Asymptotic of the maximal displacement in a branching random walk

Transcription

1 Graduate J Math , Asymptotic of the maximal displacement in a branching random walk Bastien Mallein Abstract In this article, we study the maximal displacement in a branching random walk a particle process on the real line in which particles move and reproduce independently We prove that its asymptotic behaviour consists in a rst almost sure ballistic term, a negative logarithmic correction in probability and stochastically bounded uctuations This result, proved in [20] and [1] is given here under close-to-optimal integrability conditions Borrowing ideas from [5] and [31], we provide simple proofs for this result, also deducing the genealogical structure of the individuals that are close to the maximal displacement MSC 2010 rimary 60J80; Secondary 60G50 1 Introduction A branching random walk on R is a particle process that evolves as follows It starts with a unique individual located at the origin at time 0 At each time k, each individual currently alive in the process dies, while giving birth to a random number of children, that are positioned around their parent according to an iid version of a point process We denote by T the genealogical tree of the process For any u T, we write V u for the position of u and u for the generation to which u belongs The quantity of interest is the maximal displacement M n = max V u at time n in the process One can think of the branching random walk as a combination between a Galton-Watson process and a random walk In eect, the tree T is a Galton-Waston tree, and every innite line of particles in this tree moves as in a random walk Apart from its straightforward application to biology, as a simple model of a population invading its environment, the branching random walk is linked with multiple other models in mathematics, physics and informatics For example, the branching random walk and its continuous-time counterpart, the branching Brownian motion a particle process in which individuals move according to iid Brownian motions and split into two at rate 1 are related to F-K type reaction diffusion equations In particular, the cumulative distribution function of the maximum M n converges toward a travelling-wave solution of the F-K equation see eg [13] Moreover, as the correlation between two paths u and v in the genealogical tree T grows linearly with the age of the most recent common ancestor, branching random walks provide simple examples of logarithmically correlated elds Techniques used to study the branching random walk have been successfully applied to the Gaussian free eld [6, 12, 17] Under sucient integrability conditions, the asymptotic behaviour of the maximal displacement M n is fully known Hammersley [18], Kingman [22] and Biggins [10] proved it grows almost surely at linear speed In 2009, Hu and Shi [20] exhibited a logarithmic correction in probability, with almost sure uctuations; and Addario-Berry and Reed [1] showed the tightness of the maximal displacement, shifted around its median More recently, Aïdékon [3] proved the uctuations converge in law to some random shift of a Gumbel variable, under integrability conditions that Chen [15] proved to be optimal Aïdékon and Shi gave in [5] a simple method to obtain the asymptotic behaviour of the maximal displacement They bounded the maximal displacement by computing the number of individuals that cross a linear boundary However, the integrability conditions they provided were not optimal, and the uctuations they obtained were of olog n order With a slight re- nement of their method, we compute the asymptotic behaviour up to terms of order 1 We were able to obtain the asymptotic of the maximal displacement up to a term of order 1 by choosing a bended boundary for the study of the branching random walk, a method already used by Roberts [31] for the related model of the branching Brownian motion This idea follows from a bootstrap heuristic argument, which is explained in Section 4 The close-to-optimal integrability conditions arise naturally using the wellknown spinal decomposition, introduced by Lyons in [24], and recalled in Section 2 We introduce some notation In the rest of the arti- 92

