On a memory game and preferential attachment graphs

Transcription

1 O a memory game ad preferetial attachmet graphs Hüseyi Aca School of Mathematical Scieces Moash Uiversity Melboure, VIC 3800 Australia huseyiaca@moashedu Pawe l Hitczeko Departmet of Mathematics Drexel Uiversity Philadelphia, PA 904 USA phitcze@mathdrexeledu Abstract I a recet paper Vellema ad Warrigto aalyzed the expected values of some of the parameters i a memory game, amely, the legth of the game, the waitig time for the first match, ad the umber of lucky moves I this paper we cotiue this directio of ivestigatio ad obtai the limitig distributios of those parameters More specifically, we prove that whe suitably ormalized, these quatities coverge i distributio to a ormal, Rayleigh, ad Poisso radom variable, respectively We also make a coectio betwee the memory game ad oe of the models of preferetial attachmet graphs I particular, as a by product of our methods we obtai simpler proofs although without rate of covergece of some of the results of Peköz, Rölli, ad Ross o the joit limitig distributios of the degrees of the first few vertices i preferetial attachmet graphs For provig that the legth of the game is asymptotically ormal, our mai techical tool is a limit result for the joit distributio of the umber of balls i a multi type geeralized Pólya ur model Keywords: covergece i distributio, geeralized Pólya ur, preferetial attachmet graph, memory game 00 AMS Subject Classificatio: Primary: 60F05, Secodary 05C8, 60B, 60C05 Itroductio Cosider the followig game of memory played with a deck of cards, which has bee studied by Vellema ad Warrigto [0] There are cards with differet labels, two cards for each label They are shuffled ad put i a row, all facig dow A player flips over two cards at each tur ad if the cards match, she removes the cards from the deck; otherwise she flips them over agai The goal is to fiish the game usig the smallest umber of moves Each iitial cofiguratio of the cards correspods to a permutatio of the multiset {,,,,,, } Followig Vellema ad Warrigto we assume that the player has a perfect memory ad the iitial cofiguratio is give by a permutatio chose uiformly at radom This author was partially supported by a Simos Foudatio grat #08766 His work was carried out durig a visit at Moash Uiversity i the first half of 04 He would like to thak the members of the School of Mathematical Scieces ad Nick Wormald i particular for hospitality ad support He would also like to thak Mark Wilso for suggestig the problem ad exchagig some ideas about it Both authors would like to thak Adrew Barbour for a helpful coversatio o the topic related to this paper

2 The player uses the followig optimal strategy She starts from the leftmost card ad flips over the cards from left to right Before startig roud t, if she kows the places of two idetical cards, ie if she flipped over oe of them at roud t ad oe of them before roud t, the she removes the pair at roud t Otherwise, she flips over the first card that has ot bee flipped over yet I this case, if the matchig card has bee flipped over previously, the she flips it over oce agai ad removes the pair from the deck; if ot, she flips over the ext uflipped card Note that each card is flipped over at least oce ad at most twice Thus the player fiishes i at least rouds ad at most rouds We refer the reader to [0] for more details ad refereces about the game I particular, the authors of [0] studied the expected values of three parameters: the expected legth of the game ie the total umber of moves, the expected umber of flips till the first match, ad the expected umber of lucky moves, where a lucky move refers to a move at which two ecessarily cosecutive cards of the same label are flipped for the first ad the last time i the same roud I this work we take the aalysis oe step further ad idetify the limitig distributios of those three parameters as the size of the deck,, icreases to ifiity; see Theorems 3, 4, ad 6 below for precise statemets Furthermore, we describe below a coectio betwee the memory game ad a versio of a model of the preferetial attachmet graphs itroduced i [4], mathematically formalized i [7], ad subsequetly studied i a umber of papers; see eg [5, 6, 8] ad refereces therei Thus, our methods may be used to study the preferetial attachmet models I particular, our aalysis allows us to fid the asymptotic joit degree distributio of the first k vertices of a preferetial attachmet graph, a result that was origially obtaied i a more precise versio i [6] This coectio betwee the memory game ad the preferetial attachmet graphs is made through chord diagrams, ie pairigs of poits To describe it, we say that two games are equivalet if by relabelig the cards i oe of them matchig cards receive the same labels we ca obtai the other oe Thus, each game is equivalet to a uique game i which the secod occurrece of card i precedes the secod occurrece of card j for i < j Such games called stadard i [0] correspod to chord diagrams Various eumeratio problems about chord diagrams have bee studied widely; see for example [, 9, 0,, 7] Besides their combiatorial sigificace, chord diagrams appear i various fields i mathematics, especially i topology For detailed iformatio about chord diagrams ad their topological ad algebraic sigificace we refer the reader to Chmutov, Duzhi, ad Mostovoy s book [8], ad for several other applicatios, to the paper by Aderse et al [3] ad the refereces therei As aother applicatio, Bollobás et al [7] used liearized chord diagrams to geerate a preferetial attachmet radom graph itroduced by Barabási ad Albert [4] This graph is obtaied from a discrete time radom graph process First we describe the process as suggested by Bollobás et al ad the give their alterative way of obtaiig the same graph The process starts with G, the graph with oe vertex v ad a loop o v For >, G is obtaied from G by creatig a ew vertex v ad addig a radom edge e icidet to v, where P e v i v { DG v i/ if i, / if i, ad where D G v deotes the degree of the vertex v i the graph G A loop cotributes to the degree Bollobás et al proposed the followig method to geerate G O a lie, take poits ad pair them radomly Coect the poits i each pair by a arc chord above the lie This is a radom liearized chord diagram, call it C For each arc, there is a left ad a right edpoit Proceedig from left to right,

