Maximal displacement in a branching random walk through interfaces

In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations.


Introduction
We study a branching random walk on R.This process starts with one particle located at the origin at time 0.Then, at each time k, every particle currently in the process dies, leaving a certain number of children, which are positioned around the position of their parent, according to independent versions of a point process, whose law may depend on the generation of the parent.It is supposed that each particle makes at least one child, so the process survives almost surely.
When the law of the point process (which we also refer to as branching mechanism) does not depend on the generation of the individual, and satisfies some integrability conditions, the asymptotic of the maximal displacement is fully known.Kingman [11] and then Biggins [6] proved this maximal value grows at linear speed.Later, Hu and Shi [9] proved the existence of a logarithmic correction in probability, with almost sure fluctuations ; and Addario-Berry and Reed [2] showed the tightness of this quantity around this correction.Finally, Aidékon [3] proved the convergence in law of the recentered value.
Recently, Fang and Zeitouni [16] introduced a new model of branching random walk through an interface.For all n ∈ N, they studied branching random walks with binary Gaussian branching mechanism, first with variance σ 2  1 until the generation n 2 , then with branching mechanism σ 2 2 between the generation n 2 and the generation n.They observed that the behaviour of such branching random walks depends on the chosen order for the variances, and that the logarithmic correction exhibits a phase transition.It is possible to extend their results to more general time-inhomogeneous environments.Let t ∈ (0, 1), and L and L two branching mechanisms.For all n ∈ N, the model is defined as follows: during the first p = ⌊nt⌋ generations, individuals reproduce according to independent copies of L. During the next n − p generations, they follow L. We find three possible regimes for this model.In the first case, which we call "slow regime", the rightmost individual at time n is the descendent of one of the rightmost individuals at time p and the resulting log-correction is the sum of two log-corrections of homogeneous branching random walks, one for each environment.
In the second case, the "mean regime", the behaviour is similar to the one of a homogeneous branching random walk, and the interface has no effect.
Finally, in the third case, the "fast regime", an anomalous spreading is observed.The first order is then larger than in the other two cases, and the log-correction is much higher too.In this case, the maximum of the branching random walk behaves as the maximum of a large number of independent random walks: the branching structure is forgotten.

Description of the model and notation
Let T (n) be a tree of height n; for all x ∈ T (n)   • V (n) (x) ∈ R is the position of the individual x; • |x| is the generation of x; • x k is the ancestor at generation k of x; the displacement of the children of x with respect to its position; • for all y ∈ T (n) , x ∧ y is the most recent common ancestor of x and y.
We also write M n = max |x|=n V (n) (x) the maximal displacement in a branching random walk.Let (L k , k ≤ n) be a set of branching mechanisms.A branching random walk (V (n) (x), x ∈ T (n) ) in time-inhomogeneous environment (which we refer to as BRWtie), is defined as follows: {L (x) , x ∈ T (n) } is a set of independent branching mechanisms, and L (x) has the same law as L |x| .The BRWtie is said to have branching mechanism L k at generation k.
In this article, the studied model is the branching random walk through an interface (BRWi), whose law we now define.Let t be a fixed number in (0, 1), for all n ∈ N, set p n = ⌊nt⌋, writing p if n is clearly defined in the context.Let L and L be two branching mechanisms ; the BRWi is a BRWtie with branching mechanism L during the first p generations, and L during the next n − p ones.The interface is said to be at position t.
In the sequel, it is assumed that for all branching mechanism L P(L = ∅) = 0 and In particular, under these assumptions, the population never gets extinct and grows exponentially fast.It is also assumed that the branching mechanisms are non-lattice, i.e.
We now introduce some functions linked to the branching mechanism L. For all θ > 0, set its Laplace transform and for all a ∈ R its Cramér transform.We later write κ the Laplace transform of L and κ * k the Cramér transform of L k without necessary recalling the notation.
In the context of branching random walk, we write which is the speed of a branching random walk with branching mechanism L (see Biggins [7]).Similarly, v is the speed of the branching random walk with branching mechanism L.
Let L and L be two given branching mechanisms.We make the assumption that there exist θ * > 0 and θ * > 0 such that Moreover, set which we assume to be finite.We finally add the following second order integrability condition, which is not necessary, but simplifies the proofs: Throughout the paper, the notation O P (1) stands for stochastic boundedness: let X n be a random sequence, then X n = O P (1) if for all ǫ > 0, there exists A > 0 such that P(|X n | ≥ A) ≤ ǫ for all n ∈ N.
Remark 1.1.To simplify notation, we denote in the sequel c, C two generic positive constants which may change from line to line, or even within line.Those constants are respectively small and large enough, and only depend on the branching mechanisms and the constant t > 0. .Under these assumptions, for a time-homogeneous branching random walk, the following theorem holds.Let (U (x), x ∈ T) be a branching random walk with branching mechanism L, and Theorem 1.2 (Addario-Berry and Reed, [2]).The position of the rightmost individual at time n in U is given by For BRWi, same kind of theorems holds.First, for the "slow regime", in which two logarithmic corrections adds.
For the "mean regime", in which the behaviour is no different for a timehomogeneous branching random walk, the log-correction is significantly smaller in absolute value.
In the case of the "fast regime", as the first order is modified, we need an extra notation.We suppose there exists θ > 0 such that and we write a = κ ′ (θ), b = κ ′ (θ).Observe that in this case Theorem 1.5 (Fast regime).If θ * > θ * , then the position of the rightmost individual is given by These three theorems give a full description of the possible behaviours of BRWi.This behaviour shows a phase transition for the log-correction.While the speed varies continuously with respect to the two branching mechanisms, there is a discontinuity in the second order, when θ * is getting close to θ * .

