The Genealogy of Extremal Particles of Branching Brownian Motion Louis-Pierre Arguin, Anton ...

33 Pages·2010·0.38 MB·English

The Genealogy of Extremal Particles
of Branching Brownian Motion

Louis-Pierre Arguin, Anton Bovier, Nicola Kistler

no. 477

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs-
gemeinschaft getragenen Sonderforschungsbereichs 611 an der Universität
Bonn entstanden und als Manuskript vervielfältigt worden.
Bonn, September 2010
THE GENEALOGY OF EXTREMAL PARTICLES
OF BRANCHING BROWNIAN MOTION
L.-P. ARGUIN, A. BOVIER AND N. KISTLER
Abstract. BranchingBrownianMotiondescribesasystemofparticleswhichdiffusein
spaceandsplitinto offspringaccordingto acertainrandommechanism. Invirtue ofthe
groundbreakingworkbyM.Bramsononthe convergenceofsolutionsofthe Fisher-KPP
equation to traveling waves [8, 9], the law of the rightmost particle in the limit of large
timesisratherwellunderstood. Inthiswork,weaddressthefullstatisticsoftheextremal
particles (first-, second-, third- etc. largest). In particular, we prove that in the large
t limit, such particles descend with overwhelming probability from ancestors having
−
split either within a distance of order one from time 0, or within a distance of order one
from time t. The approach relies on characterizing, up to a certain level of precision,
the paths of the extremal particles. As a byproduct, a heuristic picture of Branching
BrownianMotion“atthe edge”emerges,whichshedslightonthestill unknownlimiting
extremal process.
Contents
1. Introduction 2
2. Main results 4
2.1. The enealogy of extremal particles 4
2.2. Localization of the paths of extremal particles 6
2.3. Towards the extremal process of Branching Brownian Motion 10
3. Some properties of Brownian bridge 11
4. The genealogy of extremal particles - proofs 13
5. Localization of paths 16
5.1. The upper envelope 17
5.2. The entropic envelope 19
5.3. The lower envelope 24
5.4. The “tube” 25
Appendix A. The left tail of the maximal displacement 26
References 29
Date: August 23, 2010.
2000 Mathematics Subject Classification. 60J80,60G70, 82B44.
Key words and phrases. Travelling Waves, Branching Brownian Motion, Extreme Value Theory and
Extremal Process, Entropic Repulsion.
L.-P. Arguin is supported by the NSF grant DMS-0604869 and by the Hausdorff Center for Mathe-
matics, Bonn. A. Bovier is partially supported through the German Research Council in the SFB 611.
N. Kistler is partially supported by the Deutsche Forschungsgemeinschaft, Contract No. DFG GZ BO
962/5-3. The kind hospitality of Eurandom, Eindhoven, the Hausdorff Center for Mathematics, Bonn,
and the Technion, Haifa, where part of this work has been carried out, are gratefully acknowledged.
1
GENEALOGY OF BRANCHING BROWNIAN PARTICLES 2
1. Introduction
Branching Brownian Motion (BBM for short) is a continuous-time Markov branching
process which is constructed as follows: a single particle performs a standard Brownian
Motion x issued on some probability space (Ω, ,P) with x(0) = 0, which it continues
for an exponential holding time T independent Fof x, with P[T > t] = e t. At time T,
−
the particle splits independently of x and T into k offspring with probability p , where
k
def
∞k=1pk = 1, ∞k=1kpk = 2, and K = kk(k−1)pk < ∞. These particle continue along
independent Brownian paths starting at x(T), and are subject to the same splitting rule,
P P P
with the effect that the resulting tree X contains, after an elapsed time t > 0, a random
number, n(t), of particles located at x (t),...,x (t). Clearly, En(t) = et. With
1 n(t)
u(t,x) d=ef P max x (t) x , (1.1)
k
1 k n(t) ≤
(cid:20) ≤ ≤ (cid:21)
a standard renewal argument, first observed by McKean [25], shows that u(t,x) solves the
Kolmogorov-Petrovsky-Piscounov or Fisher (F-KPP) equation,
1
u = u +u2 u,
t xx
2 −
(1.2)
1, if x 0,
u(0,x) = ≥
0, if x < 0.