3 v v 4 v v 3 v 5 Figure : A liearized chord diagram ad the correspodig graph idetify all edpoits up to ad cotaiig the first right edpoit to form the vertex v The, startig with the ext edpoit, idetify all the edpoits up to the secod right edpoit to form v, ad so o The, for each arc, create a edge possibly a loop betwee the vertices correspodig to the left ad right edpoits of the arc; see Figure The graph G + ca be obtaied from G as follows Create oe edpoit to the right of the last edpoit i C ad the create aother edpoit i oe of the + itervals, choosig the iterval uiformly at radom Fially, joi these two edpoits by a arc ad modify the graph to obtai G + Usig this alterative descriptio, Bollobás et al [7] proved that this graph has power law degree distributio, which had bee previously observed by Barabási ad Albert [4] usig experimetal methods Power law degree distributio meas that the umber of vertices with degree k is proportioal to k γ as, for some γ > 0 Bollobás et al showed that γ 3 i particular Later Bollobás ad Riorda [6] studied the diameter of G makig use of liearized chord diagrams agai Recetly, Peköz et al [5] studied the degree of a fixed vertex i G I particular, they foud a asymptotic distributio for D G v i, alog with a rate of covergece, as I a more recet work [6], they exteded their result to the joit distributio of the first k vertices Oe cosequece of our approach is a simpler proof for the asymptotic joit degree distributio of the first k vertices for ay fixed k usig chord diagrams ad the above descriptio of G I the ext sectio we will itroduce our otatio ad preset our mai results I the subsequet sectios, we will give the proofs of these results Mai results The size of a memory game is the umber of pairs of cards a player starts with Suppose that we have a ifiite umber of cards labeled,, i pairs, ad each pair has a blue copy ad a red copy The colors are oly o oe side, ad whe a card is face dow oe caot see its color A game of size is simply a permutatio of the first pairs of cards Sice the colors are ot essetial for the game, we may assume that a blue card always precedes the red card with the same label Let T deote the set of such permutatios with pairs Thus the cardiality of T is!/ Also let T deote the subset of T such that the red card with label i comes before the red card with label i+ for every i [ ] Followig [0], we call T the set of stadard deals deoted by M i [0] It is easy to see that the cardiality of T is the product of odd positive itegers through A permutatio σ T ca be stadardized by relabelig the pairs of cards so that the red cards appear i icreasig order Such a relabelig does ot chage the game s characteristics that we are iterested i Note that there is a uique stadard deal correspodig to a permutatio i T ad coversely, there are! permutatios equivalet to a give stadard deal i T A uiformly radom game correspods to a uiformly radom elemet of T or T We are free to choose either iterpretatio for our aalyses 3

4 Red cards partitio a permutatio i T ito disjoit blocks Each block cosists of cosecutive cards i the permutatio, ad for k, the k-th block eds with the k-th red card i the permutatio The legth of a block is the umber of cards it cotais We deote by B,i the umber of blocks of legth i i a radom game with pairs of cards We have B,i i ad B,i 0 for every i > + We will use the symbol k for the fallig factorial I particular, i + 3 ii + i + will appear frequetly throughout the paper The followig result gives the joit asymptotic distributio of the block couts ad is crucial for our aalysis Theorem As B,i 4 i + 3 i d W i i, where W i are joitly Gaussia with mea zero ad covariace matrix Σ [σ ij ] give by σ ij 6 i+ 3 j+ 3 4 i+j+ 4, if i j; 4 j j+ j+ 3 4, if i j The followig theorem gives us the total umber of eve legth blocks This is closely related to the legth of the game defied as the miimum umber of moves required to fiish the game Theorem Let Y i B,i The, as Y 3 4 l d N0, σ, 3 where σ 4 l + 8 l Deote by G the legth of a radom game with pairs of cards Usig the previous theorem, we obtai the distributio of G, which is stated ext Corollary 3 As, we have G 3 l d N 0, σ /4, where σ is as i Theorem Proof This follows immediately from Theorem ad the fact that the legth of the game, G, satisfies G 3 + B,i L as observed by Vellema ad Warrigto [0, Sectio 4] Here L deotes the umber of lucky moves defied i [0] ad also below ad by Theorem 6 L is bouded i probability i 4

5 Deote by D,i the legth of the i-th block i a radom game I the radom preferetial attachmet graph described i Sectio, D,i correspods to the degree of the i-th vertex The distributio of D,i is give by Peköz et al [5] The distributio of D, is give i the followig theorem I Sectio 3 we provide a short proof for this theorem Theorem 4 Let X be a stadard Weibull radom variable with parameter, ie a radom variable whose probability desity fuctio is xe x if x > 0 ad 0 otherwise The as where D, d deotes the covergece i distributio d X, 4 The ext theorem is a geeralizatio of the previous theorem This result recovers, albeit without ay error boud, the result of Peköz et al [6, Theorem ] Theorem 5 As, for ay fixed positive iteger k, D,,, D,k d X,, X k, where the joit desity fuctio of X,, X k is give by { k x x + x x + + x k e x + +x k if x,, x k 0, fx,, x k 0 otherwise A lucky move is a move at which we pick two matchig cards by mere chace This happes whe two matchig cards are ext to each other ad either of these two cards has bee flipped over before this move We deote by L the umber of lucky moves i a radom game with pairs of cards Theorem 6 As, the umber of lucky moves L coverges i distributio to a Poisso radom variable with parameter l, that is, L d P oisl 3 Legth of the first block Recall that the legth of the first block correspods to the degree of the first vertex i the radom preferetial attachmet graph G Whe stadard deals are used, the first match occurs at the positio of the red card with value For σ T Vellema ad Warrigto deoted this positio by fσ ad they defied a sequece a, j : {π T : fπ j} They oted [0, Sectio 3] that the umbers a, j satisfy the recurrece a, j ja, j + j a, j with the boudary coditios a, ad a, j 0 if j < or j > + They used this relatio to obtai the expressio for the expected positio of the first match Their argumet quickly leads to the followig statemet 5

