Invariant Measures, Hausdorff Dimension and Dimension Drop of some Harmonic Measures on Galton-Watson Trees
aa r X i v : . [ m a t h . P R ] M a y Invariant measures, Hausdorff dimensionand dimension drop of some harmonicmeasures on Galton-Watson trees
Pierre Rousselin ∗ May 7, 2018
We consider infinite Galton-Watson trees without leaves together withi.i.d. random variables called marks on each of their vertices. We definea class of flow rules on marked Galton-Watson trees for which we are able,under some algebraic assumptions, to build explicit invariant measures. Weapply this result, together with the ergodic theory on Galton-Watson treesdeveloped in [12], to the computation of Hausdorff dimensions of harmonicmeasures in two cases. The first one is the harmonic measure of the (tran-sient) λ -biased random walk on Galton-Watson trees, for which the invariantmeasure and the dimension were not explicitly known. The second case isa model of random walk on a Galton-Watson trees with random lengths forwhich we compute the dimensions of the harmonic measure and show dimen-sion drop phenomenon for the natural metric on the boundary and anothermetric that depends on the random lengths. Keywords:
Galton-Watson tree, random walk, harmonic measure, Hausdorff dimension,invariant measure, dimension drop.
AMS 2010 subject classifications:
Primary 60J50, 60J80, 37A50; secondary 60J05,60J10.
Consider an infinite rooted tree t without leaves. The boundary ∂t of the tree t is theset of all rays on t , that is infinite nearest-neighbour paths on t that start from the rootand never backtrack. If we equip the space ∂t with a suitable metric d, we may considerthe Hausdorff dimension of ∂t with respect to d. Now assume that we run a transient nearest-neighbour random walk on the vertices of t , starting from the root of t . For ∗ LAGA, University Paris 13; Labex MME-DII n ≥
0, let Ξ n be the last vertex at height n that is visited by the randomwalk. We call Ξ the exit ray of this random walk. The distribution of Ξ, which is aprobability measure on the boundary ∂t is called the harmonic measure associated tothe random walk. One may define the Hausdorff dimension of this measure to be theminimal Hausdorff dimension of a subset of ∂t of full measure. When this Hausdorffdimension is strictly less than the one of the whole boundary, we say that the dimensiondrop phenomenon occurs. Informally, this means that almost surely, the exit ray mayonly visit a very small part of the boundary ∂t .This phenomenon was first observed by Makarov in [16] in the context of a two-dimensional Brownian motion hitting a Jordan curve. On supercritical Galton-Watsontrees, Lyons, Pemantle and Peres proved the dimension drop for the simple randomwalk in the seminal paper [12] and for the (transient) λ -biased random walk in [13].For a more concrete interpretation of this phenomenon, see [13, Corollary 5.3] and [12,Theorem 9.9] which show that with high probability, the random walk is confined to anexponentially smaller subtree. Another application of this result is given in [3], wherethe authors prove a cut-off phenomenon for the mixing time of the random walk on thegiant component of an Erdős-Rényi random graph, started from a typical point.An analogous asymptotic result was discovered by Curien and Le Gall in [4] for thesimple random walk on critical Galton-Watson trees conditioned to survive at height n ,when n is large. It was then extended to the case where the reproduction law of theGalton-Watson tree has infinite variance by Lin [9]. In these works, the asymptotic resultfor finite trees is (not easily) derived from the computation of the Hausdorff dimensionassociated to a random walk on an infinite tree. This infinite tree is the scaling limitof the reduced critical Galton-Watson tree conditioned to survive at height n ; it can beseen as a Galton-Watson tree with edge lengths and the simple random walk becomesa nearest-neighbour random walk with transition probabilities inversely proportional tothe edge lengths.One of the main goals of this work is to generalize this last model and obtain similarresults. This will be done in Section 5. To introduce our method, we need a little moreformalism. We work in the space of marked trees without leaves, that is, trees t with,on each vertex x of t , an attached non-negative real number γ t ( x ) called the mark of x in t . A marked Galton-Watson tree, characterized by its reproduction law and itsmark law (which may be degenerated), is a Galton-Watson tree with i.i.d. marks. To apositive function φ on the space of marked tree, we may associate the law of a randomray Ξ on the tree t in the following way: we first choose a vertex Ξ = i of height 1 withprobability P (Ξ = i ) = Θ t ( i ) := φ ( t [ i ]) P ν t (ø) j =1 φ ( t [ j ]) , where ν t (ø) is the number of children of the root ø and t [ i ] is the (reindexed) markedsubtree starting from i , see Figure 1. For the remaining vertices of the ray Ξ, we thenplay the same game in the selected subtree t [ i ]. The sequence of such selected subtreesis a Markov chain on the space of marked trees. Our initial measure of interest for thisMarkov chain is the law of a marked Galton-Watson tree but it is rarely invariant.2 , γ t (ø) i, γ t ( i )1 , γ t (1) ν t (ø) , γ t ( ν t (ø)). . . . . . t [1] t [ i ] t [ ν t (ø)]. . . . . . . . . . . . . . . . . . . . . . . . . . . t Θ t (1) Θ t ( i )Θ t ( ν T (ø))Figure 1: Probability transitions of the Markov chain on the space of marked trees as-sociated to the flow rule Θ.The heart of this work is Theorem 3.2, in which we give some sufficient algebraicconditions on the function φ to build an explicit invariant measure for this Markov chain.Moreover, this measure is absolutely continuous with respect to the law of the Galton-Watson tree. These algebraic conditions were inspired by the proofs of [4, Proposition 25]and [9, Proposition 8]. This allows us to use the ergodic theory on Galton-Watson treesfrom [12], to obtain quantitative and qualitative results.In Section 4, we use this tool to study the harmonic measure of the λ -biased randomwalk on supercritical Galton-Watson trees. In [13], the authors prove the existence of aunique invariant probability measure µ HARM absolutely continuous with respect to thelaw of the Galton-Watson tree, and conclude that there is a dimension drop phenomenon.However, they do not (except in the case λ = 1 in [12]) give an explicit formula for thedensity of µ HARM and the Hausdorff dimension of the harmonic measure. Interestingly,although we are dealing with a model of random walk on Galton-Watson trees withoutmarks, it is easier to express an explicit invariant measure in the more general settingof Theorem 3.2. From the explicit invariant measure, we deduce an expression for theHausdorff dimension of the harmonic measure in Theorem 4.1. This invariant measureand this Hausdorff dimension were found independently by Lin in [10]. The explicitformulas allow him to answer interesting questions, including an open question from[14]. In this work, we use our dimension formula (12) to compute numerically thedimension (and the speed, thanks to [1]) of the λ -biased random walk as a function of3 , for two different reproduction laws.Section 5 is completely independent from Section 4. Let us briefly present the modelwe study in it. Let Γ be a random variable in (1 , ∞ ). We consider a Galton-Watsontree T with reproduction law p = ( p k ) k ≥ such that p = 0 and p < x ) x ∈ T with the same law as Γ. We add an artificial parent ø ∗ ofthe root ø. Then we define the length of the edge between ø and ø ∗ as the inverse Γ − of the mark of the root. We do the same in all the subtrees starting from the children ofthe root, except that we multiply all the lengths in these subtrees by (1 − Γ − ), and wecontinue recursively, see Figure 2. We call the resulting random tree a Galton-Watsontree with recursive lengths. The Γ-height of a vertex x of T is the sum of the lengths ofall the edges in the shortest path between x and ø ∗ .1 ø ∗ • ø Γ − • (cid:16) − Γ − (cid:17) Γ − • • • • • • . . . • ν T (1). . . • (cid:16) − Γ − (cid:17) Γ − • (cid:16) − Γ − (cid:17) (cid:16) − Γ − (cid:17) Γ − •••• •
22 . . . • ν T (2) . . . • (cid:16) − Γ − (cid:17) Γ − ν T (ø) ν T (ø) •• • •• • • • . . .Figure 2: A schematic representation of a Galton-Watson tree with recursive lengths.We then start a nearest-neighbour random walk on T from its root ø with transitionprobabilities inversely proportional to the lengths of the edges between the current vertexand the neighbours. The walk is reflected at ø ∗ . This random walk is almost surelytransient, thus defines a harmonic measure on the boundary ∂T . We can again applyTheorem 3.2 to this model to find an invariant probability measure. This gives in4heorem 5.2 the Hausdorff dimension of the harmonic flow, as well as the dimensiondrop, for the so-called “natural metric” on the boundary ∂T which does not take intoaccount the lengths on the tree, only the genealogical structure.Finally, we solve the same problem with respect to another metric on ∂T which de-pends on the lengths. We associate to our tree an age-dependent process whose Malthu-sian parameter turns out to be the Hausdorff dimension of the boundary with respect tothis distance, and prove that we still have a dimension drop phenomenon in this context. Acknowledgements.
Part of this work was presented at the 47 th Saint-Flour SummerSchool of Probability. The author wishes to thank the organizers of this event. Theauthor is also very grateful to his Ph.D. supervisors Julien Barral and Yueyun Hu formany interesting discussions and constant help and support and to the anonymous refer-ees for suggesting many improvements (and for the correction of an uncountable amountof typos and other mistakes).