(
The F-KPP equation is arguably one of the simplest p.d.e. that admits traveling wave
solutions. It is well known that there exists a unique solution satisfying
u t,m(t)+x ω(x), uniformly in x, as t , (1.3)
→ → ∞
with the centering(cid:0)term, the (cid:1)front of the wave, given by
3
m(t) = √2t logt, (1.4)
− 2√2
and ω(x) the unique (up to translation) distribution function which solves the o.d.e.
1
ω +√2ω +ω2 ω = 0. (1.5)
xx x
2 −
The leading order of the front has been established through purely analytic means by Kol-
mogorov, Petrovsky and Piscounov [22]. The far more delicate issue of the logarithmic
corrections has beensettled by Bramson [8], who exploited theprobabilistic interpretation
of the F-KPP equation in terms of BBM. Both F-KPP equation and BBM have attracted
a great deal of interest ever since: the reader is referred to the very partial list [1, 2, 17, 18]
for results of analytical flavor, and to [3, 16, 20, 21, 23, 26] for more probability-oriented
work.
The large time asymptotic of the maximal displacement of BBM is a paradigm for the
behavior of extrema of random fields, which is a classical problem in probability theory.
In the case of BBM, the correlations among particles are given in terms of the genealogical
GENEALOGY OF BRANCHING BROWNIAN PARTICLES 3
def
distance: for i,j Σ = 1,...,n(t) and conditionally upon the branching mechanism,
t
∈ { }
it holds
E[x (t) x (t)] = Q (i,j), (1.6)
i j t
·
def
where Q (i,j) = sup s t : x (s) = x (s) is the time to first branching. (In spin glass
t i j
{ ≤ }
terminology, Q (i,j) is the overlap between configuration i and j). Since Q can take any
t t
value in [0,t], one might expect that the maximal displacement of BBM lies considerably
lower than in the independent, identically distributed (i.i.d.) setting. Somewhat surpris-
ingly, this is not the case: to leading order, it coincides with that of e t independent
⌊ ⌋
centered Gaussians of variance t, in which case the correct centering is well known to be
1
r(t) d=ef √2t logt. (1.7)
− 2√2
We will refer henceforth to the Gaussian i.i.d. setting as the Random Energy Model of
Derrida, or REM for short [14]. The law of the maximum of a REM is known to belong
to the domain of attraction of the Gumbel distribution, G(x) d=ef exp e √2x , see e.g.
−
−
[24]. Although BBM does not belong to this universality class (it is(cid:16)straightf(cid:17)orward to
check that G does not solve (1.5)), the distribution of its maximum is still Gumbel-like.
Indeed, denoting by
n(t)
Z(t) d=ef √2t x (t) exp √2 √2t x (t) (1.8)
k k
− − −
Xk=1(cid:16) (cid:17) (cid:16) (cid:17)
the so-called derivative martingale, Lalley & Sellke [23] proved that Z(t) converges weakly
to a strictly positive random variable Z, and established the integral representation
ω(x) = E e −CZe−√2x (1.9)
h i
(for some C > 0). This exposes the law of the maximum of BBM as a random shift of
the Gumbel distribution.
It is also known that the limiting derivative martingale has infinite mean, E[Z] = + .
∞
This affects the asymptotics to the right to the extent that
1 ω(x) x e √2x, x + , (1.10)
−
− ∼ → ∞
with meaning that the ratio of the terms converges to a positive constant in the con-
∼
sidered limit (see e.g. Bramson [9] and Harris [20]). Tails of the form (1.10) have recently
started to appear in different fields, see e.g. the studies on Spin Glasses with logarithmic
correlated potentials [5, 11]; there is thus strong evidence for the existence of a new uni-
versality class different from the Gumbel (which has tail 1 G(x) e √2x for x + ).
−
− ∼ → ∞
Despite (1.9) and (1.10), the limiting law ω still remains a rather mysterious object. To
our knowledge the behavior of the left tail of the maximal displacement is not discussed
in the literature; we provide its asymptotics in an appendix, by means of an argument
that we learned from Camillo De Lellis [12]. However, more information on ω would be
definitely desirable.