6 Propositio 7 Let + A x a, jx j, be the geeratig polyomial of the sequece a, j The, for j A x A x + xx A x, A x x 5 Furthermore, see [0, Theorem 5] for the first statemet as ad Proof Let hj be ay sequece ad set E [D, ] π 6 vard, 4 π + O 7 + H : hja, j j The, by [0, Lemma 4] where hj j is used but the same argumet works for ay sequece hj we obtai H H + jhj + hja, j If hj x j the H A x, hj + hj x j x ad we obtai 5 To prove 6 ad 7 let r The Leibitz formula ad 5 imply that It follows that A r x A r Whe r 0 we obtai A r x + j0 j xx x + A x r r j r xx j A r j+ x A r x + xx Ar+ x + rx Ar + rx A r x + rr Ar x + rr Ar x x + xx Ar+ x A r + ra r + rr Ar A A!! as was observed by Vellema ad Warrigto [0, Lemma 3] This yields the followig recurrece for the factorial momets E [D, r ] Ar A + r + rar A A r A rr + + r E [D, r ] rr Ar A A r A rr E [D, r ]

7 with the iitial coditio specified by D, Lettig r gives E [D, ] which proves 6 Whe we set r we get E [D,]!!!! π E [D, ] + E [D, ] + E [D,] + E [D, ] + Lettig g : E [D, ] / + this yields with g /3 This is solved by Hece ad thus which proves 7 g g + g E [D, ] π O vard, E [D, ] + E [D, ] E [D, ] 4 π + O, Remark Presumably, this process could be cotiued to determie the asymptotics of the factorial momets, ad possibly idetify the limitig distributio of D, For istace, for r 3, we ca get π +! E [D, 3 ] 6 Γ + / π 3/ However, computatios become progressively more complicated ad hece we use a differet approach to fid a asymptotic distributio for D, The followig simple lemma will be used throughout the paper Lemma 8 Let ad let m m be a positive iteger valued fuctio such that m o /3 We have m m e m / + Om 3 / Proof The desired equatio follows from the Taylor expasio of the logarithms o the right had side of m m m m j/ m exp l j/ 8 j j 7

8 Proof of Theorem 4 For the proof, we use the space of all deals T Let r deote the first red card i the radom permutatio If the legth of the first block is t, the the positio of r is t, i which case t blue cards precede r, oe of them beig the blue couterpart of r Now we wat to fid the umber of games such that the first red card appears at positio t To this ed, we first choose r, which ca be doe i ways The we choose ad permute the blue cards precedig r, which ca be doe i t t! ways Note that we oly choose t blue cards sice oe of the t is already kow The we permute the remaiig cards that will appear after r, ad this ca be doe i t! t+ ways Lastly, to fid the probability P D, t, we eed to divide the product by!, the size of T Hece, for t +, P D, t t t! Let t α for some positive α Usig Lemma 8, we have ad cosequetly, t! t+! t t t 9 t + t t t exp t /4 + Ot/ + Ot 3 /, D, P α + O αe α Thus, D, / coverges i distributio to a radom variable X with desity fuctio fx, where fx 0 for x < 0, ad fx xe x for x 0 I particular, we have D, F z : P z e z for ay z 0 With a little more work, we ca boud F α F α, where F is the cumulative distributio fuctio of X Defie e t t/ exp t /4 Note that the sequece {e t } t is log-cocave Now fix α > 0 ad let N α α By 9, we have Usig Lemma 8, N α F α P D, t t t t t Combiig the previous two equatios, F α N α t N α t t t t t t N α t tt t 3 exp + O 4 e t + O t/ + t 3 / I fact, by a more careful aalysis of 8, we ca write N α t N α C + α + α 3 F α e t t 8 t + Ot / t t t e t + O /

9 for a absolute costat C uiformly for all α Cosiderig e t as a Riema sum, by the log-cocavity of {e t }, it is easy to see that, for some costat C, N α t e t F α C e, where e is the maximum of {e t }, which is at most / e Usig the last two equatios, for all α > 0, F α F α C 3 + α + α 3 / for a absolute costat C 3 O the other had, for α log, F α e α O/ ad F α F α F α + F α O/ + F log O log 3/ / Proof of Theorem 5 Let t,, t k be positive itegers ad let t t + + t k Similarly to 9, we write P D, t,, D,k t k k!t t + t 3 t + + t k k + k k t! t k! t k t+k! Here t is the umber of possible positios for the blue parter of the first red card, t + t 3 is the umber of positios for the blue parter of the secod red card ad so o Simplifyig this, we get P D, t,, D,k t k t k t k m k t t + t k k + m + k k Lettig t i α i for positive αi, the right had side becomes + O e α + +α k α α + + α k, from which the theorem follows 4 Blocks of a give size I this sectio we will estimate the expected value ad the variace of B,i We eed these estimates i the ext sectios We start with the followig easy lemmas Lemma 9 For N, i,ad j we have N k k i j k k N + i + j + k N k N k i i + j N + l i j i i + j + l + 0 9