2 1 Introduction 93 cle, c, C are two positive constants, respectively small enough and large enough, which may change from line to line, and depend only on the law of random variables considered Given a sequence of random variables X n, n 1 we write X n = O 1 if the sequence is tight, ie lim sup X n K = 0 K + n 1 Moreover, we work with the convention max = and min = + For any u R, we write position a A branching random walk generation u + = maxu, 0 and log + u = log u + The main assumptions on the branching random walk T, V that we consider are the following The Galton-Watson tree T of the branching random walk is supercritical: E 1 > 1, 11 u =1 position generation and we write S = {#T = + } for the survival event of the tree We allow #{ u = 1} = + > 0, ie a given individual may have an innite number of children The displacements V u of the children u of the root are assumed to be in the so-called boundary case see [11], namely E e V u = 1 and E V ue V u = 0 u =1 u =1 12 For any branching random walk satisfying 12, we have lim n + M n /n = 0 as on S Any branching random walk with mild integrability assumption can be normalized to satisfy these inequalities, see eg Bérard and Gouéré [9] We also assume that the following integrability conditions hold: σ 2 := E V u 2 e V u < + 13 E u =1 e V u u =1 2 log V v + e V v v =1 < +, 14 which is a standard assumption to the study of the maximal displacement of a branching random walk The integrability condition 14 is in fact equivalent to [3, Equation 14] under assumption 13 see eg [28, Lemma A1] for a proof of this statement The main result of the article is the following Theorem 11 Under the assumptions 11, 12, 13 and 14, we have M n = 3 2 log n + O 1 on S b Reduction to the boundary case Fig 1: A branching random walk on R and its normalization to the boundary case The asymptotic behaviour of M n might seems surprising at rst In eect, let M n be the maximum of an exponentially large number of random walks of length n The asymptotic behaviour of M n, is given, up to a linear term, by M n = 1 2 log n + O 1 Therefore, the logarithmic correction of M n keeps track of the correlations existing in the branching random walk More precisely, we prove in the rest of this article that the term 3 2 is related to the probability for a random walk S n to make an excursion, namely cn 3/2 S n 1, S j 0, j n Cn 3/2 In fact, we observe that the trajectory taken by the individual that realizes the maximal displacement behaves as a random walk excursion More precisely, Chen [16] proved that this trajectory, suitably normalized, converges toward a Brownian excursion Using the estimates developed in this article, we also prove Section 4 the following well-known fact: the individuals with the largest displacements at time n are either close relatives, or their most recent common ancestor is a close relative of the root see Theorem 45 for a more precise statement A similar result was proved in [4, 7] for the branching Brownian motion, but a proof of this result under the assumptions on Theorem 11 does not seems to exist in the literature for branching random walks The rest of the article is organised as follows In Section 2, we introduce the so-called spinal decomposition of the branching random walk, which links

3 2 Spinal decomposition of the branching random walk 94 position generation u Given u, v U, we introduce the concatenation of u and v written uv = u1, u u, v1,, v v We dene a partial order on U by u < v u < v and v u = u a Trajectory followed by the rightmost particle position O1 O n b Outline of this trajectory generation 3 2 log n Fig 2: How the rightmost particle at large time reaches its position additive moments of the branching random walk with random walk estimates In Section 3, we give a list of well-known random walk estimates, and extend them to random walks enriched with additional random variables Section 4 is devoted to the study of the left tail of M n This result is used in Section 5 to prove Theorem 11, using a coupling between the branching random walk and a Galton-Watson process 2 Spinal decomposition of the branching random walk We introduce in this section the spinal decomposition of the branching random walk This result consists in an alternative description of a size-biased version of the law of the branching random walk Spinal decomposition of a branching process has been introduced for the rst time to study Galton-Watson processes in [25] In [24], this technique is adapted to the study of branching random walks We precise in a rst time the branching random walk notation we use In particular, we introduce the Ulam-Harris notation for plane trees, and dene the set of marked trees and marked trees with spine Using this notation, we then state the spinal decomposition in Section 22, and use it in Section 23 to compute the number of individuals satisfying given properties 21 Branching random walk notation We denote by U = n 0 Nn, where N 0 = { } the set of nite sequences of positive integers We write U = U\{ } For any u U, we denote by u the length of u For any k u, we write uk for the value of the kth element in u and u k = u1, u2,, uk for the restriction of u to its k rst elements We introduce π : u U u u 1 U, and we call πu the parent of We