Contrary to the statistics of the maximal displacement, nothing is known on a rigorous
level about the full statistics of the extremal configurations (first-, second-, third-, etc.
GENEALOGY OF BRANCHING BROWNIAN PARTICLES 4
largest) in BBM. Such a statistics is completely encoded in the extremal process, which is
the point process associated to the collection of points shifted by the expectation of their
maximum (lower order included), namely
n(t)
def def
= δ , x (t) = x (t) m(t). (1.11)
Nt xi(t) i i −
i=1
X
In fact, it is not even known whether converges to a well defined limit at all, although
t
N
we prove her (see Corollary 2.3 below) that the collection of laws is tight. On a non-
rigorous level, the situation is only slightly better, see Section 2.3 below for a discussion
of some recent work by Brunet & Derrida [13].
The extremal process oftheREM iswell known tobegiveninthelimit of largetimes by
a Poisson point process with exponential density e √2x d x on R. Given the Gumbel-like
−
behavior (1.9), one may (perhaps) be tempted to conjecture that the limiting extremal
process of BBM is a randomly shifted Poisson point process, but the work by Brunet &
Derrida mentioned above provides strong evidence against this: BBM seems to belong, as
far its full statistics of extremal particles is concerned, to a new universality class, which
is expected to describe the extrema of models “at criticalitgy”, such as the 2 dim Gauss-
−
ian free field [4], directed polymers on Cayley trees [16], or spin glasses with logarithmic
potentials [5, 11].
In this work we obtain some first rigorous results on the statistics of the extremal
particles of BBM. Although we cannot yet characterize the limiting extremal process, a
clear picture of BBM at the edge emerges from our analysis, which we believe will prove
useful for further studies.
2. Main results
2.1. The enealogy of extremal particles. The major difficulty in the analysis of the
BBM stems from the delicate dependencies among particles, which are due to the contin-
uous branching. A first, natural step towards the extremal process is to study it at the
level of the Gibbs measure, which is less sensitive to correlations. The Gibbs measure is
the random probability measure on the configuration space Σ attaching to the particle
t
k Σ the weight
t
∈
n(t)
expβx (t)
def k def
(k) = , where Z (β) = expβx (t), (2.1)
β,t t k
G Z (β)
t
j=1
X
where β > 0 is the inverse temperature. A first study of the Gibbs measure of BBM was
carried out by Derrida & Spohn [16]. Through comparisons with Derrida’s GREM, and
exploiting the Ghirlanda-Guerra identities introduced in the context of the Sherrington-
Kirkpatrick model [19], Bovier & Kurkova [7] put on rigorous ground the findings by
Derrida&Spohn, therebyprovinginparticularthat,forβ > √2,thelawofthenormalized
time to first branching under the product Gibbs measure over the replicated space Σ Σ
t t
×
GENEALOGY OF BRANCHING BROWNIAN PARTICLES 5
converges in distribution to the superposition of two delta functions,
Q (t)
i,j
lim d x = c δ ( d x)+(1 c )δ ( d x). (2.2)
β,t β,t o 0 o 1
t G ⊗G t ∈ −
→∞ (cid:18) (cid:19)
forsome β-dependent 0 < c < 1. Hence, thesupport of theGibbsmeasure isrestricted to
o
“almostuncorrelated”particles. SincetheGibbsmeasurefavoursextremal configurations,
one may wonder whether a similar result holds true also at the level of the extremal
process. In this paper, we answer this question in the affirmative. In fact, we prove
a stronger result which concerns the unnormalized time to first branching of extremal
def
particles: denoting by Σ (D) = i Σ : x (t) D the set of particles falling into the
t t i
∈ ∈
subset m(t)+D, we have: n o
Theorem 2.1 (The genealogy of extremal particles.). For any compact D R,
⊂
lim supP i,j Σ (D) : Q (i,j) (r,t r) = 0. (2.3)
t t
r t>3r ∃ ∈ ∈ −
→∞
h i
Extremal particles are therefore “essentially uncorrelated”, in the sense that they de-
scend from common ancestors which either branch off very early (in the interval (0,r))
or “very late” (in the interval (t r,t)) in the course of time. The proof of Theorem
−
2.1 is given in Section 4 and relies on results about the localization of the paths of the
extremal particles which is of independent interest. Below we explain the heuristics be-
hind Theorem 2.1, and then describe, in Section 2.2, the results on the localization of the
paths.