10 Proof The first assertio is [, formula 56] with q 0 For the secod formula ote that N k N k i N k! N k i! i + j N k i j i!n k i! j!n k i j! i i + j Hece, by 0, the left had side of is i + j k N k i + j N + i l i + j i i + j + l + k The ext lemma follows from simple algebra ad the proof is omitted Lemma 0 For f xy : 6 x+y y + 3 x+y y+ + x+y+ y+, we have f ij + f ji 4 Expected umber of blocks of a give size i + j + 4! i +!j +! It has bee show by Vellema ad Warrigto that the expected value of the umber of blocks of size i is asymptotic to 4/i + 3 as the umber of pairs of cards goes to ifiity We will eed a more quatitative versio of their result Propositio As ad for i o / E [B,i ] E [ξ i ] + 4 i O i, where ξ i is the idicator of the evet that the first block has size i Note, i particular, that ξ 0 The boud holds uiformly over i / Furthermore, for every ad i +, E [B,i ] E [ξ i ] + 4e/ i Proof We use the set of permutatios T throughout the proof For x [], let x B ad x R deote the blue ad the red cards labeled with x, respectively Recall that the block sizes are determied by the positios of the red cards For a b, let Ia,b i deote the idicator radom variable which takes the value if the followig hold i the radom permutatio of cards: i a R comes before b R, ii there is o red card betwee a R ad b R, ad iii there are exactly i blue cards betwee a R ad b R If all three of these evets hold, the the cotributio of the pair a R, b R to B,i is Thus, we have B,i ξ i + a b I i a,b 4 ad E [B,i ] E [ξ i ] + a b E [ I i a,b] E [ξi ] + E [ I i,], 5 where the last idetity follows from symmetry For computig E [ I i,], we ca first choose the positios of R ad R, the choose cards all blue to put betwee R ad R, ad fially permute everythig ad divide by the total umber of cofiguratios We write E [ I i,] E + E, 0

11 where E is the cotributio of those cofiguratios such that both B ad B appear before R, ad E is the rest Of course, E is positive if ad oly if i Now, as explaied below, E k i i!/ i kk i! i! i i!/ k Explaatio: I the sum, k + represets the positio of R This implies that the positio of R is k + i + We choose the positios of B ad B i kk ways We choose i cards out of blue cards to put i betwee R ad R, ad permute these i i! ways Next, we choose i positios from {k + i +,, } for the red parters of the blue cards sadwiched betwee R ad R, ad permute these red parters i i! ways The rest of the umerator is for permutig all the remaiig cards After simple cacellatios, we get E i+ i! i k k i i+ i Usig 0 for the right had side above ad simplifyig the resultig expressio, we get i i + E i+ i! i i+ k i+ i + 3 i i 6 Similarly, for i we have E k i i!/ i k i! i! i i!/ k i i! i k k i i i k i i! i i i i 7 i i i i Combiig 6 ad 7, we get E + E i+ i+ i + 3 i + i i i i It follows from Lemma 8 that for i o i i i + O 4 i i + i i + 8 i + 3 i 4 i uiformly over i / This, combied with 5 ad 8, implies Furthermore, for i + i i i j i i j j i j j j j j { exp Hece, applyig + x e x to the last factor i 8 we obtai i { } i + i i + ii exp + i i + e /, i which proves 3 } ii 4

12 4 Boud o the variace of the umber of blocks of a give size I this sectio we estimate varb,i The followig is sufficiet for our purposes Lemma There exists a absolute costat M such that for i o /3 Proof By 4, a,b,c,d varb,i M i 3 9 E [ B,i ] E ξ i + Ia,b i a,b E [ξ i ] + E Ia,b i Ii c,d + E Ia,b i Ii b,c + E ξ i a,b,c E [ξ i ] + Σ + Σ + Σ 3, 0 where differet letters i the sums deote distict umbers from [] [ ad] Σ k deotes the k-th sum before the last equality sig We will see that the mai cotributio to E B,i comes from Σ By Propositio, there are some absolute positive costats K ad K such that E [B,i ] lies i the iterval [K /i 3, K /i 3 ] for all i o /3 as Also, sice there are oly blocks, we have the trivial upper boud B,i Hece sice, usig 9, we have for i + Next we estimate Σ Σ 3 E [ξ i B,i ξ i ] E [ξ i ] i, E [ξ i ] P D, i i i + i i i Computatio of Σ Let A A, i be the followig evet: a { R, R, 3 R, 4 R } appear i the order R, R, 3 R, 4 R i the permutatio, b R ad R are separated by i blue cards, c 3 R ad 4 R are separated by i blue cards Let p i be the probability of A Note that, for each permutatio i A, the cotributio of the 4-tuple R, R, 3 R, 4 R to Σ is, ad we have Σ 4!p i 4 p i 4 We ow estimate p i for i O /3 Before we start, we itroduce some otatio The positios of the red cards R, R, 3 R, ad 4 R determie five disjoit itervals i a permutatio We deote the j-th iterval by I j for j 5 Depedig o which itervals the blue cards { B, B, 3 B, 4 B } lie, we have differet cases Let t u be the umber of blue cards from { B, B, 3 B, 4 B } i the iterval I u for u 4 Note that i a,b I i a,b

