a) f (x,, a). (ii) For all (x,, a) D, we have that f (x,, a) ad aa f (x,, a) > 0. (iii) Let c > 0 ad sppose (x, x ) < ( + c) max x x. Sppose i additio that (x, µ) < (x, x ). The: (x, µ) (x, x ) < µ x + c max x x. (iv) Let c > 0 ad sppose (x, x ) ( + c) max x x. The: ( f (x, (x, x ), x ) ( ) ). + c Remark Propositios (iii) ad (iv) of Lemma still hold if we iterchage x ad µ. Or ext lemma shows that (x, x ) (x, µ) as x µ. This is a importat itermediate step becase it sggests that, i the limit, we ca replace X with µ, eablig s to ivoke the law of large mbers (as show i the forthcomig Propositios ad ). Lemma For every ϵ > 0 there exists δ > 0 sch that the implicatio holds for all. µ x < δ (x, µ) (x, x ) < ϵ 7

8 We have the followig propositio. Propositio Let X = (X, X,..., X ) be a -tple of idepedet ad idetically distribted radom variables with desity fctio p, ad sppose p satisfies the coditios of Assmptio. The we have that: (X, X ) P (X, µ). Propositio Let X = (X,..., X ) be ad -tple of idepedet idetically distribted radom variables with desity fctio p, ad sppose p satisfies the coditios of Assmptio. The we have that: where is give by (7). (X, µ) P, 4. The mai reslt We are ow ready to state or mai reslt, which is essetially a corollary of Propositios ad combied. Theorem 4 Let X = (X, X,..., X ) be a -tple of idepedet ad idetically distribted radom variables with desity fctio p ad sppose p satisfies the coditios of Assmptio. The we have that (a) (X, X ) P ; (b) k c (X) P / S p ( ) x µ p(x)dx. Proof: Combiig Propositios ad, we have that (X, X ) P. This proves the part (a). To prove part (b), recall that k c (X) = (X, X ) (X, X ) (X X ), whe (X, X ) 0 ad k c (X) = 0 otherwise. Uder the hypotheses of Assmptio, we have that > 0 ad S p ( ) (x µ) p(x)dx > 0. Hece, if we ca show that (i) (X, X ) P ( ), ad (ii): (X, X ) (X X ) P S p ( ) (x µ) p(x)dx, 8

9 the we are doe (if A P a ad B P b, the A B prove (i), ote that for some c 0, we have that, P a b provided b 0). To (X, X ) ( ) = (X, X ) (X, X )+ c (X, X ). This implies that P( (X, X ) ( ) > ϵ) P( (X, X ) > ϵ ). (7) c Uder the hypotheses of the theorem the right had side of ieqality (7) teds to zero for every ϵ > 0 as (this was established i part (a)), ad hece so does the left had side. We coclde that (X, X ) P ( ). The proof of (ii) reqires some more work, bt essetially proceeds alog the same lies. The idea is to se stadard estimates (sch as a b a b which holds for all oegative a, b) to show that X imply that P µ ad (X, X ) P (X, X ) (X X ) P ( ) (X µ). The reslt the follows from the law of large mbers (Corollary ). This completes the proof. Note that i order to evalate the expressio for the critical coplig, we eed a estimate for. I geeral, this reqires solvig the itegral eqatio f() = 0. However, i the special case whe f(sp x Sp x µ ) 0, is simply give as = sp x Sp x µ. The examples i the ext sectio will deal exclsively with the latter case. 5 Applicatio of the mai reslt To illstrate the reslt of Theorem 4, we preset two examples. I the first example, we cosider distribtios of itrisic freqecies described by a family of symmetric desity fctios with fiite spport that icldes the iform desity fctio. I the secod example, we cosider a distribtio described by a asymmetric desity fctio. The prpose of this example is to show that the applicatio of or reslt is ot limited to symmetric distribtios. 9