Consider two extremal particles, say i and j, that reach, at time t, heights of about
m(t), and assume that the common ancestor of these particles branched at times well
inside the interval [0,t] (the reason for the possibility of ”very old” resp. ”very recent”
ancestries will become apparent when discussing the results on the localization of paths):
forconcreteness, assumethatQ (i,j) = t/3. Thecommonancestor oftheparticlesattime
t
t/3, will be shown by Theorem 2.5 to lie at heights at most of order √2(t/3) c(t/3)α for
−
some 0 < α < 1/2, omitting logarithmic corrections. (Anticipating, this is a phenomenon
strongly reminiscent of the entropic repulsion witnessed in the statistical mechanics of
interface models). In order for a descendant, say particle i, to be on the edge at time
t, the ancestor must thus produce a random tree of length (2/3)t where at least one
particle makes the unusually high jump √2(2/3)t + c(t/3)α. One can easily check that
this is indeed possible: there are to leading order exp +√2(t/3)α particles at levels
√2(t/3) c(t/3)α, and the probability of such a big jump in the remaining time interval
− (cid:0) (cid:1)
is of order exp √2(t/3)α , the product being thus of order one. But to have the particle
−
j reach the same levels and overlapping for t/3 of its lifetime with i amounts to finding
(cid:0) (cid:1)
within the same tree of length (2/3)t yet a second particle which makes the unusually
high jump. The probability of finding such two particles is to leading order at most
exp +√2(t/3)α exp 2√2(t/3)α , which is vanishing in the limit of large times. Of
−
course, this is valid only to leading order and for a fixed value of the overlap, the nature of
(cid:0) (cid:1) (cid:0) (cid:1)
the continuous branching compounding the difficulties, but the reasoning is in its essence
correct.
GENEALOGY OF BRANCHING BROWNIAN PARTICLES 6
2.2. Localization of the paths of extremal particles. Our approach towards the
genealogy of particles at the edge of BBM is based on characterizing, up to a certain level
of precision, the paths of extremal particles. As a first step towards a characterization, we
will prove that such paths cannot fluctuate too wildly in the upward direction. In order
to formulate this precisely, we introduce some notation. For 0 < γ < 1/2, we set
sγ 0 s t
u (s) d=ef ≤ ≤ 2 (2.4)
t,γ (t s)γ t s t .
(
− 2 ≤ ≤
The upper envelope at time t, denoted U , is defined as
t,γ
s
def
U (s) = m(t)+u (s). (2.5)
t,γ t,γ
t
Notice that U (t) = m(t).
t,γ
Theorem 2.2 (Upper Envelope). Let 0 < γ < 1/2. Let also y R, ǫ > 0 be given. There
∈
exists r = r (γ,y,ǫ) such that for r r and for any t > 3r,
u u u
≥
P k n(t) : x (s) > y +U (s), for some s [r,t r] < ǫ . (2.6)
k t,γ
∃ ≤ ∈ −
h i
Our choice of the upper envelope is not optimal. However, the proof is relatively simple
and based on the following estimate obtained by Bramson in [8, Prop. 3]:
P max x (t) m(t)+Y κ(1+Y)2exp √2Y , (2.7)
k
k n(t) ≥ ≤ −
(cid:20) ≤ (cid:21) h i
which is valid for 0 < Y < √t and where κ > 0 is a numerical constant. (Here and
henceforth, we denote by κ a positive numerical constant, not necessarily the same at
different occurrences). In fact, a slight refinement of the approach used by Bramson to
get (2.7) yields an envelope where u may be chosen of logarithmic order. It is based on
the simple observation that particles that touch the upper envelope during the interval
[r,t r] would reach at that time values that are so large that their offspring at time t
−
could easilyjumpto heights well above theestablished valueofthemaximal displacement.