13 I 5 caot cotai ay of these four blue cards Sice x B must appear before x R for ay x [], we have the followig costraits for the oegative itegers t, t, t 3, ad t 4 v t u v, for v {,, 3}; t + t + t 3 + t 4 4 u Both itervals I ad I 4 have legths i, that is, there are i positios i each iterval We deote by k ad l the legths of I ad I 3, respectively Thus, the legth of I 5 is 4 k l i i k l For a give t t, t, t 3, t 4 meetig the coditios i, let Kt be the umber of allocatios of { B,, 4 B } ito itervals I, I, I 3, ad I 4, respectig t By cosiderig the umber of blue cards i itervals ad the red parters of the blue cards i I ad I 4, we get k i l i 4 t! p i Kt t! t t t k,l i k l i t 4 4i 4 + t + t 4! i +t +t 4 t 3 i t 4!! t 4! t 3! i t t 4! t 4 i t t 4 3i k t3 + t 4 i t! i t After cacellatios ad takig the factors that do ot deped o k or l out of the ier sum, we get i+ t t 4 p i Kti! t!t 3! 4 i t t 4 t 4i+4 t t 4 k 3i k t3 + t 4 l i k l t i t t 3 i t 4 k l Usig Lemma 9, so we eed the sum l i k l i k, t 3 i t 4 i + t 3 t 4 k l k 3i k t3 + t 4 i k 3 t i t i + t 3 t 4 For the product of the secod ad the third terms i the sum, we write 3i k t3 + t 4 i k i t + t 3 t 4 i k i t i + t 3 t 4 i t i t + t 3 t 4 Usig this idetity, the sum i 3 ca be writte as i t + t 3 t 4 k i k i t t i t + t 3 t 4 Usig Lemma 9 oce agai, this becomes i t + t 3 t 4 i i t i + t t + t 3 t 4 k 3

14 Combiig the results i the previous lies ad usig the idetity 4 t t 4 t t + t 3 t 4, 4 p i t t Kti! t!t 3! i+ t t 4 i+ t t 4 4i+4 t t 4 Ktt!t 3! i t + t 3 t 4 i i t i + t t + t 3 t 4 i! i t!i + t 3 t 4! i t + t 3 t 4! i+ t t4 i+ t t 4 i + t t + t 3 t 4! i Defiig a : i + t t 4 ad usig Lemma 8 for the fallig factorials i 4, we obtai where Hece, for i o /3, we have where 4 p i i! i t + t 3 t 4! Ktt!t 3! i t t!i + t 3 t 4! i + t t + t 3 t 4! t +t 3 a + i [ ] exp + O, a + i i + O i 4 4 p i + Oi / Ktt!t 3! t +t 3 F t, 5 F t i! i t!i + t 3 t 4! Note that there exists a absolute costat γ such that for ay i ad t, t F t γ i t 3+t +t 4 i t γ i t t 3 i t + t 3 t 4! i + t t + t 3 t 4! 6 Cosequetly, the sum i 5 over the vectors t t, t, t 3, t 4 with t + t 3 3 is oly O/i 4 Also, there are three vectors t with the sum of the first ad the third compoets beig 4: t 4, 0, 0, 0, t 3, 0,, 0, ad t 3, 0,, 0 For these t we have Kt, Kt, ad Kt 3 The, 4 p i O /i O i / 4F t + F t + 4F t 3 7 Usig 6, we have 4F t + F t + 4F t 3 4 [ i! 6 i + 4! Usig Lemma 0 with i j, we get [ i i i i + + i i ] i + i + i + 3 i + i + 4! i +!, + ] i + i + 4

15 ad cosequetly, 4F t + F t + 4F t 3 i + 3 Fially, usig this equatio ad 7, we get i Σ 4 p i O i O 6 i O i + 3 i 4 8 Computatio of Σ Let A deote the evet {I, i Ii,3 } i a game with pairs of cards ad let q P A Thus we have Σ 3 q ad we eed to estimate q O the evet A, R appears before R, which appears before 3 R These three red cards determie four itervals, I through I 4, umbered i the order they appear Both I ad I 3 cotai i blue cards ad o red cards For j 3, let t j deote the umber of elemets from { B, B, 3 B } cotaied i I j We have the followig possibilities for t t, t, t 3 : t 3, 0, 0; t,, 0; t 3, 0, ; t 4,, 0; t 5,, Let K u be the umber of ways to place { B, B, 3 B } i I, I, ad I 3 respectig t u We have K, K, K 3, K 4, ad K 5 Let k deote the legth of I, so the legth of I 4 is i k We eed to put i t t 3 blue cards other tha { B, B, 3 B } i I I 3 ad we eed to put the red couterparts of these blue cards i I 4 Hece, usig Lemma 9 i the last step below, we have 5 k i i 3 q K u t!t!t 3! i t t 3! t Now, usig t u t 3 i t t 3 k k i i t t 3! 4i + t + t 3! i t t 3 i +t +t 3 5 i i! t t 3 K u t! 3 i t t i t 3 u i+ t t 3 k k i 4i+ t t 3 t i t t 3 k 5 i i! t t 3 K u t! 3 i t t i t 3 u i+ t t 3 i 4i+ t t 3 i + t t t 3 i + t t t 3 i + t t 3, we simplify the last expressio ad write 5 i q K u t!i! t t 3 3 i t t i t 3 u i+ t t 3 i i + t t 3! 5 O i 4 t t 3 i i i +t +t 3 u 5 u O i 4 t t 3 +t +t 3! 5