write u v = u min u, v = v min u, v the most recent common ancestor of u and v The set U is used to encode any individual in a genealogical tree We understand u U as the u u th child of the u u 1th child of the of the u1th child of the initial ancestor A tree T is a subset of U satisfying the three following assumptions: 1 T 2 if u T, then πu T, 3 if u T, then for any j u u, πuj T Given a tree T, the set {u T : u = n} is called the nth generation of the tree, that we abbreviate into { u = n} if the tree we consider is clear in the context For any u T, we write Ωu = {v T : πv = u} for the set of children of u We say that T has innite height if for any n N, { u = n} is non-empty Ω a Ulam-Harris notation u 0 u v u 2 u v Ω b Notation for the genealogy Fig 3: A plane tree π31 33 Given a set {ξu, u U} of iid non-negative integer-valued random variables, we write T = {u T : k < u, uk + 1 ξu k }, which is a Galton-Watson tree It is well-known see eg [8] that T has innite height with positive probability if and only if Eξ > 1

4 2 Spinal decomposition of the branching random walk 95 A marked tree is a couple T, V such that T is a tree and V : T R We refer to V u as the position of u The set of all marked trees is written T We introduce the ltration n 0, F n = σ u, V u, u n Given a marked tree T, V and u T, we denote by T u = {v U : uv T} Observe that T u is a tree Moreover, writing V u : v T u V uv V u, we note that T u, V u is a marked tree called the subtree rooted at u of T, V position u generation v M n T v, V v V u u Fig 4: The subtree of T rooted at v A branching random walk is a random marked tree T, V such that the family of point processes {V v V u, v Ωu, u T} is independent and identically distributed In a branching random walk, for any u T, T u, V u is a branching random walk independent of F u Given a tree T of innite height, we say that the sequence w N N is a spine of T if for any n N, w n T where w n is the nite sequence of the n rst elements in w A triplet T, V, w, where T is tree, V : T R and w is a spine of T is called a marked tree with spine, and the set of such objects is written T For any k 0, we write Ω k = Ωw k \{w k+1 }, the set of children of the ancestor at generation k of the spine, except its spine child We dene three ltrations on this set F n = σ u, V u, u n 21 G n = σ w k, V w k, u, V u, u Ω k 1, k n The ltration F is obtained by forgetting about the spine of the process, while The ltration G correspond to the knowledge of the position of the spine and its direct children 22 The spinal decomposition Let T, V be a branching random walk satisfying 11 and 12 For any x R, we write x for the law of T, V + x and E x for the corresponding expectation For any n 0, we set W n = ev u We observe that W n is a non-negative F n -martingale We de- ne the law x Fn = e x W n x Fn 22 V u u a Red spine of the process V u u b Information in F V u u c Information in G Fig 5: A plane marked tree with spine The spinal decomposition consists in an alternative construction of the law x, as the projection of a law on T, which we now dene We write L respectively L for the law of the point process V u, u = 1 under the law resp We observe that L = u =1 ev u L Let L n, n N be iid point processes with law L Conditionally on this sequence, we choose independently at random, for every n N, wn N such that denoting by L n = l n 1, l n N n we have h N, wn = h L n, n N e lnh = 1 {h Nn} j N n e lnj We write w for the sequence wn, n N We introduce a family of independent point processes {L u, u U} such that L w k = L k+1, and if u w u, then L u has law L For any u U such that u n, we write L u = l u 1, l u Nu We construct the random tree T = {u U : u n, 1 k u, uk Nu k 1 }, and the the function V : u T u k=1 lu k 1 uk For all x R, the law of T, x+v, w T n is written x, and the corresponding expectation is Êx This law is called the law of the branching random walk with spine We can describe the branching random walk with spine as a process in the following manner It starts with a unique individual positioned at x R at time