e
c
a
sp Ut,γ(s)= stm(t)+O(sγ,(t−s)γ) m(t)=√2t− 2√32logt
sm(t)
t
ru t−ru time
t t
2
Fig. 1. UpperEnvelope
GENEALOGY OF BRANCHING BROWNIAN PARTICLES 7
A straightforward consequence of Theorem 2.2 is that the number of particles at the
def
edge is stochastically bounded: with [y, ) = ♯ k = 1...n(t) : x (t) m(t)+y the
t k
number particles that reach at time tNvalue∞s higher{than m(t)+y (y R≥), it holds:}
∈
Corollary 2.3 (Local finiteness). To given y R,ε > 0 there exists N = N(ε,y) N
∈ ∈
and t = t (ε,y) such that for t t
o o o
≥
P[ [y, ) N] < ε. (2.8)
t
N ∞ ≥
Remark 2.4. Our proof of the local finiteness is certainly not the most straightforward.
We learned from Bramson [10] a short proof based on a contradiction of the linearity of
the maximal displacement. The corollary should, however, be seen as a first instance of
the rationale which runs throughout the paper culminating in Theorem 2.1; namely, that
characterizing the paths of extremal particles provides crucial information on the extremal
process.
Correlations among particles force the front of BBM to lie lower (by a logarithmic
factor) than the one of the REM. This has considerable impact on the finer properties of
BBM. A simple calculation reveals already that something unusual is going on: to leading
order in t, the mean number of exceedances of a level x is given by
e t x2
E[# i n(t) : x (t) > x ] exp (2.9)
i
{ ≤ } ∼ √2πt −2t
(cid:18) (cid:19)
Notethatthisquantityisnotsensitive tocorrelations. Forthelevelofthemaximuminthe
REM, x = r(t), this quantity is of order one, as t , while at the level of the maximum
↑ ∞
of BBM, x = m(t), it is of order t! This appears at first glance to contradict the claim
of the theorem above that the number of particles above m(t) stochastically bounded.
To understand why the number of particles at the level m(t) does not grow with t with
probability one is, however, easily understood with the help of Theorem 2.2. Namely,
this theorem tells us that one can additionally require that paths of extremal particles
never cross the upper envelope (up to an error which can be made as small as wished).
Precisely, extremal particles perform Brownian motion starting in zero conditioned to
reach certain values at given times. This can be reformulated in terms of a Brownian
bridge of length t starting at 0 and ending at m(t) (omitting lower orders) that is not
allowed to cross the upper envelope. This idea is omnipresent in Bramson’s paper [8], and
is used extensively in the present work. By the very definition of the upper envelope, the
situation is equivalent to a Brownian bridge (starting and ending at time t at zero) which
is not allowed to cross the curve u for most of its lifespan. The probability of such an
t,γ
event is inversely proportional to the length of the bridge, and will hence compensate the
extra t factor observed above.
ThemainconsequenceofTheorem2.2isaphenomenonwewillreferto,byaslightabuse
of terminology, as entropic repulsion. Due to the strong fluctuations of the unconstrained
paths, particles which at some point areclose to the line s sm(t) have plenty of chances
7→ t
to hit the upper envelope in the remaining time. One expects that a natural way to avoid
this is for the paths to lie well below the interpolating line for most of the time. (In other
words, a typical Brownian bridge that is conditioned to lie below the curve u for most
t,γ
of the interval of time must lie well below 0; this is not surprising in view of the fact
GENEALOGY OF BRANCHING BROWNIAN PARTICLES 8
that the conditioned Brownian bridge resembles a Bessel bridge [27]). This turns out
to be the case, the upshot being that the upper envelope identified in Theorem 2.2 can
be replaced by a lower “entropic envelope”, E, under which paths of extremal particles
lie with overwhelming probability. [Such a phenomenon is strongly reminiscent of the
entropic repulsion encountered in the statistical mechanics of membrane models, see e.g.
Velenik’s survey [31]]. To formulate this precisely, we need some notation. We consider
for the parameters C > 0 and 0 < α < 1/2 the function
Csα 0 s t/2,
def
e (s) = ≤ ≤ (2.10)
t,α C(t s)α t/2 s t.