16 I the sum above, the mai cotributio comes from u, ie whe t, t, t 3 3, 0, 0, i which case we have O i 4 I other words, we have q O i 4 The, Σ 3 q O i 4 9 Combiig 0,, 8, ad 9, we get 9 for some absolute costat M As a corollary of the previous two lemmas, we get the followig cocetratio result for B,i s Recall that B,i B,i E [B,i ] for i Corollary 3 Let l o /3 ad let ω The P B, ω,, B,l ω O/ω Proof Usig Lemma, ad Chebyshev s iequality, for ay i l, we get where M is a costat from Lemma Hece from which the corollary follows P B,i ω M/i 3 ω, l P B,i ω O/ω, i 5 Lucky moves The asymptotics of the expected value of lucky moves i a radom game of size was derived i [0, Corollary 8] Here we prove Theorem 6, which gives the limitig distributio of the umber of lucky moves as Proof of Theorem 6 Let us write L istead of L for the simplicity of otatio We will compute, asymptotically, the factorial momets E [L k ] for k Fix k ad let E be the evet that the cards umbered k +,, costitute lucky moves i a radom deal i T Clearly, E [L k ] k P E k P E Thus, we eed to fid P E asymptotically To this ed, we use the followig algorithm to geerate a radom deal with pairs of cards Start with a ucolored deck of cards ad put the cards i a row oe by oe i the followig way At the first roud put oe of the two cards labeled i a row For k, at roud k +, isert a card with label k + / ito a iterval chose uiformly at radom from the k + itervals determied by the first k cards already i the row, where a iterval refers to a space either betwee two cards, or before the first card, or after the last card After a pair of idetical cards are iserted, which happes after each eve-umbered step, color the oe o the left with blue ad the oe o the right with red We deote by a,, a k the permutatio after the k-th step of this algorithm, so a,, a deotes a radom game from T 6

17 After m pairs of cards are put dow, there are m + itervals Let them be I,, I m+ from left to right For m, let S m 0 deote the set of itervals followig red cards i a,, a m together with the first iterval, that is, S m 0 : {I i : a i is red} {I } Recursively, for t, defie S m t : {I i : a i is blue ad i S m t } Let S m : i 0 S m i be the set of good itervals A good iterval is a iterval such that if two idetical cards are iserted ito this iterval, the those two cards costitute a lucky move Note that the itervals are alterately good ad bad as log as a red card does ot iterfere The sets S m i partitio the set [m + ] ad S m S m 0 m + Evet E is equivalet to the followig evet Recursively, for j k, the two k + j s are iserted ext to each other i the algorithm ad the first oe is iserted ito a good iterval, ie ito a iterval i S k+j Note that, after such a isertio, both the umber of good itervals ad the umber of bad itervals icrease by oe, ie S k+j S k+j + Let s deote the umber of good itervals after k pairs of cards are placed, that is, s S k Sice s k +, coditioally o s, the probability of E is k j0 k+j+ s + j + k + j + s k, where the umerator o the left had side is the umber of good itervals ad the deomiator is the umber of ways to isert the two cards with labels k + j + Sice S k 0 k + ad S k i j i+ B k,j for i, we have s i 0 Hece, by Propositio ad the idetity j i+ 4 j + 3 S k i + k + i j i+ j i+ B k,j 30 j 4 j + + j + i + i +, 3 we get E [s] k + + E [B k,j ] + i j i+ i + i + i + i [ i + ] l i + i 0 j i+ 4 j + 3 Next we show that s/e [S] i probability For this, it is eough to show that for every ε > 0 P s E [s] > ε 0, as 7

18 This probability is bouded by the sum P l ε B k,j > + P i j i+ i>l j i+ ε B k,j >, where, for a radom variable X, we put X X E [X] Sice j B m,j m for every m, we have l P i j i+ B k,j > ε l i P i j B k,j > ε P j l : B k,j > ε ll + By Corollary 3 applied with ω ε /ll + the right had side above goes to zero for l o /4 Now cosider the secod probability Sice P X > t E [X] /t for a o egative radom variable X ad ay t > 0, usig Propositio we get P i>l j i+ B k,j > ε 4 ε 4 ε i>l j i+ i>l E [B k,j ] 4 ε i>l j i+ P D k, i + + c i 4 ε E [ξ j ] + 4e/ j + 3 E [D k, ] + c l By Propositio 7, E [D k, ] O so that the right had side goes to 0 as log as l with Thus l k E [L k ] k l k, ad L coverges i distributio to P oisl 6 Legth of the game I this sectio we prove Theorem ad thus also the asymptotic ormality of the legth of the game see Corollary 3 i Sectio The proof is a applicatio of Theorem which will be discussed i the ext sectio We first ote that by usig 5, 8, ad the fact that i E [ξ i], ad by lettig i be a iteger of order / log we have E [B,i ] i i i i E [B,i ] + i>i E [B,i ] i i 4 + Olog + O i + 3 i E [ξ i ] + 4 i O 3 4 l + O log, i + O i i 3 i>i where the last equality follows by the partial fractio decompositio used i 3 Hece, Y 3 4 l log B,i + O 8