5 2 Spinal decomposition of the branching random walk 96 0, which is the ancestral spine w 0 Then, at each time n N, every individual alive at generation n dies Each of these individuals gives birth to children, which are positioned around their parent according to an independent point process If the parent is w n, the law of this point process is L, otherwise the law is L The individual w n+1 is then chosen at random among the children u of w n, with probability proportional to e V u Observe that under the law, V wn+1 V w n, n 0, V u V wn, u Ω n, n 0, { T u, V u, u n N Ω n }, are respectively iid random variables, iid point processes and iid branching random walks Note that while the branching random walks are independent of G, the random variable V w n+1 V w n is correlated with V u V w n, u Ω n w 1 w 3 w 0 w 2 Fig 6: The tree of the branching random walk with spine The following result links the laws x and x The spinal decomposition is proved in [24] roposition 21 Spinal decomposition Assuming 11 and 12, for any n N, x R and u = n, we have x Fn = x Fn and x w n = u F n = ev u W n The spinal decomposition enables to compute moments of any additive functional of the branching random walk, and links it to random walk estimates, as we observe in the next section 23 Application of the spinal decomposition Let T, V be a branching random walk For any n N, we denote by A n a subset of { u = n}, such that {u A n } F n We write A n for the number of elements in A n We compute in this section the rst and second moments of this random variable Lemma 22 For any n 0 and x R, we have E x A n = e x Ê x e V wn 1 {wn A n} roof Applying the spinal decomposition, we observe that E x A n = e x E x 1 = e x Ê x 1 {u An} W n e V u W n e V u 1 {u An} = e x Ê x u = w n F n e V u 1 {u An} = e x Ê x e V wn 1 {wn A n} An immediate consequence of this result is the celebrated many-to-one lemma This equation, known at least from the early works of Kahane and eyrière [30, 21], enables to compute the mean of any additive functional of the branching random walk Lemma 23 Many-to-one lemma There exists a random walk S verifying x S 0 = x = 1 such that for any measurable positive function f, we have fv u j, j n = E x e x S n fs j, j n E x roof Let B be a measurable subset of R n Applying Lemma 22 yields E x =Êx 1 {V uj,j n B} e x V wn 1 {V wj,j n B} =E x e x S n 1 {Sj,j n B}, where S is a random walk under law x with the same law as V w j, j 0 under law x This equality being true for any measurable set B, it is true for any measurable positive functions, taking increasing limits We now compute the second moment of A n Lemma 24 For any 0 k < n, we write A 2 n,k = 1 { u v =k} 1 {u An}1 {v An}, v =n We have A 2 n = A n + n 1 A2 n,k Moreover, for any 0 k < n, EA 2 n,k e = Ê V wn 1 {wn A n}b k,n, where B k,n = u Ω k Ê v =n,v u 1 {v A n} G n roof The rst equality is immediate: A 2 n is the number of couples of individuals in A n, and we may partition this set according to the generation at which the most recent common ancestor was alive We now

6 3 Some random walk estimates 97 use the spinal decomposition in the same way as in Lemma 22, we have EA 2 n,k =Ê =Ê e V u W n e V u 1 {u An} e V wn 1 {wn A n}h k,n, where H k,n = v =n = v =n 1 { u v =k,v An} 1 { v wn =k}1 {v An} u Ω k v =n,v u 1 {v An} a robability for a random walk to hit a target Equations 32 and 33 Thus, conditioning on G n, we have EA 2 n,k e = Ê V wn 1 {wn A n}b k,n 3 Some random walk estimates We collect in this section a series of random walk estimates, and extend these results to random walk enriched with additional random variables, correlated to the last step of the walk The events that we consider are illustrated in Figure 3 More precisely, let X j, ξ j be iid random variables on R 2 such that EX 1 = 0, σ 2 := EX 2 1 0, + and Eξ < + 31 We denote by T n = T 0 + n j=1 X j, with T 0 such that for any x R, x T 0 = x = 1 We set = 0 for short The process of interest T n 1, ξ n, n 1 is an useful toy-model to study the spine of the branching random walk It has similar structure as V wn, V u V w n, u Ω n, n 0 We rst recall bounds on the value taken by T n at time n Stone's local limit theorem [32] bounds the probability for a random walk to end up in an interval of nite size For any a, h > 0, lim sup n 1/2 sup T n [y, y + h] C1 + he a 2σ 2 n + y an 1/2 32 Caravenna and Chaumont obtained in [14] a similar result for a random walk conditioned to stay nonnegative Given r n = On 1/2, there exists H > 0 such that for any 0 < a < b, lim inf inf n + n1/2 y [0,r n], x [an 1/2,bn 1/2 ] T n [x, x + H] T j y, j n 2 > 0 33 In the rest of the paper, H is a xed constant, large enough such that 33 holds for the random walk followed by the spine of the branching random walk b robability for a random walk to stay above a boundary Equations 34 and 35 c robability for a random walk to make an excursion above a boundary Lemmas 31 to 33 Fig 7: Some random walk events studied in this section We now study the probability for a random walk to stay above a given boundary f n, that is On 1/2 ɛ This result is often called in the literature the ballot theorem see eg [2] for a review on this type of results The upper bound we use here can be found in [27, Lemma 36], sup n N,y 0 n 1/2 1 + y T j f j y, j n < + 34 The lower bound is a result of Kozlov [23], n 1/2 inf n N,y [0,n 1/2 ] 1 + y T j y, j n < + 35 Using these results, we are able to compute the probability for a random walk to make an excursion above a given curve The proofs of the following results are very similar to the proofs of [5, Lemmas 41 and 43], and are postponed to the Appendix A1 Lemma 31 Let A > 0 and α < 1/2, for any n 1 we consider a function k f n k such that f n 0 = 0