10 5. Critical coplig i the limit for a class of symmetric desity fctios with fiite spport Let α >, c > 0, ad let p α(x) : R R 0 be give as p α (x) := c Γ(α+ ) ( ( ) π Γ(α+) x µ ) α, c x µ < c; 0, x µ c. (8) Note that E α (x) := R x p α(x)dx = µ, ad that p α (µ x) = p α (µ+x) for all x. I other words, p α is symmetric abot its mea, µ. Figre shows the graph of p α (µ x) o the iterval [0, c] for selected vales of α. Note that p 0 is the.5 c pα(µ x) 0.5 α = 4 α = + 4 α = ( x ) c Fig.. The graph of p α (µ x) o the iterval [0,c] for α { 4, 4, 4}. iform desity fctio o ( c, c). For all α we have that S pα = ( c, c) ad sp x Spα x µ = c. For the family of distribtios cosidered here we ca evalate f(sp x Sp x µ ) aalytically. Ideed, we have that α ( ) f( sp x µ ) := Γ (α+ ) c, α >, α 0; α(α+)(+α) Γ (α) x S pα 0, α = 0, where ( ) is the Gamma fctio. It follows that f(sp x Spα x µ ) > 0 if ad oly if α < 0. By Theorem 4, this implies that for all α 0, we have that k c (X) P c S p ( ) = x µ p(x)dx c Γ(α + )Γ(α + ) Γ (α + ) c. I particlar, for α = 0 we have that K P c 4 c = π πp 0, which coicides with (µ) the Kramoto coplig vale for the oset of partially phase-locked behavior. 0

11 Figre shows how, for the case c =, α = 0, both the critical coplig k c (x) := (x, x ) ( ), (9) x x (x, x ) ad the estimate obtaied by replacig x with µ, ˆk c (x) := (x, µ) ( ), (0) x µ (x,µ) coverge to the expected vale E X k c (X) = as the mber of oscillators π icreases. Figre shows, agai for the case c =, α = 0, the behavior of (x, x ) ad (x, µ) as a fctio of. Note that (x, µ) appears to coverge mch faster tha (x, x ). For α > 0, we have that ( ) k c (X) P π Γ(α + ) Γ(α + ) πp α (µ), which sggests that the relative differece betwee the Kramoto coplig ad the critical coplig i the sese of Defiitio diverges as α teds to ifiity, with the former (associated with the oset of global phase-lockig) tedig to c ad the latter (associated with the oset of partial phase-lockig) tedig to (x, x ) ( x ) x (x, x ) kc(x), ˆkc(x) π 0.6 (x,µ) ( x ) µ (x,µ) Fig.. The graph of k c ad ˆk c, (9) ad (0), for a give realizatio of itrisic freqecies x = (x,..., x ) ad selected vales of i the iterval [0, 0 4 ].

12 0.54 (x, {µ, x }) (x, x ) 0.5 (x, µ) Fig.. The graph of (x, x ) ad (x, µ) for a particlar realizatio of itrisic freqecies x = (x,..., x ) ad selected vales of i the iterval [0, 0 4 ]. 5. Critical coplig i the limit for a o-symmetric distribtio with fiite spport Next we cosider a simple o-symmetric desity fctio p : R R 0, p(x) := x for 0 < x, 0 elsewhere. For this distribtio we have that µ =, S p = (0, ) ad sp x Sp x µ =. Ispectio shows that f(sp x µ ) = f( x S p ) = 8. 7 Hece, by Theorem, we have that k c (X) P sp x Sp x µ S p ( ) x µ p(x)dx = sp x Sp x µ 8 π = 9 4π. 7 This is cosistet with simlatio reslts (data ot show).

13 6 Coclsio I this paper we proved a covergece reslt for the Kramoto model. I particlar, we showed that, der appropriate techical coditios o the distribtio of itrisic freqecies, the critical coplig stregth associated with the emergece of global phase-locked soltios coverges i probability as the mber of oscillators teds to ifiity. I geeral, this critical vale differs from the Kramoto coplig, which relates to the existece of partially phase-locked statioary soltios, bt for some distribtios they coicide (otably for the iform distribtio). We hope that the reslts described i this paper cotribte to a better derstadig of the asymptotic properties of the classical Kramoto model. A Algorithm for comptig the critical coplig The algorithm below comptes the soltio to Eqatio () for a give vector x = (x, x,..., x ) of itrisic freqecies, with precisio AbsTol.. a := max x x ;. := a;. AbsTol := 0 6 ; 4. Err := ; 5. := ( a); 6. while Err > AbsTol 6.. Err := 6.. if Err 0 := ; ( x x := ; 6.. else := + ; 6.4. ed 7. ed := ; ) ( x x ) ;