(
− ≤ ≤
Throughout the paper the parameter C will be fixed. For simplicity, we omit it in the
notation. The entropic envelope is
s
def
E (s) = m(t) e (s). (2.11)
t,α t,α
t −
Notice that E U .
t,α t,α
≪
Theorem 2.5 (Entropic Repulsion). Let D R be a compact set. Set D d=ef sup x D
⊂ { ∈ }
def
and D = inf x D . Let 0 < γ < α < 1/2. There exists r = r (α,D,ǫ) such that for
e e
{ ∈ }
r r and t > 3r,
e
≥
P k n(t) : x (t) m(t)+D, x (s) D +U (s)
k k t,γ s [r,t r]
∃ ≤ ∈ ≤ ∀ ∈ − (2.12)
h
but x (s) D +E (s) < ǫ .
s [r,t r] k t,α
∃ ∈ − ≥
i
Such path-localizations as in Theorem 2.5 evidently cannot hold true for times which
are close to 0 or t, and this is the reason why “very old” resp. “very recent” ancestries
are indeed not only possible, but, as we believe (see Section 2.3 below) also crucial for the
peculiar properties of the extremal process of BBM.
e
c
spa Ut,γ(s)= stm(t)+O(sγ,(t−s)γ) m(t)=√2t− 2√32logt
E (s)= sm(t) O(sα,(t s)α)
t,α t − −
re t−re time
t
Fig. 2. Entropic Repulsion
Remark 2.6. Energy/entropy considerations provide a straightforward explanation of the
mechanism underlying Theorem 2.5: at any given time s [r,t r] (for r large enough
∈ −
GENEALOGY OF BRANCHING BROWNIAN PARTICLES 9
but finite), there are simply not enough particles at heights sm(t) e (s) for their
≥ t − t,α
offspring to be able to make large jumps allowing them to reach at time t the edge.
Entropic repulsion is instrumental in the proof of Theorem 2.1. For technical reasons,
we need yet another piece of information about paths of extremal particles: namely that
they cannotlietoolow, which isagaintobeexpected fromanenergy/entropy perspective.
We introduce, for c > 0 and 1/2 < β < 1, the curves
csβ 0 s t/2
lc (s) d=ef ≤ ≤ (2.13)
t,β c(t s)β t/2 s t,
(
− ≤ ≤
and the lower envelope
s
def
L (s) = m(t) l (s). (2.14)
t,β t,β
t −
Of course that L E . Again the parameter c will be fixed and dropped in the
t,β t,β
≪
notation.
Theorem 2.7 (Lower Envelope). Let D R be a compact set. Set D d=ef sup x D .
⊂ { ∈ }
Let 0 < α < 1/2 < β < 1. Let by E and L be the curves defined by (2.11) and (2.14),
t,α t,β
respectively. Then there exists r = r (α,β,D,ǫ) such that, for r r and for any t > 3r,
l l l
≥
P k n(t) : x (t) m(t)+D, x (s) D +E (s), ,
k k t,α r s t r
∃ ≤ ∈ ≤ ∀ ≤ ≤ − (2.15)
h
but : x (s) D +L (s) < ε.
s t r k t,β
∃ ≤ − ≤
i
Theorems 2.5 and 2.7 are proven in Section 5. The two theorems provide an explicitly
characterized tube, the space-time region between lower and entropic envelopes, where
paths of extremal particles spend most of their time with overwhelming probability.
Corollary 2.8. Let 0 < α < 1/2 < β < 1. Let D R be a compact set and ǫ > 0 be
⊂
given. There exists r = r (α,β,D,ε) such that for r r and t > 3r,
1 1 1
≥
P k n(t) : r s t r : x (t) m(t)+D, x (s) > D +E (s)
k k t,α
∃ ≤ ∃ ≤ ≤ − ∈
(2.16)
h
or r s t r : x (t) m(t)+D,x (s) D+L (s) < ǫ .
′ k k ′ t,β ′
∃ ≤ ≤ − ∈ ≤
i
where E (s) is defined in (2.11) and L (s) in (2.14).
t,α t,β