7 4 Bounding the tail of the maximal displacement 98 and fn j sup max j n j α, f nn f n j n j α A There exists C > 0 such that for all y, z, h > 0, Tn f n n + y [z h, z], T j f n j y, j n C 1 + y n1/2 1 + h n 1/2 1 + z n 1/2 n 3/2 Similarly, we obtain a lower bound for this quantity Lemma 32 Let λ > 0, for any 0 k n we write f n k = λ log n k+1 n+1 There exists c > 0 such that for any y [0, n 1/2 ], Tn f n n y + H, c 1 + y T j f n j y, j n n 3/2 The proofs of these two lemmas are based on the following observation: a random walk excursion above a curve can be divided into three parts, as in Figure 8 The rst pink third of the curve is a random walk staying above the line The last orange third is the same process, reversed in time Finally, the middle blue curve has to connect the two extremities and thus is a random walk hitting a boundary Therefore, we expect T n [z y, z + h y], T j y, j n T j y, j n/3 }{{} n 1/2 T n/3 [0, h] }{{} n 1/2 T j z, j n/3, }{{} n 1/2 which explains the decay in n 3/2 obtained in Lemmas 31 and 32 Finally, we are able to bound the probability for the random walk to make an excursion, while the additional random variables remain small This proof, very similar to [3, Lemma B2] is also postponed to Appendix A2 Lemma 33 With the same notation as in the previous lemma, we set τ = inf{k n : T k f n k y + ξ k+1 } There exists C > 0 such that for all y 0, Tn f n n y + H, τ < n, T j f n j y, j n C 1 + y ξ1 0 + E ξ n 3/ a A random walk excursion b Its decomposition into three parts Fig 8: Decomposition of a random walk excursion 4 Bounding the tail of the maximal displacement Let T, V be a branching random walk satisfying 11, 12, 13 and 14, and M n its maximal displacement at time n We write m n = 3 2 log n The main result of the section is the following estimate on the left tail of M n Theorem 41 Under the assumptions 11, 12, 13 and 14, there exist c, C > 0 such that for n 1 and y [0, n 1/2 ], c1 + ye y M n m n + y C1 + ye y Note that the upper bound in Theorem 41 is given by [19, Lemma 44] In fact, this theorem is a particular case of [29, Theorem 41] A natural way to compute an upper bound for M n m n + y would be a direct application of the Markov inequality We have M n m n + y E 1 {V u mn+y} E e Sn 1 {Sn m n+y}, by Lemma 23 Therefore, as S n is a centrer random walk, we have M n m n + y + e mn y e h S n m n y [h, h + 1] h=0 C n3/2 e y n 1/2, by 32 But this computation is not precise enough to yield Theorem 41 To obtain a more precise upper bound, instead of computing the number of individuals that are at time

8 4 Bounding the tail of the maximal displacement 99 n greater than m n, we compute for any generation k n the number of individuals alive at generation k such that the probability that one of their children is greater than m n is larger than a given threshold In other words, instead of counting the mean numbers of grey individuals at generation n in Figure 4, we count the number of red individuals If we assume Theorem 41 to be true, for an individual u alive at generation k, the median of the position of its largest descendant is close to V u + m n k Therefore, we compute see Figure 6 the number of individuals that cross for the rst time at time k n the boundary k f n k := 3 2 log n k+1 n+1 m n m n k generation position Fig 9: The boundary of the branching random walk Lemma 42 Assuming 11, 12 and 13, there exists C > 0 such that for any y 0, u n : V u f n u + y C1 + ye y roof For all k n, we write Z n k y = u =k 1 {V u fnk+y}1 {V uj<f nj+y,j<k} the number of individuals crossing for the rst time the curve j f n j at time k By Lemma 23, we have E Z n k y =E [ e S k 1 {Sk f nk+y}1 {Sj f nj+y,j<k} ] e fnk y S k f n k + y, S j f n j + y, j < k We condition this probability with respect to the last step S k S k 1 to obtain S k f n k + y, S j f n j + y, j < k = Eφ k 1 S k S k 1, with φ k x = Sk f n k + y x, S j f n j + y, j k Applying Lemma 31, there exists C > 0 such that for all k n and x R, φ k x C1 {x 0} 1+y1+x 2 k +1 3/2 Thus, u n : V u f n u + y n Z n k y 1 n EZ n k y C1 + ye y n n+1 3/2 ES k S k k+1 3/2 n k+1 3/2 As we have, for any n N, As a consequence, we obtain u n : V u f n u + y C1 + ye y n/2 2 3/2 k 3/2 + n k=n/2 2 3/2 n k+1 3/2 C1 + ye y This lemma directly implies the upper bound in Theorem 41 roof of the upper bound in Theorem 41 As f n n = 3 2 logn log n 2, we observe that M n m n + y u n : V u f n u + y 2 We apply Lemma 42 to conclude the proof To obtain