14 B Proofs B. Proof of Lemma Proof:. To prove Propositio (i), let (x,, a) D. The by defiitio we have that > max x a ad the reslt follows by ispectio: a f (x,, a) = ( x a max ( x a ( ( x a ) ( ( x a ) ( ( x a) ) ) ) } {{ } =: f (x,,a). ) )( ( x a ( Propositio (ii) agai follows by ispectio. Ideed, we have that ) ), ( x a ) ), f (x,, a) = aa f (x,, a) = ( ( x a ( ( ) ( x a ) ) ) ) 5, > 0. To prove Propositio (iii), ote that, der the give hypotheses, max This implies that x µ < (x, µ) < (x, x ) < ( + c) max x x. (x, µ) (x, x ) < ( + c) max = max x x max x µ, x x max µ x + c max x µ + c max x x, x x, as reqired. Fially, let c > 0 ad sppose (x, x ) (+c) max x x. Propositio (iv) follows by ispectio: 4

15 f (x, (x, x ), x ) := This cocldes the proof. max ( ( x x (x, x ) ( ) ) ( ) x x ) (x, x ) = ( ( + c) ). B. Proof of Lemma Proof: Defie λ := sp x,y Sp x y ad ote that λ < by Assmptio. We distigish two cases: (i) x i = x for all (i, ); (ii) x i x for some (i, ). To prove case (i), let ϵ > 0 be give ad defie δ := ϵ. Uder the give hypotheses, it follows from the defiitio of that (x, µ) = µ x ad (x, x ) = 0. Now sppose µ x < δ. The by costrctio we have that (x, µ) (x, x ) = µ x < δ = ϵ, as reqired. To prove case (ii), let x i x for some (i, ) ad sppose (x, x ) > (x, µ) (i case (x, x ) = (x, µ) there is othig to prove, while i case (x, x ) < (x, µ) the reslt follows by aalogy with the preset case). Let ϵ > 0 be give ad defie δ := ϵ mi{, ( ( ) ) + ϵ }. We λ distigish two cases: (ii-a) (x, x ) < ( + ϵ (ii-b) ) max λ x x ; (x, x ) ( + ϵ ) max λ x x. First we cosider case (ii-a). Sppose µ x < δ. Uder the give hypotheses, it follows from Lemma, Propositio (iii), that (x, µ) (x, x ) < µ x + ϵ λ max x x < ϵ + ϵ = ϵ, as reqired. Next we cosider case (ii-b). Firstly, ote that der the give hypotheses f (x,, µ) is strictly cocave ad differetiable o (max x µ, ). This implies that 0 < (x, x ) (x, µ) f (x, (x, x ), µ) <. f (x, (x, x ), µ) (see Figre B. for a illstratio). Secodly, ote that, restricted to the iterval I := [mi{µ, x }, max{µ, x }], the fctio f (x, (x, x ), ) is Lipschitz. I particlar, we have that f (x, (x, x ), µ) f (x, (x, x ), x ) < max a I af (x, (x, x ), a) µ x. 5

16 Sice aa f (x, (x, x ), a) > 0 by Lemma, Propositio (ii), it follows that max a I a f (x, (x, x ), a) = max a {µ, x } a f (x, (x, x ), a). Now recall that f (x, (x, x ), x ) = 0 by defiitio of (x, x ). It follows that (x, x ) (x, µ) max a {µ, x } a f (x, (x, x ), a) f (x, (x, x ), µ) µ x.(b.) Direct applicatio of Lemma, Propositios (i), (ii), ad (iv) yields: a f (x, (x, x ), µ) f (x, (x, x ), µ), f (x, (x, x ), µ), f (x, (x, x ), x ) ( ( + ϵ ) ). λ Combied with (B.) this gives s (x, x ) (x, µ) max {, ( ( + ϵ λ = ( ( + ϵ λ ) ) µ x ) ) } µ X, Now i coclsio sppose µ x < δ. The by costrctio we have that (x, µ) (x, x ) < ϵ. This proves the case (x, µ) < (x, x ). To prove the case (x, µ) > (x, x ), iterchage µ ad x as sggested by Remark. Ispectio shows that all the ieqalities still hold. This cocldes the proof. B. Proof of Propositio Proof: What we eed to show is that, for all ϵ > 0, P( (X, X ) (X, µ) > ϵ) 0 as. Let ϵ > 0 ad defie δ(ϵ) := ϵ mi{, ( ( ) ) + ϵ }, where, as before, λ := sp x,y Sp x y. The by Lemma we have λ that P( (X, X ) (X, µ) > ϵ) P( µ X δ(ϵ)). Moreover, by Hoeffdig s ieqality (Theorem ) we have that P( µ X δ(ϵ)) exp ( δ(ϵ) ). λ 6