Heuristics
Biggins [7] proved that in the case of a time-homogeneous branching random walk with branching mechanism L, for all a ∈ R, e −nκ * (a) represents either an approximation of the number of individuals in a neighbourhood of na at time n, or the probability that there exists one such individual, depending on whether κ * (a) > 0 or not.
We now consider a BRWi with branching mechanisms L and L. For all a ≤ v, there are at time p about e −pκ * (a) individuals close to pa.Each of these individuals begins an independent time-homogeneous branching random walk with branching mechanism L. The probability to have an individual to the right of (n − p)b at generation n − p for one of those branching random walks is of order e −(n−p) κ * (b) .As a consequence there are individuals in a neighbourhood of pa + (n − p)b at time n with an ancestor at time p close to pa if tκ * (a) + (1 − t) κ * (b) < 0, and a < v.
We then expect that the first order for the maximal displacement in the BRWi is given by v = max with To describe the logarithmic correction in M n , it is enough to study the path which leads an individual to the rightmost position.Set (a, b) ∈ A attaining the maximum in (11).The decomposition into three regimes is made in the following way: if κ * (a) < 0 (or equivalently a < v), then v > tv + (1 − t) v and we are in the fast regime.In this case, the path followed by the individual which realise the maximal displacement at time n stay away from the boundary of the process most of the time, and the logarithmic correction is small.
If κ * (a) = 0, we are either in the mean or the slow regime.Set If v T = v, we are in the mean case.The restriction κ * (a) ≤ 0 is not important in the optimization problem (11).As a consequence, the path leading to the rightmost position at time n stay close to the boundary of the process at all time, as in the homogeneous case, so the logarithmic correction are the same.
In the slow regime, v T > v, therefore, the restriction κ * (a) ≤ 0 is crucial in (11).For the individuals, this means that being close to the boundary at time p gives an important advantage to the descendants.The rightmost individual at generation n is a descendant of one of the rightmost individuals at generation p.As a consequence, the logarithmic correction is the sum of the two timehomogeneous corrections obtained in the two phases of the process.
The next section presents some preliminary lemmas; Section 3 contains the proof of Theorem 1.3, Section 4 the proof of Theorem 1.4 and Section 5 the proof of Theorem 1.5.An extension of these results to multiple interfaces is discussed in the final section.

Some computations in the branching random walk
In this section, we make some computations that are valid for a general tieBRW with branching mechanism L k at generation k written (V (x), x ∈ T).We recall that κ k (θ) is the Laplace transform of L k .

The many-to-one lemma
Let θ ∈ R + ; we suppose that for all k ≤ n, κ k (θ) < +∞.We denote by (X k , k ≥ 0) a sequence of independent random variables such that for all measurable nonnegative function f For all n ≥ 0, set S n = n k=1 X k .The many-to-one lemma links moments of some functionals of the branching random walk and of (S n , n ≥ 0).This result has been stated in many forms, going back at least to the article of Kahane and Peyrière [10].
Lemma 2.1 (Many-to-one lemma).Let n ∈ N and f : R n → R be a measurable non-negative function, we have Proof.This result can be proved by induction.For n = 1, the statement is exactly the definition of the law of S 1 = X 1 .We assume that the statement is true for some n ∈ N. Let f : R n+1 → R be a measurable non-negative function.Using the branching structure, we get where Finally, using the induction hypothesis, we obtain which ends the proof.
Remark 2.2.Under nice integrability conditions, As a consequence, writing