a lower bound for M n m n + y, we apply the Cauchy-Schwarz inequality: if X is an integer-valued non-negative random variable, we have X 1 EX2 EX 2 A good control on the second moment of the number of individuals staying below f n until time n is obtained by considering only the individuals that do not make too many children too high To quantify this property, we set ξu = log v Ωu 1 + V u V v + e V u V v for any u T For n N and y, z 0, we dene the sets A n y = { u n : V u j f n j + y, j u }, A n y = { u = n : u A n y, V u f n n + y H} B n y, z = { u n : ξu j z + fnj+y V uj 2, j u } We write Y n y, z = A n y B n y, z, and take interest in Y n y, z = #Y n y, z, the number of individuals that made an excursion below f n, while making not too many children We also set Y 2 n,k y, z = u = v =n 1 { u v =k} 1 {u Yny,z}1 {v Yny,z}, and we compute the mean of this quantity

9 4 Bounding the tail of the maximal displacement 100 Lemma 43 Assuming 11, 12 and 13, there exists C > 0 such that for y, z 0 and 0 k n, EY n y, z C1 + ye y and E Y 2 n,k y, z n+1 C 3/2 1 + ye z y k+1 3/2 n k+1 3/2 In particular, E Y n y, z 2 C1 + ye z y roof We rst apply Lemma 22, we have E Y n y, z =Ê e V wn 1 {wn Y ny,z} Ê e V wn 1 {wn A ny} n + 1 3/2 e H y V wn f n n + y H, V w j f n j + y, j n C1 + ye y by Lemma 31 To bound the mean of Y 2 n,k, applying Lemma 24 we can write E Y 2 n,k y, z = e Ê V wn 1 {wn Y ny,z}z k,n, 41 where Z k,n = u Ω k Ê v =n,v u 1 {v Y ny,z} G n We observe that by denition of, for any k < n and u Ω k, the subtree T u, V u is a branching random walk independent of G n Therefore, setting C y n,k = { v = n k 1 : V v fn n + y H, V v j f n j + k y, j n k 1 we have Ê v =n,v u 1 {v Yny,z} G n E V u Thus, applying Lemma 22, we have E V u #C y n,k e V u Ê V u e V wn k 1 1 {wn k 1 Cn,k} y n + 1 3/2 e V u y V u w n k 1 C y n,k #C y n,k } as w n B n y, z Conditioning on the value taken at time j by V w j f n j y, we have Ê φf n k + y V w k /21 {wn A ny} + C φ h+1 2 wn A n y, f n k + y V w k [h, h + 1] C C h=0 + h/2 1+y1+h 1+h h + 2e k+1 3/2 n k+1 3/2 h=0 1+y, k+1 3/2 n k+1 3/2 applying Lemma 31 on the intervals [0, j] and [j, n] We conclude that E Y 2 n,k y, z n+1 C 3/2 1 + ye z y k+1 3/2 n k+1 3/2 Moreover, using again Lemma 24, we have E Y n y, z 2 n 1 = EY n y, z + E Y 2 n,k y, z C1 + ye z y We now bound from below the mean of Y n y, z Lemma 44 Assuming 11, 12 and 14, there exist c > 0 and Z 1 such that for all y [0, n 1/2 ], z Z, and n N, EY n y, z c1 + ye y roof Let n N, y [0, n 1/2 ] and z 1 By Lemma 22, we have [ ] EY n y, z Ê e V wn 1 {wn Y ny,z} n 3/2 e y wn Y n y, z To bound this probability, we observe rst that w n A n y B n y, z = w n A n y w n A n y B n y, z c, and w n A n y c1 + yn 3/2 by Lemma 32 Moreover, by Lemma 33, there exists C > 0 such that C n+13/2 n k+1 3/2 1 + f n k y V u + e V u y, by Lemma 31 again As x x + is 1-Lipschitz, we obtain Z k,n C n+13/2 1+f nk+y V w k + n k+1 3/2 e V w k y+ξw k w n A n y B n y, z c C 1 + y n 3/2 ξw 0 z + Êξw 0 z + 2 By 14, we have Êξw < +, therefore by dominated convergence, we have Ce ξwk φf n k + y V w k, lim sup n 3/2 where φ : x 1 + x + e x z + For k < n, 41 yields n N,y y w n A n y B n z c = 0, E Y 2 n,k y, z thus for all z 1 large enough we have w CÊ e ξw k V w n n A n y B n z c1 + yn 3/2 φf n k + y V w k 1 {wn Y ny,z} Cn + 1 3/2 e z y Ê φ fnk+y V w These two lemmas are used to complete the proof k 2 1 {wn A ny}, of Theorem 41

10 5 Concentration estimates for the maximal displacement 101 Lower bound in Theorem 41 By the Cauchy-Schwarz inequality, we have Y n y, z 1 EY ny, z 2 EY n y, z 2 Using Lemmas 43 and 44, there exists z 1 such that Y n y, z 1 c1 + ye y 2 C1 + ye z y c1 + ye y As a consequence, we conclude M n m n + y G n y, z c1 + ye y Lemma 43 can also be used to prove that with high probability, individuals in the branching random walk above m n are either close relatives or their most recent common ancestor is not far from the root of the process Theorem 45 Under the assumptions 11, 12, 13 and 14, we have u, v = n : lim lim sup V u m n, V v m n = 0 R + n + and u v [R, n R] roof Let n 1 We introduce for any y, z 0 the random variable X n y, z = 1 {V u mn}1 {u Any B ny,z c } By Lemma 22, we have EX n y, z =Ê e V wn 1 {V wn m n}1 {wn A ny B ny,z c } n + 1 3/2 V wn m n, w n A n y B n y, z c C1 + y 3 ξw0 z + Ê ξw 0 z 2 + using Lemma 33 We observe that for any y, z, R 0: u, v = n : V u, V v mn, and u v [R, n R] u < n : V u f n u + y + X n y, z 0 + k [R, n R] : Y 2 n,k y, z 0 n R C1 + ye y + EX n y, z + E Y 2 n,k y, z, k=r using Lemma 42 We apply Lemma 43, we obtain u, v = n : lim sup V u m n, V v m n n + and u v [R, n R] C 1 + ye y y 3 χz yez y R 1/2 where χz = ξw 0 z + Ê ξw 0 z 2 + satis- es lim z + χz = 0 by 14 We set z = log log R