Following Neveu ([17]), we represent our trees as special subsets of the finite words onthe alphabet N ∗ := { , , . . . } . The set of all finite words is denoted by U and equals thedisjoints union F ∞ k =0 ( N ∗ ) k , where we agree that ( N ∗ ) := { ø } is the set containing onlythe empty word ø. The length of a word x is the unique integer k such that x belongsto ( N ∗ ) k and is denoted by | x | . The concatenation of the words x and y is denoted by xy . A word x = ( x , x , . . . , x | x | ) is called a prefix of a word y = ( y , y , . . . , y | y | ) wheneither x = ø or | x | ≤ | y | and x i = y i for any i ≤ | x | . We denote by (cid:22) this partial orderand by x ∧ y the greatest common prefix of x and y . The parent of a non-empty word x = ( x , x , . . . , x | x | ) is x ∗ := ( x , x , . . . , x | x |− ) if | x | ≥
2; otherwise it is the emptyword ø. We also say that x is a child of x ∗ . A (rooted, planar, locally finite) tree t is asubset of U such that ø ∈ t (in this context, we call ø the root of t ) and for any x ∈ t : • if x = ø, then x ∗ ∈ t ; • there exists a unique non-negative integer, denoted by ν t ( x ) and called the numberof children of x in t , such that for any i ∈ N , xi is in t if and only if 1 ≤ i ≤ ν t ( x ).The tree t is endowed with the undirected graph structure obtained by drawing an edgebetween each vertex and its children. A leaf of t is a vertex that has no child. In thiswork, we are interested in leafless trees, that is infinite trees in which the number ofchildren cannot be 0. We denote by T the set of all such trees.We will need to add an artificial parent ø ∗ of the root. Let U ∗ be the set U ⊔ { ø ∗ } and let us agree that ø ∗ is the only strict prefix of ø and has length −
1. Likewise, forany tree t in T , we let t ∗ := t ⊔ { ø ∗ } and denote by T ∗ the set of all such trees.Let U ∞ := ( N ∗ ) N ∗ be the set of all infinite words. The k -th truncation of an infiniteword ξ is the finite word composed of its k first letters and is denoted by ξ k . When a5nite word u is a truncation of an infinite word ξ , we still say that u is a prefix of ξ . Fortwo distinct infinite words ξ and η , we may again consider their greatest common prefix ξ ∧ η ∈ U . A ray in a tree t is an infinite word ξ such that each of its truncations belongsto t . One may also think of a ray as an infinite non-backtracking path in t starting fromthe root. The set of all rays in t is called the boundary of t and denoted by ∂t .Let us denote by [ x ] t the cylinder of all rays passing through x in t , that is the set ofall rays ξ in ∂t such that ξ | x | = x . The boundary ∂t of a tree t is always endowed withthe topology generated by all the cylinders [ x ] t , for x in t .A (unit) flow on t is a function θ from t to [0 , θ (ø) = 1 and for any x ∈ t , θ ( x ) = ν t ( x ) X i =1 θ ( xi ) . We may define a flow θ M on t from a Borel probability measure M on the boundary ∂t by setting θ M ( x ) = M ([ x ] t ). A monotone class argument shows that the mapping M θ M is a one-to-one correspondance and we will abuse notations and write θ forboth the flow on t and the associated Borel probability measure on ∂t . Consider an infinite leafless rooted tree t as before, and a metric d on its boundary. For ξ in ∂t and a non-negative number r , let B ( ξ, r ) be the closed ball centered at ξ withradius r for the metric d. We always make the assumption that the balls of the metricspace ( ∂t, d) are exactly the cylinders [ x ] t , for x in t . An example of a metric satisfyingthis assumption is the natural distance d U ∞ defined by, ∀ η = ξ ∈ ∂t, d U ∞ ( ξ, η ) = e −| ξ ∧ η | . (1)We will use this metric in Section 4 and the first half of Section 5. At the end of thispaper, we will need to consider another (random) metric.For δ >
0, and a subset E of ∂t , a δ -cover of E is a denumerable family ( E i ) i ≥ ofsubsets of ∂t such that for any i ≥
1, the diameter diam E i (with respect to d) is lesseror equal to δ and E ⊂ S i ≥ E i . Now, let s be a non-negative real number. First wedefine the (outer) measure H sδ by H sδ ( E ) = inf n ∞ X i =1 diam( E i ) s : ( E i ) i ≥ is a δ -cover of E o . (2)The s -dimensional Hausdorff measure of E (with respect to d ) is H s ( E ) = lim δ → H sδ ( E ) ∈ [0 , ∞ ] . (3)As usual, the Hausdorff dimension of a subset E of ∂t isdim H E = inf { s ≥ H s ( E ) = 0 } = sup { s ≥ H s ( E ) = + ∞} . θ on the leafless tree t is defined bydim H θ := inf { dim H ( B ) : B Borel subset of ∂t, θ ( B ) = 1 } . In this work, we will mainly deal with well-behaved flows that satisfy the following(strong) property, which allows us to compute their (Hausdorff) dimensions.
Definition 2.1.
Let θ be a Borel probability measure on ( ∂t, d). If there exists d θ in[0 , ∞ ] such that, for θ -almost every ξ in ∂t , the limitlim r ↓ − log θ B ( ξ, r ) − log r exists and equals d θ , one says that the flow θ is exact-dimensional, of dimension dim d θ = d θ .It is well-known in the Euclidean space that, when a Borel probability measure µ isexact-dimensional of dimension d µ , then all reasonable notions of dimensions coincideand are equal to d µ (see [19, Theorem 4.4]). In particular, the (upper) Hausdorff dimen-sion of µ equals its dimension. On the boundary of a tree, with our assumption on themetric d, this is also true (it is in fact easier than the Euclidean case). The interestedreader may want to read [15, Section 15.4] for a simplification of (2) on the boundaryof a tree, then [18] and [5] for density theorems (in [18], the authors consider centeredcovering measures as an alternative to Hausdorff measures, but in our case, they are thesame), and finally, [6, Proposition 10.2], for how to relate local and global dimensions ofa measure, using density theorems. Example . For the the natural distance d U ∞ , a flow θ on t is exact-dimensional ofdimension d θ if and only iffor θ -almost every ξ , lim n →∞ n ( − log)( θ ( ξ n )) = d θ . (4) We now introduce the marks, which can be thought of as additional data on the verticesof the tree, in the form of non-negative real numbers. A (leafless) marked tree is a tree t ∈ T together with a collection ( γ t ( x )) x ∈ t of elements of R + .We define the (local) distance between two marked trees t and t ′ byd m (cid:0) t, t ′ (cid:1) = X n ≥ − r − δ n (cid:0) t, t ′ (cid:1) , where δ n is defined by δ n (cid:0) t, t ′ (cid:1) = ( t and t ′ (without their marks) do not agree up to height n ;min (1 , sup {| γ t ( x ) − γ t ′ ( x ) | : x ∈ t, | x | ≤ n } ) otherwise.It is a Polish space, which we denote by T m and equip with its Borel σ -algebra.7or a marked tree t and a vertex x ∈ t , we let t [ x ] := { u ∈ U : xu ∈ t } be the reindexed subtree starting from x with marks γ t [ x ] ( y ) = γ t ( xy ) , ∀ y ∈ t [ x ] . A (consistent) flow rule is a measurable function Θ on a Borel subset B of T m suchthat for any tree t in B ,1. Θ t is a flow on t ;2. for any vertex x in t , Θ t ( x ) > x in t , t [ x ] is in B and for all y in t [ x ], Θ t [ x ] ( y ) = Θ t ( xy ) / Θ t ( x ).Let µ be a Borel probability measure on T m . We say that Θ is a µ -flow rule if µ ( B ) = 1.When the µ -flow rule does not depend on the marks, it is essentially the same notion offlow rule as in [12] and [15, chapter 17], except that we allow it to only be defined on aset of full measure and that we impose that Θ t ( x ) > t and any vertex x in t . These choices were made in order to simplify the statements of the next subsection.The set of all marked trees with a distinguished ray is denoted by T m,r := { ( t, ξ ) : t ∈ T m , ξ ∈ ∂t } . It is endowed with the same distance as before except that for n ≥
0, we impose that δ n (( t, ξ ) , ( t ′ , ξ ′ )) is equal to 1 if the two distinguished rays ξ and ξ ′ do not agree up toheight n . It is again a Polish space. The shift S on this space is defined by S ( t, ξ ) = ( t [ ξ ] , Sξ ) , where Sξ is the infinite word obtained by removing the first letter of ξ .Let µ be a Borel probability measure on T m and Θ be a µ -flow rule. We build aprobability measure on T m,r , denoted by µ ⋉ Θ, or E µ when the µ -flow rule Θ is fixed,by first picking a tree T at random according to µ , and then choosing a ray Ξ of T according to the distribution Θ T .The sequence of trees ( T [Ξ n ]) n ≥ is then a discrete time Markov chain with values in T m , initial distribution µ and transition kernel P Θ given by P Θ ( T, T ′ ) = ν T (ø) X i =1 Θ T ( i ) { T ′ = T [ i ] } . If the law of T [Ξ ] is still µ , we say that µ is a Θ-invariant measure. When µ is Θ-invariant (for the Markov chain), the system ( T m,r , µ ⋉ Θ , S ) is measure-preserving.8 .4 Ergodic theory on marked Galton-Watson trees We now consider a reproduction law p = ( p k ) k ≥ , that is a sequence of non-negativereal numbers such that P ∞ k =0 p k = 1, together with a random variable Γ with valuesin R + . We will assume throughout this work that p = 0, p < m := P ∞ k =0 kp k is finite.A (Γ , p )-Galton-Watson tree is a random tree T such that the root ø has k childrenwith probability p k , has a mark γ T (ø) = Γ ø with the same law as Γ and, conditionally onthe event that the root has k children, the trees T [1], T [2], . . . , T [ k ] are independent andare (Γ , p )-Galton-Watson trees. In particular, all the marks are i.i.d. We denote themby (Γ x ) x ∈ T . The distribution of a (Γ , p )-Galton-Watson on T m is denoted by GW . Thebranching property is still valid in this setting of marked trees: if T is a random markedtree distributed as GW , then for all integers k ≥
1, Borel sets A , A , . . . , A k of T m ,and Borel sets B of R + , P (cid:0) ν T (ø) = k, Γ ø ∈ B, T [1] ∈ A , . . . , T [ k ] ∈ A k (cid:1) = p k P (cid:0) Γ ∈ B (cid:1) k Y i =1 GW (cid:0) A i (cid:1) . We now recall the main ingredients (all of which will be used throughout this work)of the ergodic theory on Galton-Watson trees developed in [12, Section 5] (see alsoSection 17.5 of [15]). Since our marked Galton-Watson trees still satisfy the branchingproperty, all these results still apply in our setting of marked trees. Remark that in thetwo following statements, the omission of the extra hypotheses in [12, Section 5] is dueto our slightly modified definition of a flow rule (we impose that for GW -almost anytree t , we have Θ t ( x ) > x in t ). Lemma 2.1 ([12, Proposition 5.1]) . Let Θ and Θ ′ be two GW -flow rule. Then Θ T and Θ ′ T are either almost surely equal or almost surely different. Theorem 2.2 ([12, Proposition 5.2]) . Assume there exists a Θ - stationary probabilitymeasure µ Θ which is absolutely continuous with respect to GW . Then µ Θ is equivalent to GW and the associated measure-preserving system is ergodic. Moreover, the probabilitymeasure µ Θ is the only Θ -stationary probability measure which is absolutely continuouswith respect to GW . The flow Θ T is almost surely exact-dimensional, and its dimension,with respect to the metric d U ∞ is dim d U∞ Θ T = E µ Θ [ − log Θ T (Ξ )] . (5)The ergodic theory Lemma 6.2 in [12] (see also [15, Lemma 17.20]) will be used severaltimes. We recall it for the reader’s convenience. Lemma 2.3. If S is a measure-preserving transformation on a probability space, g is afinite and measurable function from this space to R , and g − Sg is bounded from belowby an integrable function, then g − Sg is integrable with integral . UNIF is defined as in [12] Section 6; it does not depend on themarks. For any tree t , we denote Z n ( t ) := { x ∈ t : | x | = n } . Let T be a (Γ , p )-Galton-Watson tree. By the Seneta-Heyde theorem, there exists asequence of positive real numbers ( c n ) n ≥ which only depends on p such that almostsurely f W ( T ) := lim n →∞ Z n ( T ) c n exists in (0 , ∞ ). Furthermore, lim n →∞ c n +1 /c n = m . Hence we may define the GW -flowrule UNIF by UNIF T ( i ) = f W ( T [ i ]) P ν t (ø) j =1 f W ( T [ j ]) , for j = 1 , , . . . , ν t (ø)and f W almost surely satisfies the recursive equation f W ( T ) = 1 m ν t (ø) X j =1 f W ( T [ i ]) . (6)Theorem 7.1 in [12] is still valid with the same proof in our setting of marked trees. Proposition 2.4 (Dimension drop for non uniform flow rules, [12, Theorem 7.1]) . Withthe same hypotheses as Theorem 2.2, if P (Θ T = UNIF T ) < , then, almost surely dim Θ T is strictly less than log m , which equals dim H ∂T . Let T be a (Γ , p )-Galton-Watson tree. Let φ be a measurable positive function on aBorel set of T m which has full GW -measure. We define a GW -flow rule Θ byΘ T ( i ) = φ ( T [ i ]) P ν T (ø) j =1 φ ( T [ j ]) , ∀ ≤ i ≤ ν T (ø) . Lemma 3.1.