Bounds for the probability of the existence of certain paths in the branching random walk
In this section, we apply the many-to-one lemma to compute the moments of the number of individuals which remain in some specific paths.These moments allow to bound the probability that M n is above a certain level.For all x ∈ T, we denote by H(x) = (V (x j ), j ≤ |x|) the path followed by x.For all n ∈ N and k ≤ n, let φ n (k) ∈ R be the deterministic path to follow.Let y ≥ 0 and θ > 0, we suppose that for all k ≤ n, κ k (θ) < 0. We write S the time-inhomogeneous random walk obtained using the many-to-one lemma with the parameter θ, and In a first step, we compute the probability there exists an individual which crosses at some time the upper frontier (φ n (k) + y, k ≤ n).To do so, denote the set of paths which can be followed by individuals crossing for the first time at time k the upper frontier, and the probability that S belongs to this set.Proof.By the Markov inequality and the many-to-one lemma This lemma can be used to prove the upper bound in Theorem 1.2 Corollary 2.4.Let (U (x), x ∈ T) be a branching random walk with branching mechanism L such that ( 6), (7) and (2) are satisfied.We denote There exists C > 0 such that for all y > 0 Proof.Observing that it is enough to only prove the second inequality.Applying Lemma 2.3 to the branching random walk U and the function φ n , there exists C > 0 such that where and ξ n,k (y) = P(S k > φ n (k) + y, S j ≤ φ n (j) + y, y < k), for S a classical random walk.By use of Corollary 2.9, proved in the next section, there exists (n + 1) < +∞, so there exists C > 0 such that In a second step, we bound from below the probability there exists an individual to the right of φ n (n) + y at time n.To do so, let the admissible interval sequence, in which we are looking for individuals which arrive in a neighbourhood of φ n (n) at time n.We write, for all n ∈ N and k ≤ n It is then assumed that there exists C = C(y) > 0 and c n,k such that for all We also suppose that there exists a constant Φ such that and we finally denote The following estimate holds: Lemma 2.5.For all y ≥ 0, there exists C > 0 such that for all n ∈ N we have This lemma can be used to obtain the lower bound for M hom n in Theorem 1.2, but we skip this proof, very similar to the one used for the study of the mean regime.
Proof.We write Y n (y) = |x|=n 1 {H(x)∈Ω φ n (y)} .Using the many-to-one lemme, we have We now bound from above the second moment of Y n (y) using the branching structure using the Markov property, where and U k,n is a branching random walk with the same law as the descendants of an individual at generation k in the branching random walk V , i.e. with branching mechanism L k+j at generation j.By use of the many-to-one lemma by assumption (12).Moreover, using assumption (13), there exists As a consequence, we obtain Using once again the many-to-one lemma, we obtain with ρ n,k the quantity defined in (14).Therefore, for any y ≥ 0, there exists C(y) > 0 such that Finally, by the Cauchy-Schwarz inequality, we have Lemmas 2.3 and 2.5 enable us to bound from above and from below the probability there are individuals in a neighbourhood of φ n (n), as soon as it is possible to obtain bounds for the probabilities ω, π and ξ.In the next part, we state some results which will allow us to obtain those, for a branching random walk through an interface.

Some random walks estimates
We first recall here some classical results for random walks, then state some extensions that applies to branching random walks through an interface.Let (X n , n ∈ N) be i.i.d.non-lattice random variables (i.e.there exists no (a, b) ∈ R 2 such that X 1 ∈ aZ + b a.s.) such that E(X 1 ) = 0 and 0 < Var(X 1 ) < +∞.
For all n ∈ N, denote T n = n j=1 X j .This first lemma, which is often called the ballot theorem, computes the probability for a random walk to stay above a constant.It is found in Kozlov [12], see also [1] for a review on such results.Lemma 2.6.There exist 0 < c < C such that for all y ∈ R + and n ∈ N, we have The next lemma is a consequence of the Stone local limit theorem [15], which bounds the probability for a random walk to end up in a window of finite size.Lemma 2.7.There exist 0 < c < C, and L > 0 such that for all h ≥ L and n ≥ 1 We now state an extension of Lemma 2.6, to estimate the probability for a random walk to stay above a barrier of the form −j α at any step j ≤ n, for some 0 < α < 1  2 .Lemma 2.8.Let α ∈ (0, 1  2 ), there exists C > 0 such that for all y > 0 and n ≥ 1, we have The proof of this result is postponed to Appendix A. We end this section with a bound for the probability for a random walk to make an excursion over some given line.A corollary, which can be found under a modified form in [5], Annex A, compute the probability the random walk makes a bridge above some barrier.
Corollary 2.9.Let X, Y, Z be three independent random variables such that We denote (X k ), (Y k ) and (Z k ) three sequences of i.i.d.random variables with respectively the same law as X, Y and Z.For all p, q, r ∈ N 3 , we denote Z l else .
There exist c, C > 0 such that for all y, h > 0, and p, q, r ∈ and moreover Proof.These bounds are obtained by dividing the event into three parts.We begin with the upper bound.First, using the Markov property at time p, we have where By use of Lemma 2.6, We write W j = W n−p − W n−p−j the time-reversal random walk, using the Markov property at time r, and then Lemmas 2.8 and 2.7.
To obtain the second upper bound, it is enough to condition with respect to the value of The lower bound is obtained using Lemma A.2 in [5], and similar arguments as above.