and y = log χz, letting R + concludes the proof 5 Concentration estimates for the maximal displacement We prove in this section that Theorem 41 implies Theorem 11 To do so, we use the fact that on the survival event S, the size of the population in the process grows at exponential rate, as in a Galton-Watson process More precisely, we use the following result, which is a straightforward consequence of [26, Lemma 29] Lemma 51 Let T, V be a branching random walk satisfying 11 and 14 There exists a > 0 and ρ > 1 such that as on S, for all n 1 large enough, # { u = n : V u na} ρ n roof of Theorem 11 We recall that m n = 3 2 log n, we prove that lim y + lim sup M n m n y, S = 0 n + Using the upper bound of Theorem 41, we have lim sup M n m n + y C1 + ye y 0 n + y + To complete the proof, we have to strengthen the lower bound of Theorem 41, which states there exists c > 0 such that n N, y [0, n 1/2 ], M n m n +y c1+ye y To do so, we observe that by Lemma 51, there exists a > 0 and ρ > 1 such that as for any k large enough, there are at least ρ k individuals above ka On this event, each individual starts an independent branching random walk, and the largest of their maximal displacement at time n k is smaller than M n Therefore for any y 0 and k 1, we have M n m n y yielding #{ u = k : V u ka} < ρ k + 1 M n k m n y + ka ρk, lim sup M n m n y n + #{ u = k : V u ka} < ρ k + 1 cka y + e ka ρk We conclude the proof choosing y = ka 1 and letting k + A roof of the random walk estimates A1 roof of Lemmas 31 and 32 We recall that the proof of these lemmas is based on the decomposition of a random walk excursion into three parts, each of which being bounded using the estimates obtained in Section 3

11 A roof of the random walk estimates 102 roof of Lemma 31 We denote by p = n/3 Applying the Markov property at time p, we have y T n f n n [z h, z], T j f n j, j n y T j Aj α, j p Tn f sup n n [z h, z], x x f np T j f n p + j, j n p We set T k = T n p T n p k, which is a random walk with the same law as T Note that for any x R; Tn p f n n [z h, z], x T j f n p + j, j n p Tn p f n n [x + z h, x + z], T j z + f n n f n n j, j n p z Tn p f n n [x h, x], T j Aj α, j n p A1 We apply again the Markov property at time p, we have Tn p f n n [x h, x], z T j Aj α, j n p z T j Aj α, j p sup T n 2p [x, x + h] x R Consequently, using 32 and 34, we obtain y T n f n n [z, z + h], T j f n j, j n 1 + y n1/2 1 + h n 1/2 1 + z n 1/2 n 3/2 roof of Lemma 32 The proof is very similar to the previous one Let p = n/3, we apply the Markov property at time p, to obtain, for any ɛ > 0 and n 1 large enough, y T n f n n + H, T j f n j, j n y T p [p 1/2, 2p 1/2 ], T j ɛ, j p Tn f inf n n + H, x x [p 1/2,2p 1/2 ] T j f n p + j, j n p Setting again T k = T n p T n p k, for any x R we have Tn p f n n + H, x T j f n p + j, j n p Tn p f n n [x, x + H], T j f n n f n n j, j n p Tn p f n n [x h, x], T j f n n f n n j, j n p We apply again the Markov property at time p, for x [p 1/2, 2p 1/2 ] we have Tn p f n n [x h, x], T j Aj α, j n p Tp [p 1/2, 2p 1/2 ], T j f n n f n n j, j p inf T j 0, T p f n n [x H, x] z [p 1/2,2p 1/2 ] Thus, we apply 33 and 35 to conclude the proof A2 roof of Lemma 33 We now extend Lemma 31 to random walks enriched by additional random variables The idea behind the proof is that a random walk excursion is typically at distance On 1/2 from the boundary Therefore, as soon as Eξ+ 2 is nite, we expect similar upper bounds for the branching random walk To prove Lemma 33, we rst prove the following result, that is a direct consequence of [3, Lemma C1] Lemma A1 With the notation of Lemma 31, there exists C > 0 such that for any y 0, k n : Tk f n k y + ξ k, T j f n j y, j n C 1 + y ξ1 0 + E ξ n 1/ roof Let n N and y 0 For k < n we compute π k = y T k f n k + ξ k, T j f n j, j n = E y 1{Tk f nk+ξ k,t j f nj,j k+1}φ k T k+1, with φ k x = x T j f n k j, j n k 1, by Markov property Thus, by 34 we have π k CE y 1 {Tk f nk+ξ k,t 1+T k f nk j f nj,j k}, n k+1 1/2 Then, conditioning on X k, ξ k and applying Lemma 31, we have π k 1+y1+ξk CE 1 {ξk 0} +ξ X k 2 + k k + k+1 3/2 n k+1 1/2 We denote by X, ξ a random vector with the same law as X k, ξ k We have k n : Tk f n k y + ξ k, T j f n j y, j n C1 + ye Observing that n/2 we conclude that n 1 1 {ξ 0} 1 + ξ + n 1 π k 1+ξ X 2 + k k 3/2 n k+1 1/2 1+ξ X 2 + k k 3/2 C1 + ξ X +, k n : Tk f n k y + ξ k, T j f n j y, j n C 1+y n 1/2 ξ1 0 + E ξ roof of Lemma 33 We use Lemma A1 to prove this result Recalling that τ = inf{k < n : T k f n k + ξ k+1 } = inf{k < n : T k+1 f n k + ξ k+1 + X k+1 },