Let Θ and Θ ′ be two flow rules defined respectively by φ and φ ′ as above.Then, Θ T = Θ ′ T almost surely if and only if the functions φ and φ ′ are proportional.Proof. This is the same proof (except the last sentence) as [12, Proposition 8.3], so weomit it.We now assume that the marks and the function φ have their values in the samesub-interval J of (0 , ∞ ) and that there exists a measurable function h from J × J to J such that, almost surely, φ ( T ) = h (cid:16) Γ ø , ν t (ø) X i =1 φ ( T [ i ]) (cid:17) .
10n words, φ is an observation on the tree T that can be recovered from the mark of theroot Γ ø and the sum of such observations on the subtrees T [1], . . . , T [ ν T (ø)].We now make algebraic assumptions on the function h . These assumptions, as well asthe next theorem, are inspired by the proofs of [4, Proposition 25] and [9, Proposition 8]. symmetry ∀ u, v ∈ J, h ( u, v ) = h ( v, u ); associativity ∀ u, v, w ∈ J, h ( h ( u, v ) , w ) = h ( u, h ( v, w )); position of summand ∀ u, v ∈ J, ∀ a > , h ( u + a, v )( u + a ) v = h ( u, a + v ) u ( a + v ) . Here are examples of such functions.1. J = (0 , ∞ ) and h ( u, v ) = αuv , for some α > c > J = ( c, ∞ ) and h ( u, v ) = uvu + v − c ;3. for d ≥ J = (0 , ∞ ) and h ( u, v ) = uvu + v + d .We treat the second case. By writing h ( u, v ) = (cid:16) u − + v − − cu − v − (cid:17) − = (cid:16) u − (cid:16) − cv − (cid:17) + v − (cid:17) − , and noticing that (1 − cv − ) >
0, we see that h ( u, v ) > (cid:16) c − (cid:16) − cv − (cid:17) + v − (cid:17) − > c. Symmetry is clear, so is the last property because h ( u, v ) / ( uv ) only depends on the sumof u and v . Associativity follows from the following identity : h ( h ( u, v ) , w ) = (cid:16)(cid:16) u − + v − − cu − v − (cid:17) + w − − c (cid:16) u − + v − − cu − v − (cid:17) w − (cid:17) − = (cid:16) u − + v − + w − − c (cid:16) u − v − + u − w − + v − w − (cid:17) + c u − v − w − (cid:17) − . We will use examples 2 and 3 in the next section, with deterministic marks equal to1 and example 2 in Section 5 with random marks. The first example is illustrated by
UNIF , when α = 1 /m and the marks are all equal to one. Another example with randommarks is given by the flow rule UNIF Γ in Proposition 5.5.For u in J , define κ ( u ) = E (cid:20) h (cid:16) u, X ν e T (ø) i =1 φ ( e T [ i ]) (cid:17)(cid:21) , where e T is a (Γ , p )-Galton-Watson tree independent of T . Theorem 3.2.
Assume that C := E [ κ ( φ ( T ))] < ∞ . Then the probability measure µ withdensity C − κ ( φ ( T )) with respect to GW is Θ -invariant. roof. We use the same notations as in the previous discussion. Let f be a non-negativemeasurable function on the set of marked trees and let Ξ be a random ray on T dis-tributed according to Θ T . We need to show that E [ f ( T [Ξ ]) κ ( φ ( T ))] = E [ f ( T ) κ ( φ ( T ))] . We compute the left-hand side. By conditioning on the value of ν T (ø), we get E [ f ( T [Ξ ]) κ ( φ ( T ))] = ∞ X k =1 p k k X i =1 E h f ( T [ i ]) { Ξ = i } κ ( φ ( T )) i . Since the other terms only depend on T , we can replace { Ξ = i } by its conditionalexpectation given T , which is φ ( T [ i ]) P kj =1 φ ( T [ j ]) . The function κ ( φ ( T )) being also symmetrical in T [1], T [2], . . . , T [ k ], all the terms inthe sum are equal: E [ f ( T [Ξ ]) κ ( φ ( T ))] = ∞ X k =1 p k k E (cid:20) f ( T [1]) φ ( T [1]) P kj =1 φ ( T [ j ]) κ (cid:16) h (cid:0) Γ ø , k X j =1 φ ( T [ j ]) (cid:1)(cid:17)(cid:21) . The definition of κ and Tonelli’s theorem give E [ f ( T [Ξ ]) κ ( φ ( T ))]= ∞ X k =1 p k k E " f ( T [1]) φ ( T [1]) P kj =1 φ ( T [ j ]) h (cid:18) h (cid:18) Γ ø , X kj =1 φ ( T [ j ]) (cid:19) , X ν e T (ø) r =1 φ ( e T [ r ]) (cid:19) . (7)We now use the assumptions on h . We first use symmetry and associativity to obtain h (cid:18) h (cid:18) Γ ø , X kj =1 φ ( T [ j ]) (cid:19) , X ν e T (ø) r =1 φ ( e T [ r ]) (cid:19) = h (cid:18) h (cid:18) Γ ø , X ν e T (ø) r =1 φ ( e T [ r ]) (cid:19) , φ ( T [1]) + X kj =2 φ ( T [ j ]) (cid:19) . Since we never used the value of the mark of the root of e T , we might as well decide thatit is Γ ø , so that h (cid:18) Γ ø , X ν e T (ø) r =1 φ ( e T [ r ]) (cid:19) = φ ( e T ) . Notice that e T has the same law as T and is independent of T [1], T [2], . . . , T [ k ]. Thefirst and third condition on the function h now imply h (cid:18) φ ( e T ) , φ ( T [1]) + X kj =2 φ ( T [ j ]) (cid:19) = h (cid:18) φ ( T [1]) , φ ( e T ) + X kj =2 φ ( T [ j ]) (cid:19) φ ( e T ) (cid:16)P kj =1 φ ( T [ j ]) (cid:17) φ ( T [1]) (cid:16) φ ( e T ) + P kj =2 φ ( T [ j ]) (cid:17) . T for the tree obtained by replacing the subtree T [1] by e T in T . The random tree T has the same law as T and is independent of T [1]. Furthermore, φ ( e T ) φ ( e T ) + P kj =2 φ ( T [ j ]) = φ ( T [1]) P kj =1 φ ( T [ j ]) . Plugging the last two equalities in equation (7), we obtain E [ f ( T [Ξ ]) κ ( φ ( T ))] = ∞ X k =1 kp k E " f ( T [1]) φ ( T [1]) P kj =1 φ ( T [ j ]) h (cid:18) φ ( T [1]) , X kj =1 φ ( T [ j ]) (cid:19) . By symmetry, for all i ≤ k , we have E " f ( T [1]) φ ( T [1]) P kj =1 φ ( T [ j ]) h (cid:18) φ ( T [1]) , X kj =1 φ ( T [ j ]) (cid:19) = E " f ( T [1]) φ ( T [ i ]) P kj =1 φ ( T [ j ]) h (cid:18) φ ( T [1]) , X kj =1 φ ( T [ j ]) (cid:19) , so that k E " f ( T [1]) φ ( T [1]) P kj =1 φ ( T [ j ]) h (cid:18) φ ( T [1]) , X kj =1 φ ( T [ j ]) (cid:19) = E (cid:20) f ( T [1]) h (cid:18) φ ( T [1]) , X kj =1 φ ( T [ j ]) (cid:19)(cid:21) . To conclude the proof, we remove the conditioning on the value of ν T (ø) = ν T (ø) andreplace h ( φ ( T [1]) , P kj =1 φ ( T [ j ])) by its conditional expectation given T [1], which equals κ ( φ ( T [1])). λ -biased randomwalk Let t ∗ ∈ T ∗ and λ >
0. The λ -biased random walk on t is the Markov chain whosetransition probabilities are the following : P t ( x, y ) = x = ø ∗ and y = ø; λλ + ν t ( x ) if y = x ∗ ;1 λ + ν t ( x ) if y is a child of x ;0 otherwise.13or x in t , we write P tx for a probability measure under which the process ( X n ) n ≥ isthe Markov chain on t , starting from x , with transition kernel P t .Let T be a (Γ , p )-Galton-Watson tree, where Γ is deterministic and will be fixed lateron. Recall that m denotes the expectation of our reproduction law and that we assumeit to be finite. From [11], we know that the λ -biased random walk on T is almost surelytransient if and only if λ < m , which we assume from now on. Since the walk is transientthe random exit times defined by et n := inf { s ≥ ∀ k ≥ s, | X k | ≥ n } are P T ø -almost surely finite. We call Ξ := ( X et n ) n ≥ the harmonic ray et denote by HARM T its distribution.We need to introduce the conductance to state our result. For any tree t in T , β ( t ) := P t ø ( ∀ k ≥ , X k = ø ∗ ) . This is the same as the effective conductance between ø ∗ and infinity in the tree t whenwe put on each edge ( x ∗ , x ) the conductance λ −| x | . Notice that for any x ∈ t , we alsohave β ( t [ x ]) = P tx ( ∀ k ≥ , X k = x ∗ ) . We denote, for x ∈ t ∗ , τ x := inf { k ≥ X k = x } , with inf ∅ = ∞ . Then, applying successively the Markov property at times 1 and τ ø , weget β ( t ) = P t ø ( τ ø ∗ = ∞ ) = ν t (ø) X i =1 P t (ø , i ) P ti ( τ ø ∗ = ∞ )= ν t (ø) X i =1 P t (ø , i ) (cid:16) P ti ( τ ø = ∞ ) + P ti ( τ ø < ∞ ) P t ø ( τ ø ∗ = ∞ ) (cid:17) = ν t (ø) X i =1 P t (ø , i ) ( β ( t [ i ]) + (1 − β ( t [ i ])) β ( t )) . Some elementary algebra, together with the fact that1 − ν t (ø) X i =1 P t (ø , i ) = P t (ø , ø ∗ ) , lead to the recursive equation: β ( t ) = P ν t (ø) i =1 P t (ø , i ) β ( t [ i ]) P t (ø , ø ∗ ) + P ν t (ø) i =1 P t (ø , i ) β ( t [ i ]) . (8)14n our special case of the λ -biased random walk, this equation becomes: β ( t ) = P ν t (ø) i =1 β ( t [ i ]) λ + P ν t (ø) i =1 β ( t [ i ]) . (9)For all 1 ≤ i ≤ ν t (ø), the harmonic measure of i is: P ø (Ξ = i ) = P t (ø , ø ∗ ) P t ø ∗ (Ξ = 1) + P t (ø , i ) P ti (Ξ = i ) + ν t (ø) X j =1 j = i P t (ø , j ) P tj (Ξ = i )= P ø (Ξ = i ) (cid:16) P t (ø , ø ∗ ) + ν t (ø) X j =1 (1 − β ( t [ j ])) P t (ø , j ) (cid:17) + P t (ø , i ) β ( t [ i ]) . As a consequence, for all 1 ≤ i ≤ ν t (ø), HARM t ( i ) = P ø (Ξ = i ) = P t (ø , i ) β ( t [ i ]) P ν t (ø) j =1 P t (ø , j ) β ( t [ j ]) . (10)For the λ -biased random walk, this becomes, HARM t ( i ) = β ( t [ i ]) P ν t (ø) j =1 β ( t [ j ]) . (11)This equation describes HARM as a GW -flow rule. We are now ready to use the ma-chinery described in Section 3. Set J = (0 , ∞ ) if λ ≥ J = (1 − λ, ∞ ) if λ < β ( T ) > − λ comes from the Rayleighprinciple and the fact that p <
1, comparing the conductance of the whole tree to theone of a tree with a unique ray. Set, for u and v in J , h ( u, v ) = uvu + v + λ − . Notice that we are in the setting of the second example of Section 3 if λ < λ ≥
1, so that h fulfills the algebraic assumptions stated in Section 3. Byequation (9), β ( t ) = h (cid:16) , X ν t (ø) i =1 β ( t [ i ]) (cid:17) . So we set Γ x := 1 for all x ∈ T . For u ∈ J , let κ ( u ) := E (cid:20) h (cid:18) u, X ν e T (ø) i =1 β ( e T [ i ]) (cid:19)(cid:21) = E u P ν e T (ø) i =1 β ( e T [ i ]) λ − u + P ν e T (ø) i =1 β ( e T [ i ]) , where e T is an independent copy of T .We need to prove that E [ κ ( φ ( T ))] < ∞ . By [1, Lemma 4.2]: E (cid:20) λ − β ( T ) (cid:21) < ∞ . T and e T , that E β ( T ) P ν e T (ø) i =1 β ( e T [ i ]) λ − β ( T ) + P ν e T (ø) i =1 β ( e T [ i ]) ≤ E β ( T ) P ν e T (ø) i =1 β ( e T [ i ]) λ − β ( T ) = E (cid:20)X ν e T (ø) i =1 β ( e T [ i ]) (cid:21) E (cid:20) β ( T ) λ − β ( T ) (cid:21) ≤ m E (cid:20) λ − β ( T ) (cid:21) < ∞ . We may now use Theorem 3.2.
Theorem 4.1.
The probability measure µ HARM of density C − κ ( β ( T )) , with respect to GW , where κ is defined by κ ( u ) = E u P ν e T ( ø ) j =1 β ( e T [ i ]) λ − u + P ν e T ( ø ) j =1 β ( e T [ i ]) , and C is the renormalizing constant, is HARM -invariant. The dimension of
HARM T equals almost surely d λ = log( λ ) − C − E log(1 − β ( T )) β ( T ) P ν e T ( ø ) i =1 β ( e T [ i ]) λ − β ( T ) + P ν e T ( ø ) i =1 β ( e T [ i ]) . (12) Proof.
The only statement we still need to prove is the formula for the dimension.We write µ := µ HARM for short and E µ [ · ] := E [ · C − κ ( β ( T ))]. By formula (5) andequation (11), d λ = E µ (cid:20) log 1 HARM T (Ξ ) (cid:21) = E µ " log P ν T (ø) i =1 β ( T [ i ]) β ( T [Ξ ]) . Using equation (9), we see that X ν T (ø) i =1 β ( T [ i ]) = λβ ( T )1 − β ( T ) . Therefore, d λ = log λ + E µ [ − log(1 − β ( T )) + log( β ( T )) − log( β ( T [Ξ ]))] . We are done if we can prove that log( β ( T )) − log( β ( T [Ξ ])) is integrable with integral0 with respect to E µ . By invariance and Lemma 2.3, it is enough to show that it isbounded from below by an integrable function. We compute, using again formula (9) : β ( T ) β ( T [Ξ ]) = 1 + β ( T [Ξ ]) − P ν T (ø) i =1 , i =Ξ β ( T [ i ]) λ + β ( T [Ξ ]) + P ν T (ø) i =1 , i =Ξ β ( T [ i ]) ≥ λ + β ( T [Ξ ]) ≥ λ + 1 , x β ( T [Ξ ]) − x + 1 x + λ + β ( T [Ξ ])is increasing on [0 , ∞ ).To conclude this section, we recall that a very similar formula was independentlydiscovered by Lin in [10], using the same invariant probability measure. Its work showsthat such a formula, though it might look complicated, does indeed yield very interestingresults. We will now use our formula to conduct numerical experiments about the dimension d λ as a function of λ .It was asked in [14] whether d λ is a monotonic function of λ , for λ in (0 , m ). To thebest of our knowledge, this question is still open. We were not able to find a theoreticalanswer. However, using formula (12) together with the recursive equation (9), we are ableto draw a credible enough graph of d λ versus λ , for a given (computationally reasonable)reproduction law.The idea is the following. Fix a reproduction law p (such that p = 0, p < m ) and a bias λ in (0 , m ). For any non-negative integer n , and a Galton-Watson tree T , let β n ( T ) = P T ø (cid:16) τ ( n ) < τ ø ∗ (cid:17) , where τ ( n ) is the first hitting time of level n by the λ -biased random walk ( X n ) n ≥ . Since the family of events { τ ( n ) < τ ø ∗ } isdecreasing, we have β ( T ) = P T ø (cid:16) \ n ≥ n τ ( n ) < τ ø ∗ o(cid:17) = lim n →∞ β n ( T ) . Using the Markov property as in (9) yields the recursive equation β n +1 ( T ) = P ν T (ø) i =1 β n ( T [ i ]) λ + P ν T (ø) i =1 β n ( T [ i ]) . By definition, β ( T ) is equal to one. Hence, we may use the following algorithm tocompute the law of β n := β n ( T ): • the law of β is the Dirac measure δ ; • for any n ≥
0, assuming we know the law of β n , the law of β n +1 is the law of therandom variable P νi =1 β ( i ) n λ + P νi =1 β ( i ) n , where ν , β (1) n , β (2) n , ... are independent, ν has the law p and each β ( i ) n has the law β n . 17 . . . . . . λ = 0 . . . . λ = 1 0 0 . . . . . . λ = 1 . Figure 3: The apparent density of the conductance β , for p = p = p = and λ in { . , , . } .Using the preceding algorithm, after n iteration, we obtain the law of β n ( T ). Since β n ( T ) → β ( T ), almost surely, we also have convergence in law. Remark . The preceding discussion shows that the law of β is the greatest (for thestochastic partial order) solution of the recursive equation (9). In [14, Theorem 4.1],for λ = 1, the authors show that the only solutions to this recursive equation are theDirac measure δ and the law of β . However, their proof cannot be adapted to the moregeneral case λ ∈ (0 , m ). That is why, here, we had to choose for our initial measure, theDirac measure δ .For the numerical computations, we discretize the interval [0 ,
1] and apply the preced-ing algorithm with some fixed final value of n . See Figure 3 for an example of what onecan obtain with 100 iterations and a discretization step equal to 1 / ν X j =1 β ( e T [ j ]) = λ e β − e β , (13)where e β is an independent copy of β = β ( T ). Recalling that the constant C in (12) isthe expectation of κ ( β ), we obtain d λ = log( λ ) − E " log(1 − β ) β e βλ − β + e β − β e β E " β e βλ − β + e β − β e β . (14)From there, computing d λ from a dicretized approximation of the law of β is straight-forward.From [10], we know that d λ goes to E [log( ν )] (the almost sure dimension of thevisibility measure, see [12, Section 4]) as λ goes to 0, and to log m as λ goes to m .We also compute numerically the speed. Recall from [1], that the speed of the λ -biasedtransient random walk is given by ℓ λ = E " ( ν − λ ) β λ − β + P νj =1 β j E " ( ν + λ ) β λ − β + P νj =1 β j , . . . . . . . . . / / λ d λ . . . . . . . . . .