Slow regime in the branching random walk through an interface : proof of Theorem 1.3
Recall that (V (n) (x), x ∈ T (n) ) is a BRWi, with branching mechanism L during the p = ⌊nt⌋ first steps, and L in the n − p next ones.In this section, it is assumed that θ * < θ * , the characterization of the slow regime.We denote Theorem 1.3 can then be proved using the estimates obtained for a timehomogeneous branching random walk.
Proof of Theorem 1.3.Let ǫ > 0. We begin with a proof of the lower bound.The first p steps of the BRWi are the same steps as those of a time-homogeneous branching random walk, with branching mechanism L. Consequently, using the Theorem 1.2, there exists y 1 > 0 such that In the same way, if there is an individual x 0 to the right of vp − 3 2θ * log p − y 1 at generation p, its descendants reproduce using a branching mechanism L. Using again Theorem 1.2, there exists y 2 > 0 such that with probability greater than 1 − ǫ 2 a descendant of x 0 at generation n is to the right of m n − y 1 − y 2 .
To conclude, there exists y = y 1 + y 2 such that In a second step, we study the upper bound.We denote, for all k ≤ p Using Corollary 2.4, there exits C > 0 such that so for all y > 0 large enough, with high probability, individuals do not hit this barrier.We compute the number of individuals which remain below this boundary and arrive in a neighbourhood of φ n (p) + y at generation p. Set By the many-to-one lemma, we have for all thanks to the upper bound in Corollary 2.9.
Let ( U (x), x ∈ T) be a time-homogeneous branching random walk with branching mechanism L. Using once again Corollary 2.4, we observe there exists K > 0 such that for all y > 0 We denote by M n−p a random variable with the same law as max |x|=n−p U (x), and b n = v(n − p) − 3 2 θ * log(n − p).We finally bound from above the probability that there is an individual at time n to the right of m n + y by looking at the position of its ancestor at time p.We have Using these two bounds, we finally deduce

Behaviour in the mean regime : proof of Theorem 1.4
In this section, we consider a BRWi verifying θ * = θ * =: θ, so we are in the mean regime.As a consequence, in this case Set where we denote k ∧ p = min(k, p) and (k − p) + = max(k − p, 0).Using lemmas in Section 2, we prove that with high probability no individual crosses that boundary.

Lemma 4.1.
There exists C > 0 such that for all y ∈ R and n ∈ N Proof.First, observe that Then, using Lemma 2.3 and by use of Corollary 2.9, we obtain ≤ C(1 + y)e −θy .
WE will now prove that with positive probability, there exists an individual which remain below the curve φ n (k) + y at time k ≤ n and arrives at time n above m n .