19 ad it suffices to show that the first term o the right had side is asymptotically ormal For j let V j j i W i ad V i W i where W i are defied i Theorem Pick ay ε > 0 Pick N 0 such that for every N N 0 we have P V V N > ε/ < ε/3 This is possible sice the variace of V V N goes to zero as N ad hece V V N goes to zero i probability Take ay x R ad ay N N 0 ad pick 0 such that for 0 P B,i x + ε/ P V N x + ε/ < ε/3 ad i N P B,i x ε/ P V N x ε/ < ε/3 i N This is possible sice by Theorem there is a joit covergece, ad thus also covergece of ay fiite sums Now, for 0 we have P B,i x P B,i + B,i x i i N i>n P B,i x + ε/ + P B,i > ε/ i N i>n P V N x + ε/ + ε/3 + P B,i > ε/ i>n P V x + ε + ε/3 + P B,i > ε/ i>n Similarly, P B,i x P B,i x, B,i ε/ i i i>n P B,i x ε/ P B,i > ε/ i N i>n P V N x ε/ ε/3 P B,i > ε/ i>n P V x ε ε/3 P B,i > ε/ i>n As we will show below, 0 ad N 0 ca be chose so that P B,i > ε/ ε/3 3 i>n 9

20 for 0 ad N N 0 We the have P V x ε ε P B,i x P V x + ε + ε Lettig ε 0 ad usig the fact that the distributio of V is cotiuous we coclude that P B,i x P V x Fially, ote that V d N0, σ where σ var W i σ i,i + i i 4 i i i,j 3 4 l 4 4 l l 4 4 l 4 4 l + 8 l 3 i i i<j σ i,j i + 3 j j 4j l 4 π 4 i,j j j i 48 i + j i + j π 48 7 l 8 It remais to prove 3 Pick i > N of order /3 η where 0 < η < / The left had side of 3 is bouded by P > ε + P > ε N<i i B,i i>i B,i The first probability by Chebyshev s iequality is at most 6 ε var B,i 6 ε varb,i + covb,i, B,j N<i i N<i i N<i<j i It follows from Lemma ad the choice of i that the first sum is O/N By Cauchy Schwartz covb,i, B,j varb,i varb,j / O i 3/ j 3/ Thus, the secod sum is of order at most i 3/ j 3/ c i 3/ i / O N i>n j>i i>n so that the first probability i 33 is O/N The secod probability, by Markov ad 3 is bouded by [ 4 ε E ] [ ] B,i 8 ε E 8 ε 4e / E [ξ i ] + ii + i + i>i 8 ε P D, > i + O i>i B,i i>i O/ + O i η /6 o, 0

21 by our choice of i Thus, for a give ε > 0 each of the probabilities i 33 ca be made smaller tha ε/6 by choosig N 0 ad 0 sufficietly large This implies 3 7 Proof of Theorem We cosider the followig dyamic versio of the game For k, at step k, we isert the pair of cards labeled k ito the game so that the positio of the blue card is chose uiformly at radom from k positios first ad the the red card goes to the ed of the row We ca represet the evolutio of blocks as a geeralized Pólya ur model with ifiitely may types of balls: 0,,, Drawig a ball of type i correspods to isertig the blue card of the ew pair i a existig block of legth i Drawig a type zero ball represets a situatio whe the blue ad red cards of the arrivig pair are both put at the ed thus creatig a block of size The evolutio of the ur is as follows: we start with oe ball of type 0 If, at ay time, a ball of type i, i is draw it is replaced by oe ball of type ad oe ball of type i + If a ball of type 0 is draw it is retured to the ur alog with oe ball of type ad the process is cotiued i the same maer After draws, there will be oe ball of type 0 ad balls of other types With this iterpretatio, the umber of blocks of legth i whe cards have bee played, B,i, is the umber of balls of type i i the ur after the th draw ad their asymptotic distributios have bee studied widely i the literature I fact, if ot for the ball of type zero it would be exactly the situatio of outdegrees i a radom plae recursive tree studied by Jaso i [4] see Theorem 3 i particular ad i a much more geeral settig i [3] except that his i + ad i +, i + 3 are our i, i +, ad i +, respectively To deal with this type 0 ball, we wish to apply Theorem i [] I that work, a type 0 ball is referred to as a immigratio ball Let us call the vector a m,k K k0 represetig the umber of balls added to the ur after a immigratio ball is draw the immigratio vector Our immigratio vector a m,k K k0 see item a o p 646 i [] is time idepedet ad is give by [a m,0,, a m,k ] [0, 0,, 0,, 0] Similarly, the additio or replacemet matrices are time idepedet, ad o radom, ad the umber of balls i the ur icreases whe a o-immigratio ball is draw Thus, our situatio falls ito the secod of the three cases described i [, Sectio 3] ad i this case, Theorem i [] states that the immigratio is asymptotically egligible ad the ur has the same asymptotics as without immigratio The ur without immigratio was studied by Jaso [3, 4] who reduced ifiitely may types to fiitely may ad used a superball trick see [3, Remark 4] After these reductios, i Jaso s otatio ad icludig the immigratio balls, if ξ i ξ 0,i, ξ,i,, ξ M,i is the replacemet colum vector whe a ball of type i is draw, the with δ i,j deotig the Kroecker delta, ξ 0,j δ,j ie ξ 0 0, 0,, 0, 0 For i < M ξ i,j δ,j iδ i,j + i + δ i+,j 0,, 0, 0, i, i +, 0,, 0, where i is o the i + st positio we start eumeratio at 0 Fially, ξ M,j δ,j + δ M,j 0,, 0,, 0,