81 11 / λ ℓ λ Figure 4: The dimension and the speed of the λ -biased random walk on a Galton-Watsontree as functions of λ , for p = p = 1 / ν , β , β , . . . are independent and ν has law p , while for each i , β i has law β ( T ).Using first symmetry and then (13), one obtains E " ( ν ± λ ) β λ − β + P νj =1 β i = E (cid:16)P νj =1 β j (cid:17) ± λβ λ − β + P νj =1 β j = E λ (cid:16) e β ± β ∓ β e β (cid:17) λ − β + e β − e ββ , where we have denoted β = β and P νj =1 β i = λ e β/ (1 − e β ). Finally, we may express thespeed as ℓ λ = E " β e βλ − β + e β − β e β E " β + e β − β e βλ − β + e β − β e β . (15)We also recall that, in the case λ = 1, it was shown in [12] that the speed of the randomwalk equals ℓ = E (cid:20) ν − ν + 1 (cid:21) . We have made the numerical computations in two cases, the first one is when thereproduction law is given by p = p = 1 /
2, see fig. 4 and the second one is for p givenby p = p = p = 1 /
3, see fig. 5.These figures suggest that the speed and the dimension are indeed monotonic withrespect to λ . Furthermore, the speed looks convex, while the dimension seems to beneither convex nor concave. We generalize a model of trees with random lengths (or resistances) that can be foundin [4, Section 2] and [9, Section 2]. It appeared as the scaling limit of the sequence19 . . . . . . . . . . . . / λ d λ . . . . . . . . . . . .
81 15 / λ ℓ λ Figure 5: The dimension and the speed of the λ -biased random walk on a Galton-Watsontree as functions of λ , for p = p = p = 1 / T n /n ) n ≥ , where T n is a reduced critical Galton-Watson tree conditioned to survive atthe n th generation.In both [4] and [9], the marks have the law of the inverse of a uniform random variableon (0 , p = 1 in [4], whereas in [9] it is given by p k = k ≤ α Γ ( k − α ) k ! Γ (2 − α ) otherwise, (16)where α is a parameter in (1 , Γ is the standard Gamma function.Here, we assume that the marks are in (1 , ∞ ), and as before, that p = 0 and p < t be a marked leafless tree with marks in (1 , ∞ ). We associate to each vertex x in t , the resistance, or length, of the edge ( x ∗ , x ): r t ( x ) = γ t ( x ) − Y y ≺ x (cid:16) − γ t ( y ) − (cid:17) . Informally, the edge between the root and its parent has length γ t (ø) − ∈ (0 , t [1], t [2], . . . , t [ ν t (ø)] by (cid:0) − γ t (ø) − (cid:1) and wecontinue recursively see Figure 2. We run a nearest-neighbour random walk on the treewith transition probabilities inversely proportional to the lengths of the edges (furtherneighbours are less likely to be visited). To make this more precise, the random walk in t ∗ , associated to this set of resistances has the following transition probabilities: Q t ( x, y ) = x = ø ∗ and y = ø; γ t ( xi ) / (cid:16) γ t ( x ) − P ν t ( x ) j =1 γ t ( xj ) (cid:17) if y = xi for some i ≤ ν t ( x )( γ t ( x ) − / (cid:16) γ t ( x ) − P ν t ( x ) j =1 γ t ( xj ) (cid:17) if y = x ∗ ;0 otherwise.20hen we reindex a subtree, we also change the resistances to gain stationarity. For x ∈ t and y ∈ t [ x ], we define r t [ x ] ( y ) := r t ( xy ) Q z ≺ x (1 − γ t ( z )) − . This is consistent with the marks of the reindexed subtree t [ x ] and does not change theprobability transitions of the random walk inside this subtree. For x in t ∗ , let Q tx be aprobability measure under which the process ( Y n ) n ≥ is the random walk on t startingfrom x with probability transitions given by Q t . To prove that this walk is almostsurely transient, we use Rayleigh’s principle and compare the resistance of the wholetree between ø ∗ and infinity to the resistance of, say, the left-most ray. If, for n greateror equal to one, we denote by r n ( t ) the resistance in the ray between ø ∗ and the vertex x n := 11 · · · | {z } n times , we have that1 − r n ( t ) = (cid:16) − γ t (ø) − (cid:17) (cid:16) − γ t (1) − (cid:17) · · · (cid:16) − γ t ( x n ) − (cid:17) , (17)so the resistance of the whole ray is less or equal to 1. In particular, it is finite and sois the resistance of the whole tree. We denote by HARM Γ t the law of the exit ray of thisrandom walk. For x in t ∗ , let τ x = inf { k ≥ Y k = x } , with inf ∅ = ∞ and β ( t ) := Q t ø ( τ ø ∗ = ∞ ) . Applying equations (8) and (10) to this model, we obtain:
HARM Γ t ( i ) = γ t ( i ) β ( t [ i ]) P ν t (ø) i =1 γ t ( j ) β ( t [ j ]) , ∀ i ≤ ν t (ø) , (18) γ t (ø) β ( t ) = γ t (ø) P ν t (ø) j =1 γ t ( j ) β ( t [ j ]) γ t (ø) − P ν t (ø) j =1 γ t ( j ) β ( t [ j ]) . (19)Let φ ( t ) := γ t (ø) β ( t ) = γ t (ø) Q t ø ( τ ø ∗ = ∞ ) . (20)In fact, φ ( t ) is the effective conductance between ø ∗ and infinity. From the identity (17),the Rayleigh principle and the law of parallel conductances, whenever t has at least tworays, φ ( t ) >
1. Thus we can write φ ( t ) = h (cid:16) γ t (ø) , ν t (ø) X i =1 φ ( t [ i ]) (cid:17) , (21)with h ( u, v ) := uv/ ( u + v −
1) for all u and v in J := (1 , ∞ ).Now, let T be a (Γ , p )-Galton-Watson tree, where Γ almost surely belongs to (1 , ∞ ), p = 0 and p <
1, so that T almost surely has infinitely many rays.21 .2 Invariant measure and dimension drop for the natural distance We set for all u > κ ( u ) := E " h (cid:16) u, ν e T (ø) X j =1 φ ( e T [ j ]) (cid:17) = E u P ν e T (ø) j =1 φ ( e T [ j ]) u − P ν e T (ø) j =1 φ ( e T [ j ]) . (22)where e T is an independent copy of T . We will be able to use Theorem 3.2 if we canprove that κ ( φ ( T )) is integrable. To this end, one needs some information about thelaw of Γ and about p . The following criterion is certainly not sharp but it might sufficein some practical cases. For its proof, we rely on ideas from [4, Proposition 6]. Proposition 5.1.
Assume that there exist two positive numbers a and C such that forall numbers s in (1 , ∞ ) , P (Γ ≥ s ) ≤ Cs − a . Then, E [ φ ( T )] and E [ κ ( φ ( T ))] are finitewhenever one of the following conditions occurs:1. a > ;2. a = 1 and P k ≥ p k k log k < ∞ ;3. < a < and P k ≥ p k k − a < ∞ .Proof. From the fact that for all real numbers u and v greater that 1, h ( u, v ) < u , wededuce that E [ κ ( φ ( T ))] is finite as soon as E [ φ ( T )] is, and we also conclude in the firstcase.Let M be the set of all Borel probability measures on (1 , ∞ ]. For any distribution µ in M , let Ψ( µ ) be the distribution of h (Γ , P νi =1 X i ), where ν , Γ and X , X , . . . areindependent, each X i having distribution µ and ν having distribution p . To handle thecase where µ ( {∞} ) >
0, we define by continuity h ( u, ∞ ) = u for all u >
1. Consider forany s ∈ (1 , ∞ ), F µ ( s ) := µ [ s, ∞ ] , with F µ ( s ) = 1 if s ≤
1. On M , the stochastic partialorder (cid:22) is defined as follows: µ (cid:22) µ ′ if and only if there exists a coupling ( X, X ′ ) insome probability space, with X distributed as µ , X ′ distributed as µ ′ such that X ≤ X ′ almost surely. This is equivalent to F µ ≤ F µ ′ . From the identity h ( u, v ) − h ( u, v ′ ) = ( v − v ′ ) u ( u − u + v − u + v ′ − , (23)we see that Ψ is increasing with respect to the stochastic partial order.Let us denote by ϕ the distribution of φ ( T ) and by γ the distribution of Γ. SinceΨ( δ ∞ ) = γ and Ψ( ϕ ) = ϕ , we have ϕ (cid:22) Ψ n ( γ ) for all n ≥