12 A roof of the random walk estimates 103 we have y T n f n n + H, τ < n, T j f n j, j n y T n f n n + H, τ n/2, T j f n j, j n + y T n f n n + H, τ n/2, n], T j f n j, j n To bound the rst term, we apply the Markov property at time n/2, we have y T n f n n + H, τ n/2, T j f n j, j n y τ < n/2, T j f n j, j n/2 Tn f sup n n + H, z z R T j f n n/2 + j, j n/2 C 1 + y ξ1 + X n 3/ E ξ 1 + X 1 2 +, by Lemma A1 and A1 We denote by T j = T n T n j and ξ j = ξ n j To bound the second term, we observe that y T n f n n + H, τ n/2, n], T j f n j, j n T n f n n + y [0, H], T j f n n f n n j, j n, k < n/2 : T k f n n f n n k ξ k 1 T n f n n + y [0, H], T j f n n f n n j, j n, k < n/2 : T k f n n f n n k ξ k 1 C 1 + y ξ1 + X n 3/ E ξ 1 + X , using the previous inequalities, which concludes the proof References [1] L Addario-Berry and B Reed Minima in branching random walks Ann robab, 373: , 2009 [2] L Addario-Berry and B A Reed Ballot theorems, old and new In Horizons of combinatorics, volume 17 of Bolyai Soc Math Stud, pages 935 Springer, Berlin, 2008 [3] E Aïdékon Convergence in law of the minimum of a branching random walk Ann robab, 413A: , 2013 [4] E Aïdékon, J Berestycki, É Brunet, and Z Shi Branching Brownian motion seen from its tip robab Theory Related Fields, :405451, 2013 [5] E Aïdékon and Z Shi Weak convergence for the minimal position in a branching random walk: a simple proof eriod Math Hungar, 611-2:43 54, 2010 [6] L- Arguin Extrema of log-correlated random variables: rinciples and Examples 2016 arxiv: [7] L- Arguin, A Bovier, and N Kistler Genealogy of extremal particles of branching Brownian motion Comm ure Appl Math, 6412: , 2011 [8] K B Athreya and E Ney Branching processes Dover ublications, Inc, Mineola, NY, 2004 Reprint of the 1972 original [Springer, New York; MR ] [9] J Bérard and J-B Gouéré Survival probability of the branching random walk killed below a linear boundary Electron J robab, 16:no 14, , 2011 [10] J D Biggins The rst- and last-birth problems for a multitype age-dependent branching process Advances in Appl robability, 83: , 1976 [11] J D Biggins and A E Kyprianou Fixed points of the smoothing transform: the boundary case Electron J robab, 10:no 17, , 2005 [12] M Bramson and O Zeitouni Tightness of the recentered maximum of the two-dimensional discrete Gaussian free eld Comm ure Appl Math, 651:120, 2012 [13] M D Bramson Maximal displacement of branching Brownian motion Comm ure Appl Math, 315:531581, 1978 [14] F Caravenna and L Chaumont Invariance principles for random walks conditioned to stay positive Ann Inst Henri oincaré robab Stat, 441:170190, 2008 [15] X Chen A necessary and sucient condition for the nontrivial limit of the derivative martingale in a branching random walk Adv in Appl robab, 473:741760, 2015 [16] X Chen Scaling limit of the path leading to the leftmost particle in a branching random walk Theory robab Appl, 594:567589, 2015 [17] J Ding and O Zeitouni Extreme values for twodimensional discrete Gaussian free eld Ann robab, 424: , 2014 [18] J M Hammersley ostulates for subadditive processes Ann robability, 2:652680, 1974 [19] Y Hu How big is the minimum of a branching random walk? Ann Inst Henri oincaré robab Stat, 521:233260, 2016 [20] Y Hu and Z Shi Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees Ann robab, 372:742789, 2009 [21] J- Kahane and J eyrière Sur certaines martingales de Benoit Mandelbrot Advances in Math, 222:131145, 1976

13 A roof of the random walk estimates 104 [22] J F C Kingman The rst birth problem for an age-dependent branching process Ann robability, 35:790801, 1975 [23] M V Kozlov The asymptotic behavior of the probability of non-extinction of critical branching processes in a random environment Teor Verojatnost i rimenen, 214:813825, 1976 [24] R Lyons A simple path to Biggins' martingale convergence for branching random walk In Classical and modern branching processes Minneapolis, MN, 1994, volume 84 of IMA Vol Math Appl, pages Springer, New York, 1997 [25] R Lyons, R emantle, and Y eres Conceptual proofs of L log L criteria for mean behavior of branching processes Ann robab, 233: , 1995 [26] B Mallein Branching random walk with selection at critical rate 2015 arxiv: [27] B Mallein Maximal displacement in a branching random walk through interfaces Electron J robab, 20:no 68, 40, 2015 [28] B Mallein Genealogy of the extremal process of the branching random walk 2016 arxiv: [29] Bastien Mallein Maximal displacement in a branching random walk through interfaces Electron J robab, 20:no 68, 40, 2015 [30] J eyrière Turbulence et dimension de Hausdor C R Acad Sci aris Sér A, 278:567569, 1974 [31] M I Roberts A simple path to asymptotics for the frontier of a branching Brownian motion Ann robab, 415: , 2013 [32] C Stone A local limit theorem for nonlattice multi-dimensional distribution functions Ann Math Statist, 36:546551, 1965 Bastien Mallein address: bastienmallein@mathuzhch Universität Zürich, Switzerland