17 f (x,, µ) f (x,, x ) f(x,, {µ, x }) f(x,(x, x ),µ) f (x, (x, x ),µ) f (x, (x, x ), µ) (x, µ) (x, x ) This implies that max x x Fig. B.. Illstratio with the proof of Lemma. P( (X, X ) (X, µ) > ϵ) exp ( δ(ϵ) ). (B.) λ We observe that the right had side of (B.) teds to 0 as ad does so for every ϵ > 0. We coclde that (X, X ) P (X, µ). B.4 Proof of Propositio Proof: Let f be give by (5). Observe that f is strictly decreasig: f(ξ) ξ = ξ= S p ( ( ) ) x µ p(x)dx < 0. Sppose f(sp x Sp x µ ) > 0. Sice f() < 0 for large, by cotiity there exists > sp x Sp x µ sch that f( ) = 0. Let ϵ > 0 be small (i 7

18 particlar, let ϵ < max x Sp x µ ). The by Corollary, otig that Z i := X i Xi, i =,...,, are idepedet ad idetically distribted radom variables, we have that f (X, ϵ, µ) P f( ϵ) > 0 ad f (X, + ϵ, µ) P f( + ϵ) < 0. By cotiity of f ad f this implies that P( (X, µ) > ϵ + δ) 0 for all δ > 0 as. Sice this is tre for every ϵ it follows that (X, µ) P. Note that = by defiitio. Now sppose istead f(sp x Sp x µ ) 0 ad defie := sp x Sp x µ. Let ϵ > 0. Agai by Corollary we have that f ( + ϵ, µ) P f( + ϵ) < 0. By the same argmet as above this implies that P( (X, µ) > δ + ϵ) 0 for all δ > 0 ad all ϵ > 0 as. What remais to be show is that P( (X, µ) < δ) 0 for all δ > 0 as. Recall that (X, µ) > max X µ. It is easy to see that, der the coditios of Assmptio, P(max X µ < δ) 0 for all δ > 0 as. This implies that P( (X, µ) < δ) 0 for all δ > 0 as, as reqired. Note that = by defiitio. This cocldes the proof. Refereces [] Y. Kramoto. Self-etraimet of a poplatio of copled oliear oscillators. I H. Araki, editor, Iteratioal Symposim o Mathematical Problems i Theoretical Physics, volme 9 of Lectre Notes i Physics. Spriger, 975. [] Y. Kramoto. Chemical Oscillatios, Waves ad Trblece. Spriger, 984. [] S. Strogatz. From Kramoto to Crawford: explorig the oset of sychroizatio i poplatios of copled oscillators. Physica D, 4: 0, 000. [4] J. Acebró, L. Boilla, C. Pérez Vicete, F. Ritort, ad R. Spigler. The Kramoto model: a simple paradigm for sychroizatio pheomea. Reviews of Moder Physics, 77:7 85, 005. [5] Y. Maistreko, V. Popovych, O. Brylko, ad P. Tass. Mechaism of Desychroizatio i the Fiite-Dimesioal Kramoto Model. Physical Review Letters, 9(8), 004. [6] Y. Maistreko, O. Popovych, ad P. Tass. Desychroizatio ad Chaos i the Kramoto Model, volme 67 of Lectre Notes i Physics [7] S. Strogatz ad R. Mirollo. Stability of icoherece i a poplatio of copled oscillators. Joral of Statistical Physics, 6(-4):6 65, 99. [8] M. Verwoerd ad O. Maso. Global phase-lockig i fiite poplatios of phase-copled oscillators. SIAM Joral o Applied Dyamical Systems, 7():4 60,

19 [9] M. Verwoerd ad O. Maso. O comptig the critical coplig coefficiet for the Kramoto model o a complete bipartite graph. SIAM Joral o Applied Dyamical Systems, 8():47 45, 009. [0] G. Grimmett ad D. Stirzaker. Probability ad Radom Processes. Oxford Uiversity Press, editio, 005. [] R. Mirollo ad S. Strogatz. The spectrm of the locked state for the Kramoto model of copled oscillators. Physica D, 05:4966, 005. [] R. Mirollo ad S. Strogatz. The Spectrm of the Partially Locked State for the Kramoto Model. Joral of Noliear Sciece, 7(4):09 47,