Lemma 4.2. There exists
Proof.We use Lemma 2.5 with functions b n (k) = 0 and There exists C > 0 and y > 0 such that In this formula, which can be bounded using Corollary 2.9 .
Moreover, using this same corollary, we define and observe that for all u ≥ 0 Recall also that ρ n,k = E e θS k −(k∧p)κ(θ)−(k−p) + κ(θ) 1 {Sj≤φn(j)+y,j≤k} , by decomposition of the expectation in two parts, whether or not S k ≥ kv − 3 2θ log k.In the same way, for all k > p .
As a consequence which is bounded uniformly in n.
These two results give an upper and a lower bound for the position of the rightmost individual at time n.We now prove Theorem 1.4.
For the lower bound, we denote by (U (x), x ∈ T) and ( U (x), x ∈ T) two independent branching random walks with branching mechanism respectively L and L. Lemma 4.2 proves that there exists c > 0 such that for all n ≥ 0 and thanks to (1) there exists h ∈ N and y 1 ∈ R such that Let n ∈ N, we decompose the branching random walk V (n) as follows : • it begins with the h first steps of the branching random walk U , which gives, with probability at least (1 − ǫ 3 ), N individuals to the right of y 1 ; • at least N independent branching random walks with the same law as V (n−⌈ h t ⌉) all starting to the right of y, with probability at least 1 − ǫ 3 at least one of them makes a child to the right of m n−⌈ h t ⌉ + y 1 , • at most k steps of one branching random walk U starting from m n−⌈ h t ⌉ + y 1 , which gives, with probability at least 1− ǫ 3 a child at position As a consequence, there exists y > 0 such that which ends the proof.
As usual, m n is defined by and we begin by proving there is no individual to the right of m n at time n with high probability.
Lemma 5.1.There exists C > 0 such that for all y > 0 we have This result is the same as the one we could expect for large deviations in a simple random walk.As a consequence, there is no need to keep track of the branching structure here.
Proof.By use of the many-to-one lemma, we have and using Stone's local limit theorem we obtain that As a consequence, using the Markov inequality, which yields and this conclude the proof.
We now look for a lower bound for the position of the rightmost individual.To do so, we prove that there are individuals which stay, at any time k ≤ n in a neighbourhood of the path Lemma 5.2.There exists c > 0 such that Proof.Let A > 0, and Applying Lemma 2.5 with function φ n , b n (k) = −a n (k) = r n (k) yields that there exists y > 0 such that .
We recall that I n (j, y) = [φ n (j) − r n (j), φ n (j) + r n (j) + y], and By Lemma 2.7, we have Secondly, c k,n is defined in a way such that but using once again Lemma 2.7, we have In a third step, we compute and if k > p using Lemma 2.7 Using inequality (15), we obtain Adding up these inequalities, there exists c y > 0 such that , which ends the proof.
Those last two lemmas are enough to prove Theorem 1.5, using the same argument as in the proof of Theorem 1.4.
Proof of Theorem 1.5.Let ǫ > 0. Using Lemma 5.1, there exists y > 0 large enough such that for all n ∈ N P(M n ≥ m n + y) ≤ ǫ.
For the lower bound, we use exactly the same cutting argument as in the study of the mean regime.In a few steps, a large number of individuals are created, each of them having a descendent at generation n to the right of m n with positive probability.The few remaining steps do not change the order of the position.As a consequence, there exists y > 0 such that which ends the proof.

Concluding remarks
We have exhibited here a phase transition, already known to Zeitouni and Fang [16] in the case of binary Gaussian branching random walks.We have Using the strict convexity of κ * and κ * we notice that this decomposition is unique.We write θ = (κ * ) ′ (a) and θ = ( κ * ) ′ (b).Using Lagrange theory, we may notice that θ ≤ θ and θ < θ only if κ * (a) = 0.
With these notations, we resume the results in the following way This result is easily extended to a succession of interfaces.Let 0 We consider a branching random walk with K − 1 interfaces at positions (α i , i ≤ K).In other word, for all n ∈ N, V (n) is a branching random walk with branching mechanism L i at any generation k such that α i−1 ≤ k n < α i , for all i ≤ K. Writing Using once again Lagrange theory, this quantity is equal to and if the maxima are reached respectively at (a We denote by L the quantity 2φ p .
We finally have, under suitable integrability hypothesis Heuristically, the reason for this result is that the optimal path leading to the rightmost individual sticks to the frontier of the branching random walk whenever θ k increases (slow interface).As a consequence, the total logarithmic correction is the sum of the logarithmic corrections in each of the "eras" with constant optimal parameter.The logarithmic parameter in one of these eras depend on whether the optimal path stay close to the frontier at the beginning or at the end of the era (mean regime) or not (fast regime).
Finally, observe that the integrability conditions we asked in (13) are far from being optimal.These are used to compute the second moments, but we could have restricted our sets of individuals to those with few children.This was done in Aidékon [3], using the spinal decomposition.However, such truncation arguments require more technical details which we skipped.

A Proof of the Ballot extension
Proof of Lemma 2.8.Let y ≥ 0 and α < 1  2 .We compute the probability for a random walk to stay above a frontier −j α − y.To do so, we decompose this path into a sequence of hitting times on different levels.In the sequel, S k = min j≤k S j , and for all z ≥ 0, T z = inf{k ≥ 0 : S k ≤ −z}.
Let 0 < λ < 1 and k ∈ N.For all n ∈ N, by using the Markov property at time T (2k) α +y ∧ λn, we have As a consequence, choosing λ < 1  3 and k large enough such that K < 1, and then p ≥ − log n log K , this last inequality can be rewritten which ends the proof.