1. We are done if we can showthat Ψ n ( γ ) has a finite first moment for some n ≥ µ be in M and s ∈ (1 , ∞ ), F Ψ( µ ) ( s ) = P Γ ≥ s, ν X i =1 X i ≥ s (cid:18) Γ − − s (cid:19)! ≤ P Γ ≥ s, ν X i =1 X i ≥ s ! = P (Γ ≥ s ) P ν X i =1 X i ≥ s ! = F γ ( s ) X k ≥ p k P k X i =1 X i ≥ s ! ≤ F γ ( s ) X k ≥ kp k F µ (cid:18) sk (cid:19) . (24)We may apply it to γ , to get Z ∞ F Ψ( γ ) ( s ) d s ≤ X k ≥ kp k Z k F γ ( s ) d s + Z ∞ k F γ ( s ) F γ (cid:18) sk (cid:19) d s ! = X k ≥ kp k Z k F γ ( s ) d s + k Z ∞ F γ ( s ) F γ ( ks ) d s ! . In the second case, where F γ ( s ) ≤ Cs − and P k ≥ p k k log k < ∞ , this is enough toconclude.In the third case, we need to play this game a little bit longer. Let N ≥ a ( N + 1) >
1. Notice that this implies that aN ≤
1. Iteratingon the inequality (24) and applying it to γ , we get F Ψ N ( γ ) ( s ) ≤ X k ,k ,...,k N ≥ k k · · · k N p k p k · · · p k N F γ (cid:16) s (cid:17) F γ (cid:16) sk (cid:17) F γ (cid:16) sk k (cid:17) · · · F γ (cid:16) sk k · · · k N (cid:17) . Hence, we may write an upper bound of R ∞ F Ψ N ( γ ) ( s ) d s as X k ,...,k N ≥ k · · · k N p k · · · p k N h I ( k ) + I ( k , k ) + · · · + I N ( k , . . . , k N )+ J ( k , . . . , k N ) i , where I ( k ) := Z k F γ ( s ) d s ≤ C − a k − a ;23or r between 2 and N , I r ( k , . . . , k r ) := Z k ...k r k ··· k r − F γ ( s ) F γ ( s/k ) · · · F γ (cid:0) s/ ( k · · · k r − ) (cid:1) d s = k · · · k r − Z k r F γ ( s ) F γ ( sk r − ) · · · F γ ( sk r − · · · k ) d s ≤ k · · · k r − C r log( k r ) k − a ( r − r − · · · k − a if r = N and aN = 1; k · · · k r − − ar C r k − arr k − a ( r − r − · · · k − a otherwise; ≤ e Ck − a . . . k − ar , where e C is the positive constant defined by e C = max ≤ r ≤ N ( C r / (1 − ar )) if aN < (cid:16) max ≤ r ≤ N − ( C r / (1 − ar )) , C N sup k ≥ (cid:0) k a − log( k ) (cid:1)(cid:17) if aN = 1.Finally, J ( k , . . . , k N ) := Z ∞ k ··· k N F γ ( s ) F γ ( s/k ) · · · F γ ( s/ ( k · · · k N )) d s ≤ C N +1 k − a · · · k − aN Z ∞ s − a ( N +1) d s = C N +1 a ( N + 1) − k − a · · · k − aN , by our assumption that a ( N + 1) > P k ≥ p k k − a < ∞ ensures that all the above sums are finite. Example . If the law of Γ − is uniform on (0 ,
1) (as in [4] and [9]), we have, for any s in (1 , ∞ ), P (Γ ≥ s ) = s − and the previous proposition shows that E [ κ ( φ ( T ))] is finiteif P k =1 p k k log k < ∞ . If the reproduction law is the same as in [9], that is, is given by(16), then by a well-known equivalent on gamma function ratios (see for instance [8]),we have p k = α Γ (2 − α ) Γ ( k − α ) Γ ( k + 1) ∼ k →∞ α Γ (2 − α ) k − − α , with α in (1 , P k ≥ p k k log( k ) is finite and so is E [ κ ( φ ( T ))].With some more knowledge of p and/or the law of Γ, it could be possible to describemore precisely the law of φ ( T ). See for instance [9, Proposition 5] or [4, Proposition 6].However, in general, it is often very difficult to establish properties (for instance, absolutecontinuity) of probability measures defined by distributional recursive equations like (21).We now apply Theorem 3.2 to our problem and prove that the dimension drop phe-nomenon occurs when the metric is the natural distance d U ∞ , defined by (1). Theorem 5.2.
Let T be a (Γ , p ) -Galton-Watson tree. Let φ ( T ) and κ be defined respec-tively by (20) and (22) . Assume that C := E [ κ ( φ ( T ))] is finite. Then, the probabilitymeasure of density C − κ ( φ ( T )) with respect to GW is invariant and ergodic with respectto the flow rule HARM Γ . he dimension of the measure HARM Γ T on ∂T with respect to d U ∞ equals almost surely C − E " log − Γ − ø − Γ − ø φ ( T ) ! κ ( φ ( T )) . (25) It is almost surely strictly less than log m unless both p and the law of Γ are degenerated.Proof. The first part of the theorem is a direct consequence of Theorem 3.2.We prove the formula for the dimension in the same way as in the previous theorem anduse the same notations. Write µ for the probability measure with density C − κ ( φ ( T ))with respect to GW . Then by formula (5), invariance of µ and equality (18) the dimen-sion of HARM Γ T equals almost surelydim d U∞ HARM Γ T = E µ " log 1 HARM Γ T (Ξ ) = E µ " log P ν T (ø) i =1 φ ( T [ i ]) φ ( T [Ξ ]) . From formula (19), we deduce that ν T (ø) X i =1 φ ( T [ i ]) = φ ( T )(1 − Γ − )1 − Γ − φ ( T ) , so that almost surely,dim HARM Γ T = E µ h log(1 − Γ − ) − log(1 − Γ − φ ( T )) + log( φ ( T )) − log( φ ( T [Ξ ])) i . As before, we need to prove that log( φ ( T )) − log( φ ( T [Ξ ])) is bounded from below byan integrable random variable to conclude. Using again formula (19), we get φ ( T ) φ ( T [Ξ ]) = 1 + φ ( T [Ξ ]) − P ν T (ø) i =1 , i =Ξ φ ( T [ i ])1 − Γ − + Γ − φ ( T [Ξ ]) + Γ − P ν T (ø) i =1 , i =Ξ φ ( T [ i ]) ≥ − Γ − + Γ − φ ( T [Ξ ]) . Hence, since 1 − Γ − ≤ − φ ( T [Ξ ]) = β ( T [Ξ ]) ≤
1, we have φ ( T ) φ ( T [Ξ ]) ≥ . To prove the dimension drop, i.e. the fact that almost surely dim
HARM Γ < log m ,we do not use the formula (25) since we know so little about the distribution of φ ( T ).Instead, we compare the flow rule HARM Γ to the uniform flow UNIF defined in Section 2.By Proposition 2.4 and Lemma 3.1 we only need to prove that if there exists a positivereal number K such that, for GW -almost every tree t , f W ( t ) = K × φ ( t ), then both thereproduction law and the mark law are degenerated.25nder this assumption, by the recursive equation (6) we have almost surely, φ ( T ) = 1 m ν T (ø) X i =1 φ ( T [ i ]) . Plugging it into the recursive equation (19), we first obtain that φ ( T ) = m Γ ø φ ( T )Γ ø + mφ ( T ) − , so that almost surely φ ( T ) = 1 m (( m − ø + 1) . In turn, using again (19), this implies that1 m [( m −
1) Γ ø + 1] = Γ ø P ν T (ø) i =1 1 m [( m − i + 1]Γ ø − P ν T (ø) i =1 1 m [( m − i + 1] . Now, if we denote by S the random variable P ν T (ø) i =1 1 m [( m − i + 1], elementary algebraleads to the second degree polynomial equation( m − + Γ ø (2 − m − S ) + S − , whose discriminant is equal to ( S − m ) . Hence, we always haveΓ ø = m + S − ± ( S − m )2( m − . We must choose the solution Γ ø = ( S − / ( m − ø = 1 m ν T (ø) X i =1 Γ i + 1 m − (cid:18) ν T (ø) m − (cid:19) . which, by independence of ν T (ø), Γ ø , Γ , Γ , . . . , imposes that both p and the law of Γare degenerated. Note that in the previous theorem, the dimension is computed with respect to the naturaldistance d U ∞ . This distance does not take into account the marks (Γ x ) x ∈ T , so we do notcompute the same dimension as in [4] and [9], where the distance between two pointsin the tree is the sum of all the resistances (or lengths) of the edges between these twopoints. 26o make this definition more precise, let us define, for x ∈ T , the Γ-height of x : | x | Γ := X y (cid:22) x (cid:16) Y z ≺ y (cid:16) − Γ − z (cid:17) (cid:17) Γ − y . We then have 1 − | x | Γ = Y y (cid:22) x (cid:16) − Γ − y (cid:17) . For two distinct rays η and ξ , letd Γ ( ξ, η ) := 1 − | ξ ∧ η | Γ . Notice that, for any rays ξ and η , and all integer n ≥
1, we have:d Γ ( ξ, η ) ≤ − | ξ n | Γ ⇐⇒ η ∈ [ ξ n ] T , (26)where we recall that [ ξ n ] t is the set of all rays whose ξ n is a prefix.We will compute the dimension of HARM Γ T with respect to this distance d Γ and showthat in this case too, we observe a dimension drop phenomenon, but we begin with moregeneral statements. We want to build a theory similar to [12, Sections 6 and 7] in oursetting of trees with recursive lengths with the length metric d Γ .We will need the following elementary lemma. Lemma 5.3.