22 The replacemet matrix A 0 which has vectors ξ j as its colums is A 0 H M M M where H was the otatio used i [] Notice that if A is a matrix obtaied from A 0 by removig its first colum ad its first row, the it is exactly the matrix cosidered by Jaso [4] for the radom plae recursive tree with i shifted by sice he starts eumeratio at i 0 ad ours, after discardig balls of type zero, starts at i It follows from his argumet see [4, proof of Theorem 3] ad Remark b below that the eigevalues of A are:,,, M ad, cosequetly, that holds with covariaces of W i give by j i σ i,j k0 l0 k+l k + l + 4 i j k l, k + l + 4! k + l + k + 3!l + 3! k + 3l + 3 As was show i [, Propositio ] this expressio simplifies to This proves Theorem Remarks: a It is temptig to icorporate the ball of type 0 i the geeral framework of Jaso, ie to cosider A 0 as the replacemet matrix A additioal eigevalue is 0 so we are still i the regime Reλ < λ so that Jaso s geeral theory [3] would apply Accordigly, for the joit distributio of B,i i 0 we would have ad B,i i0 0, B,0, B,i 4 i + 3 i 4 i + 3 i, as as d W i i0, as, where W i are joitly Gaussia with covariaces σ i,j give by if i, j ad σ ij 0 if i 0 or j 0 I particular W 0 is degeerate as it should be The formal difficulty is that coditio A6 of [3] is ot satisfied: there are two equivalece classes {0} ad {,, } ad the first oe is domiatig while the secod is ot a ball of type zero is ever put i uless it is draw Probably Jaso s theory could be exteded to iclude such cases, but we do ot kow this b Jaso foud the eigevalues ad eigevectors of A by directly solvig the system of liear equatios Alteratively, as he himself suggested oe ca proceed by iductio Ideed, after expadig A 0 λi with respect to the first row ad the repeatedly expadig the resultig cofactors with respect to their last rows we see that the characteristic polyomial of A 0 is M p A0 λ M λλ λ + j j0

23 The eigevalues the may be computed by solvig the system A 0 λiv 0 For example, whe λ this gives v 0 0 ad which is solved by v M M v M, λv M i v i jv j λ + jv j, j < M, Mv M λ v M, v j j + 3 j + v j + 3j + 4 M + j+ j + j + M v M Note that the fractio o the right had side ca be writte as M + j + j + v M M+ j+ M v M, j M ad choosig v M /M + ormalizes v so that j v j This is exactly what Jaso gives see [4, 56] except that our j is his i + The other eigevectors left ad right ca be computed i a similar fashio We skip further details Refereces [] H Aca, P Hitczeko O the covariaces of outdegrees i radom plae recursive trees J Applied Probab, to appear [] H Aca ad B Pittel Formatio of a giat compoet i the itersectio graph of a radom chord diagram, preprit arxiv: [3] JE Aderse, RC Peer, CM Reidys, ad MS Waterma Topological classificatio ad eumeratio of RNA structures by geus J Math Biol 67 03, o 5, 6-78 [4] A-L Barabási ad R Albert Emergece of scalig i radom etworks, Sciece , [5] N Berger, C Borgs, J T Chayes, ad A Saberi, Asymptotic behavior ad distributioal limits of preferetial attachmet graphs A Probab 4 04, 40 [6] B Bollobás ad O Riorda The diameter of a scale free radom graph, Combiatorica 4 004, o, 5 34 [7] B Bollobás, O Riorda, J Specer, G Tusády The degree sequece of a scale-free radom graph process, Radom Structures Algorithms 8 00, o 3, [8] S Chmutov, S Duzhi, ad J Mostovoy Itroductio to Vassiliev kot ivariats, Cambridge Uiversity Press, 0 3

24 [9] R Cori ad M Marcus Coutig o-isomorphic chord diagrams, Theoret Comput Sci , o, [0] [0] S Dulucq, J-P Peaud Cordes, arbres et permutatios, Discrete Math [] P Flajolet ad M Noy Aalytic combiatorics of chord diagrams, Formal Power Series ad Algebraic Combiatorics, th Iteratioal Coferece, FPSAC00, Moscow , Spriger, Berli [] R L Graham, D E Kuth, O Patashik Cocrete Mathematics d ed Addiso Wesley, Bosto 994 [3] S Jaso Fuctioal limit theorems for multitype brachig processes ad geeralized Pólya urs Stochastic Processes, Appl 0 004, [4] S Jaso Asymptotic degree distributio i radom recursive trees Radom Structures Algorithms 6 005, [5] E Peköz, A Rölli, ad N Ross Degree asymptotics with rates for preferetial attachmet radom graphs, A Appl Probab 3 03, o 3, 88 8 [6] E Peköz, A Rölli, ad N Ross Joit degree distributios of preferetial attachmet radom graphs, preprit, [7] J Riorda The distributio of crossigs of chords joiig pairs of poits o a circle, Math Comp 9 975, 5- [8] N Ross Power laws i preferetial attachmet graphs ad Stei s method for the egative biomial distributio Adv i Appl Probab 45 03, [9] A Stoimeow Eumeratio of chord diagrams ad a upper boud for Vassiliev ivariats, J Kot Theory Ramificatios 7 998, 93-4 [0] D J Vellema ad G S Warrigto What to expect i a game of memory Amer Math Mothly 0 03, [] L X Zhag, F Hu, S H Chug ad W S Cha Immigrated ur models theoretical properties ad applicatios A Statist 39 0,