Let f and g be two positive non-increasing functions defined on (0 , . Let ( r n ) be a decreasing sequence of positive numbers converging to . Assume that f ( r n ) g ( r n ) −−−→ n →∞ ℓ ∈ [0 , ∞ ) and f ( r n +1 ) f ( r n ) −−−→ n →∞ . Then, we have lim r ↓ f ( r ) /g ( r ) = ℓ .Proof. Let ε > n be large enough so that for all n ≥ n , f ( r n ) g ( r n ) f ( r n +1 ) f ( r n ) ≤ ℓ + ε and f ( r n +1 ) g ( r n +1 ) f ( r n ) f ( r n +1 ) ≥ ℓ − ε. Then, using the assumption that ( r n ) is decreasing to 0, for all r ≤ r n , there exists n ≥ n such that r n +1 < r ≤ r n and we have ℓ − ε ≤ f ( r n +1 ) g ( r n +1 ) f ( r n ) f ( r n +1 ) = f ( r n ) g ( r n +1 ) ≤ f ( r ) g ( r ) ≤ f ( r n +1 ) g ( r n ) = f ( r n ) g ( r n ) f ( r n +1 ) f ( r n ) ≤ ℓ + ε. Proposition 5.4 (dimension of a flow rule) . Let Θ be a GW -flow rule such that thereexists a Θ -invariant probability measure µ which is absolutely continuous with respectto GW . Then, almost surely, the probability measure Θ T is exact-dimensional on themetric space ( ∂T, d Γ ) , with deterministic dimension dim d Γ Θ T = dim d U∞ Θ T E µ [ − log(1 − Γ − ø )] . (27)27 roof. We first prove that, for GW -almost every tree t , for Θ t -almost every ray ξ ,lim n →∞ − log Θ t ( ξ n ) − log(1 − | ξ n | Γ ) = dim d U∞ Θ T E µ [ − log(1 − Γ − )] . (28)The numerator equals n − X k =0 − log Θ t ( ξ k +1 )Θ t ( ξ k ) , so, by the ergodic theorem (for non-negative functions), recalling that µ ⋉ Θ is ergodicand µ is equivalent to GW , for GW -almost every t and Θ t -almost every ξ ,1 n n − X k =0 − log Θ t ( ξ k +1 )Θ t ( ξ k ) −−−→ n →∞ E µ [ − log Θ T (Ξ )] = dim d U∞ Θ T ∈ (0 , log m ] . (29)For the denominator, we have, for any ξ in ∂t and any n ≥ n + 1 ( − log)(1 − | ξ n | Γ ) = 1 n + 1 n X i =0 − log(1 − γ t ( ξ i ) − ) . Again by the pointwise ergodic theorem, we have, for GW -almost every t and Θ t -almostevery ξ , 1 n + 1 ( − log) (cid:16) − | ξ n | Γ (cid:17) −−−→ n →∞ E µ [ − log(1 − Γ − )] ∈ (0 , ∞ ] . (30)Thus, the convergence (28) is proved.Now we want to show the following ( a priori stronger) statement: for GW -almostevery t and Θ t -almost every ξ ,lim r ↓ − log Θ t B ( ξ, r ) − log r = dim d U∞ Θ T E µ [ − log(1 − Γ − )] , (31)where B ( ξ, r ) is the closed ball of center ξ and radius r in the metric space ( ∂t, d Γ ).Let t be a marked tree and ξ be a ray in t such that (29) and (30) hold. Denote, for n ≥ r n = 1 − | ξ n | Γ . The sequence ( r n ) is positive, decreasing, and converges to 0 by(30). For r in (0 , f ( r ) = − log Θ t B ( ξ, r ) and g ( r ) = − log( r ). The functions f and g are positive and non-increasing. Furthermore, f ( r n +1 ) f ( r n ) = − log Θ t ( ξ n +1 ) − log Θ t ( ξ n ) = 1 + − log Θ t ( ξ n +1 )Θ t ( ξ n ) − log Θ t ( ξ n ) = 1 + − n log Θ t ( ξ n +1 )Θ t ( ξ n ) − n log Θ t ( ξ n ) . Using (29), we obtain lim n →∞ f ( r n +1 ) /f ( r n ) = 1, and conclude by the preceding lemma.We now associate to the random marked tree T an age-dependent process (in thedefinition of [2, chapter 4]). For any x ∈ T , let Λ x := − log (cid:0) − Γ − x (cid:1) be the lifetime ofparticle x . Informally, the root lives for time Λ ø , then simultaneously dies and gives birth28o ν T (ø) children who all have i.i.d. lifetimes Λ , Λ , . . . , Λ ν T (ø) , and then independentlylive and produce their own offspring and die, and so on. We are interested in the number Z u ( T ) of living individuals at time u >
0, that is Z u ( T ) := n x ∈ T : X y (cid:22) x ∗ Λ y < u ≤ X y (cid:22) x Λ y o . The
Malthusian parameter of this process is the unique real number α > E (cid:2) e − α Λ ø (cid:3) = 1 m . (32)We now assume that P ∞ k =1 p k k log k is finite. Under this assumption, we know from [7,Theorem 5.3] that there exists a positive random variable W Γ ( T ) such that E [ W Γ ( T )] =1 and almost surely, lim u →∞ e − uα Z u ( T ) = W Γ ( T ) . (33)By definition, we obtain the recursive equation W Γ ( T ) = e − α Λ ø ν T (ø) X j =1 W Γ ( T [ i ]) . (34)We now go back to our original tree with recursive lengths. Equations (32), (33) and(34) become E [(1 − Γ − ) α ] = 1 /m ; (35)lim ε → ε α Z − log( ε ) ( T ) = W Γ ( T ); (36) W Γ ( T ) = (1 − Γ − ) α ν T (ø) X j =1 W Γ ( T [ i ]) . (37)We define the GW -flow rule UNIF Γ by UNIF Γ T ( i ) = W Γ ( T [ i ]) P ν T (ø) j =1 W Γ ( T [ j ]) , ∀ ≤ i ≤ ν T (ø) . Proposition 5.5 (dimension of the limit uniform measure) . Assume that P k ≥ p k k log k is finite. Then, both the dimension of UNIF Γ T and the Hausdorff dimension of the bound-ary ∂T , with respect to the metric d Γ , are almost surely equal to the Malthusian parameter α .Proof. We can use Theorem 3.2, with h ( u, v ) = uv and the marks equal to ((1 − Γ − x ) α ) x ∈ T (or a direct computation) to show that the probability measure with den-sity W Γ with respect to GW is UNIF Γ -invariant. So we may apply Proposition 5.4 toobtain that the dimension of UNIF Γ with respect to the metric d Γ equalsdim d Γ UNIF Γ T = dim d U∞ UNIF Γ T E [ − log(1 − Γ − ) W Γ ( T )] . See also Section 3.4 of the preliminary Saint-Flour 2017 lecture notes by Remco Van Der Hoffstad. d U∞ UNIF Γ = E h(cid:16) − log W Γ ( T [Ξ ]) + log ν T (ø) X j =1 W Γ ( T [ j ]) (cid:17) W Γ ( T ) i = E h(cid:16) log((1 − Γ − ) − α ) + log W Γ ( T ) − log W Γ ( T [Ξ ]) (cid:17) W Γ ( T ) i = α E h − log(1 − Γ − ) W Γ ( T ) i , provided we can show that the term (log W Γ ( T ) − log W Γ ( T [Ξ ])) W Γ ( T ) is boundedfrom below by an integrable random variable.To prove this, we first use the recursive equation (37), to obtainlog W Γ ( T ) − log W Γ ( T [Ξ ]) = α log(1 − Γ − ) + log P ν t ø i =1 W Γ ( T [ i ]) W Γ ( T [Ξ ]) ! . Since Ξ is one of the children of the root, we havelog P ν t ø i =1 W Γ ( T [ i ]) W Γ ( T [Ξ ]) ! ≥ . Using again (37), we obtain (cid:16) log W Γ ( T ) − log W Γ ( T [Ξ ]) (cid:17) W Γ ( T ) ≥ α log(1 − Γ − )(1 − Γ − ) α ν T (ø) X i =1 W Γ ( T [ i ]) ≥ − e ν T (ø) X i =1 W Γ ( T [ i ]) , where, for the last inequality, we have used the fact that the minimum of the function x x α log( x ) on the interval (0 ,
1) is − / ( αe ). Since E [ P ν T (ø) i =1 W Γ ( T [ i ])] = m < ∞ ,this concludes the proof that dim d U∞ UNIF Γ = α E [ − log(1 − Γ − ) W Γ ( T )].We remark that E [log(1 − Γ − ) W Γ ( T )] is finite, because dim d U∞ UNIF Γ ≤ log m . Thuswe havedim d Γ UNIF Γ T = dim d U∞ UNIF Γ T E [ − log(1 − Γ − ) W Γ ( T )] = α E [log(1 − Γ − ) W ( T )] E [ − log(1 − Γ − ) W Γ ( T )] = α. We now know that the Hausdorff dimension of the boundary ∂T (with respect to d Γ )is almost surely greater or equal to α , so we just need the upper bound. Recall thedefinition ((2) and (3)) of the Hausdorff measures. We let A ε := { x ∈ T : 1 − | x | Γ ≤ ε < − | x ∗ | Γ } , whose number of elements is Z − log( ε ) ( T ). By the limit (36), we have H αε ( ∂T ) ≤ X x ∈ A ε (diam Γ [ x ] T ) α ≤ ε α Z − log( ε ) ( T ) a.s. −−−→ ε → W Γ ( T ) , so H α ( ∂T ) ≤ W Γ ( T ) < ∞ , which concludes the proof.30 roposition 5.6 (dimension drop for other flow rules) . Assume that P k ≥ p k k log k isfinite. Let T be a (Γ , p ) -Galton-Watson tree and Θ be a GW -flow rule such that Θ T and UNIF Γ T are not almost surely equal and there exists a Θ -invariant probability measure µ absolutely continuous with respect to GW . Then the dimension of Θ with respect to thedistance d Γ is almost surely strictly less than the Malthusian parameter α .Proof. First, we remark that if E µ [ − log(1 − Γ − )] is infinite, then the Hausdorff dimen-sion of Θ T with respect to the distance d Γ is almost surely equal to 0, so there is nothingto prove.So we assume that E µ [ − log(1 − Γ − )] is finite. Let Ξ be a random ray in ∂T withdistribution Θ T . Using formula (5) and conditioning on the value of Ξ givesdim d U∞ (Θ T ) = E µ h ν T (ø) X i =1 − Θ T ( i ) log(Θ T ( i )) i < E µ h ν T (ø) X i =1 − Θ T ( i ) log UNIF Γ T ( i ) i , where, for the strict inequality we have used Shannon’s inequality together with the fact(Lemma 2.1) that almost surely, Θ T is different from UNIF Γ T . This upper bound is equalto E µ h α ( − log(1 − Γ − )) + log W Γ ( T ) − log W Γ ( T [Ξ ]) i . Once again, all that is left to prove is that the last two terms are bounded from belowby an integrable random variable, and this is the case, becauselog W Γ ( T ) W Γ ( T [Ξ ]) ≥ α log(1 − Γ − ) , and by our assumption that E µ [log(1 − Γ − )] is finite. Cancelling out this term inequation (27), we finally obtain dim d Γ Θ T < α .Before we state and prove the main theorem of this subsection, we want to know whenthe dimension (with respect to d Γ ) of the harmonic measure equals 0. Lemma 5.7.
Let T be a (Γ , p ) -Galton-Watson marked tree. Assume that E [ κ ( φ ( T ))] and P k ≥ p k k are finite. Then, we have E [log(1 − Γ − ø ) κ ( φ ( T ))] < ∞ ⇐⇒ E [log(1 − Γ − ø )] < ∞ . Proof.
By Tonelli’s theorem, the definition of κ , and the associativity property of thefunction h , we have E [log(1 − Γ − ) κ ( φ ( T ))] = E " log(1 − Γ − ) h (cid:16) Γ ø , h (cid:0) ν T (ø) X i =1 φ ( T [ i ]) , ν e T (ø) X j =1 φ ( e T [ j ]) (cid:1)(cid:17) , where e T is a (Γ , p )-Galton-Watson tree, independent of T . Since for any u and v greaterthan 1, h ( u, v ) >
1, the direct implication is proved. For the reciprocal implication,31ecall that for u and v in (1 , ∞ ), h ( u, v ) < v , hence h (cid:16) Γ ø , h (cid:0) ν T (ø) X i =1 φ ( T [ i ]) , ν e T (ø) X j =1 φ ( e T [ j ]) (cid:1)(cid:17) < h (cid:16) ν T (ø) X i =1 φ ( T [ i ]) , ν e T (ø) X j =1 φ ( e T [ i ]) (cid:17) . The right-hand side of the previous inequality is integrable. Indeed, h (cid:16) ν T (ø) X i =1 φ ( T [ i ]) , ν e T (ø) X j =1 φ ( e T [ i ]) (cid:17) = (cid:0)P ν T (ø) i =1 φ ( T [ i ]) (cid:1)(cid:0) P ν e T (ø) j =1 φ ( e T [ j ]) (cid:1)P ν T (ø) i =1 φ ( T [ i ]) + P ν e T (ø) j =1 φ ( e T [ j ]) − ≤ ν T (ø) X i =1 φ ( T [ i ]) (cid:0)P ν e T (ø) j =1 φ ( e T [ j ]) (cid:1) φ ( T [ i ]) + P ν e T (ø) j =1 φ ( e T [ j ]) − E [ ν T (ø)] E [ κ ( φ ( T ))] , which is finite by assumption. Thus, using the fact thatΓ ø and h (cid:16) ν T (ø) X i =1 φ ( T [ i ]) , ν e T (ø) X j =1 φ ( e T [ i ]) (cid:17) are independent , we have E [log(1 − Γ − ) κ ( φ ( T ))] ≤ E [log(1 − Γ − )] E [ ν T (ø)] E [ κ ( φ ( T ))] , which proves the reciprocal implication of the lemma.Putting everything together, we finally obtain the dimension drop for the flow rule HARM Γ , with respect to the distance d Γ . Theorem 5.8.
Let T be a (Γ , p ) -Galton-Watson marked tree, with metric d Γ on itsboundary. Assume that both E [ κ ( φ ( T ))] and P k ≥ p k k log k are finite. Then, almostsurely, the flow HARM Γ T is exact-dimensional, of deterministic dimension dim d Γ HARM Γ ( T ) = E [log(1 − Γ − ø φ ( T )) κ ( φ ( T ))] E [log(1 − Γ − ø ) κ ( φ ( T ))] − , (38) except in the case E [ − log(1 − Γ − ø )] = ∞ , where it is . This deterministic dimensionis strictly less than the Malthusian parameter α (which is almost surely the Hausdorffdimension of the boundary ∂T with respect to the distance d Γ ) as soon as the mark lawand the reproduction law are not both degenerated.Proof. The formula for the Hausdorff dimension is just a rewriting using equations (27)and (25). All that is left to prove is that if there exists a positive real number K , such32hat, for GW -almost every tree t , W Γ ( t ) = K × φ ( t ), then both the mark law and thereproduction law are degenerated.We assume that the latter assertion holds, and we proceed similarly as in the proof ofTheorem 5.2. From the recursive equation (37) for W Γ , we deduce that almost surely ν T (ø) X i =1 φ ( T [ i ]) = (1 − Γ − ) − α φ ( T ) , (39)and plugging it into the recursive equation (19) for φ , we obtain that, almost surely, φ ( T ) = Γ ø (1 − (1 − Γ − ) α +1 ) . This implies that each φ ( T [ i ]) depends only on Γ i and ν T (ø) X i =1 φ ( T [ i ]) = Γ ø (1 − Γ − ) − α + 1 − Γ ø , so, by independence, P ν T (ø) i =1 φ ( T [ i ]) must be constant, which imposes that ν T (ø) mustbe constant (equal to m ) and that the law of φ ( T ) is degenerated. From (39), we nowsee that this implies that (1 − Γ − ) α = 1 /m and the law of Γ ø is degenerated.To conclude this work, we want to check that our formula (38) is consistent with theformula obtained in [4]. From now on, we work under the following hypotheses:1. the reproduction law is given by p = 1;2. the common law of the marks is the law of U − , where the law of U is uniform on(0 , Remark . The function denoted by t κ ( φ ( t )), in [4, Proposition 25] is slightlydifferent (it differs by a factor 1 /
2) from our function also denoted by κ ( φ ( t )).Under these hypotheses, Curien and Le Gall proved that the dimension (with respect tothe metric d Γ ) of the harmonic measure is almost surely (see [4, Proposition 4]):dim d Γ HARM Γ ( T ) = 2 E " log (cid:18) φ + φ φ (cid:19) φ e φ e φ + φ + φ − E (cid:20) φ φ φ + φ − (cid:21) , (40)where φ , φ and e φ are independent copies of φ ( T ). For short, we write U = Γ − , φ = φ ( T ), φ = φ ( T [1]) and φ = φ ( T [2]).We first show that E (cid:20) − log (cid:18) − U φ − U (cid:19) κ ( φ ) (cid:21) = 2 E " log (cid:18) φ + φ φ (cid:19) φ e φ e φ + φ + φ − . (41)Recall from the proof of Theorem 5.2 that, by stationarity, E [log( φ ) κ ( φ )] = E [log( φ ( T [Ξ ])) κ ( φ )] .
33y the recursive formula (21),1 − U φ − U = U − φ + φ + U − − φφ + φ , thus we obtain E (cid:20) log (cid:18) − U φ − U (cid:19) κ ( φ ) (cid:21) = E (cid:20) log (cid:18) φφ + φ (cid:19) κ ( φ ) (cid:21) = E (cid:20) log (cid:18) φ ( T [Ξ ]) φ + φ (cid:19) κ ( φ ) (cid:21) = E (cid:20) φ φ + φ log (cid:18) φ φ + φ (cid:19) κ ( φ ) + φ φ + φ log (cid:18) φ φ + φ (cid:19) κ ( φ ) (cid:21) = 2 E (cid:20) φ φ + φ log (cid:18) φ φ + φ (cid:19) κ ( φ ) (cid:21) , by symmetry. Let e T be a (Γ , p )-Galton-Watson tree such that the mark of the root is U − , and e T [1] and e T [2] are independent of T [1] and T [2]. Write e φ for the conductanceof e T and e φ i = φ ( e T [ i ]) for i = 1 ,
2. By Tonelli’s theorem and the definition of κ , we have E (cid:20) log (cid:18) − U φ − U (cid:19) κ ( φ ) (cid:21) = 2 E (cid:20) φ φ + φ log (cid:18) φ φ + φ (cid:19) h (cid:16) h (cid:16) U − , φ + φ (cid:17) , e φ + e φ (cid:17)(cid:21) = 2 E (cid:20) φ φ + φ log (cid:18) φ φ + φ (cid:19) h (cid:16) h (cid:16) U − , e φ + e φ (cid:17) , φ + φ (cid:17)(cid:21) = 2 E (cid:20) φ φ + φ log (cid:18) φ φ + φ (cid:19) h (cid:16) e φ, φ + φ (cid:17)(cid:21) = 2 E " φ φ + φ log (cid:18) φ φ + φ (cid:19) e φ ( φ + φ ) e φ + φ + φ − , where, between the first and the second line, we have used the associativity and thesymmetry of the function h . The proof of (41) is complete.Now, we want to show that E [ − log(1 − U ) κ ( φ )] = E (cid:20) φ φ φ + φ − (cid:21) . (42)Here, we rely heavily on the fact that U is uniform on (0 , g : [1 , ∞ ) → R + such that g ( x ) and g ′ ( x ) are both o ( x a )for some a in (0 , ∞ ), we have E [ g ( φ + φ )] = E [ φ ( φ − g ′ ( φ )] + E [ g ( φ )] . (43)As before, let φ , φ , e φ and e φ be independent copies of φ ( T ), independent of U . Let ψ : (1 , ∞ ) → (1 , ∞ ) be defined by ψ ( x, y, z ) = h ( x, h ( y, z )) = xyzxy + yz + xz − x − y − z + 1 .
34y definition of κ , we have E [ − log(1 − U ) κ ( φ )] = E h − log(1 − U ) ψ (cid:16) U − , φ + φ , e φ + f φ (cid:17)i . For x , y , z in (1 , ∞ ), let ψ ( x, y, z ) = ψ ( x, y, z ) + y ( y − ∂ y ψ ( x, y, z ) = x y z ( xy + xz + yz − x − y − z + 1) . Reason conditionally on U , e φ and e φ and apply the identity (43) to the function y ψ ( x, y, z ), to obtain E [ − log(1 − U ) κ ( φ )] = E h − log(1 − U ) ψ (cid:16) U − , φ , e φ + e φ (cid:17)i . Playing the same game again, we obtain E [ − log(1 − U ) κ ( φ )] = E h − log(1 − U ) ψ (cid:16) U − , φ , e φ (cid:17)i , with the function ψ defined by ψ ( x, y, z ) = ψ ( x, y, z ) + z ( z − ∂ z ψ ( x, y, z )= ( xyz ) " xyz ( xy + xz + yz − x − y − z + 1) − xy + xz + yz − x − y − z + 1) . Fix y and z in (1 , ∞ ) and let, for u in (0 , ψ ( u ) = ψ ( u − , y, z )= y z " yz [( yz + 1 − y − z ) u + ( y + z − − yz + 1 − y − z ) u + ( y + z − = ( a + b ) (cid:20) a + b )( au + b ) − au + b ) (cid:21) , with a = ( yz + 1 − y − z ) and b = ( y + z − Z − log(1 − u ) ψ ( u ) d u = a + bb = yzy + z − , so that, by independence of U , φ and e φ , E h − log(1 − U ) ψ (cid:16) U − , φ , f φ (cid:17)(cid:12)(cid:12)(cid:12) φ , e φ i = φ e φ φ + e φ − , which completes the proof of (42), and the verification of the consistency of formula (40)with (38). 35 eferences [1] Elie Aïdékon, Speed of the biased random walk on a Galton-Watson tree , Probab.Theory Related Fields (2014), no. 3-4, 597–617. MR 3230003[2] K. B. Athreya and P. E. Ney,
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