Moments of Moments and Branching Random Walks

We calculate, for a branching random walk Xn(l)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_n(l)$$\end{document} to a leaf l at depth n on a binary tree, the positive integer moments of the random variable 12n∑l=12ne2βXn(l)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2^{n}}\sum _{l=1}^{2^n}e^{2\beta X_n(l)}$$\end{document}, for β∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in {\mathbb {R}}$$\end{document}. We obtain explicit formulae for the first few moments for finite n. In the limit n→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document}, our expression coincides with recent conjectures and results concerning the moments of moments of characteristic polynomials of random unitary matrices, supporting the idea that these two problems, which both fall into the class of logarithmically correlated Gaussian random fields, are related to each other.


Moments of Moments: Characteristic Polynomials of Random Matrices
In recent years there has been significant progress towards understanding the value distribution of the maximum of the logarithm of the characteristic polynomial of a random unitary matrix and of related log-correlated processes [1][2][3][4][5][6]11,[17][18][19][20][21][22][24][25][26][27]29]. Let the maximum value of P N (A, θ) around the unit circle. It was conjectured in [19,20] that where the law of the fluctuating term m N (A) was postulated to be the same as that of the sum of two independent Gumbel random variables in the limit N → ∞. The leading order of (3) was verified by Arguin et al. [2], and Paquette and Zeitouni [26] determined (3) to subleading order. At the time of writing, the strongest result in the literature is due to Chhaibi et al. [11], who proved tightness 1 of the family of random variables 2 The maximum conjecture (3) was motivated by a heuristic analysis in [20] of the random variable the 2βth moment of the absolute value of the characteristic polynomial with respect to the uniform measure on the unit circle. In particular, determining the moments of Z N (A, θ) with respect to Haar measure on the unitary group is central to the analysis, and such an average is referred to as representing the moments of moments of P N (A, θ). Specifically, the moments of moments are defined by where the external average E[·] is with respect to the Haar measure on U (N ) 3 . In [20] it was conjectured that, as N → ∞, MoM U (N ) (k, β) is given asymptotically by where G(s) is the Barnes G-function, and c(k, β) is some (unspecified) function of the moment parameters k, β. For integer k, β, it was proved in [9] that MoM U (N ) (k, β) is a polynomial in the matrix size, N , of degree k 2 β 2 − k + 1, in line with (7).
Using a Riemann-Hilbert analysis, Claeys and Krasovsky [12] computed MoM U (N ) (2, β) for Re(β) > −1/4, and connected c (2, β) to a solution of a Painlevé equation. By so doing, they verified (7) for k = 2 and all Re(β) > −1/4. Fahs [16] subsequently extended this approach 4 to general k ∈ N, although he did not determine c(k, β) for k > 2. Additionally, Claeys and Krasovsky, and Fahs, also determined that the behaviour at the critical point kβ 2 = 1 (still for k ∈ N) is of the form for some positive coefficient α(k, β) as N → ∞ (see [12,16] for further details). One of the key ideas that underpins much of the progress outlined above is that the Fourier series representing log P N (A, θ) exhibits a hierarchical structure typical of problems associated with logarithmically correlated Gaussian fields. This structure is exemplified by the branching random walk. Understanding this connection is currently a focus of research in the area. Our aim here is to examine it in the context of the moments of moments by calculating the quantity in the theory of the branching random walk that is analogous to (6). Specifically, we will show that the analogue of the moments of moments for the branching random walk is asymptotically described by a formula that is the direct analogue of (7). Additionally, the fact that log P N (A, θ) has a central limit theorem [23] for large N is important to our analysis.
We also remark in passing that the characteristic polynomials of random unitary matrices play an important role in modelling the value distribution of the Riemann zeta-function on its critical line [23]. There are analogues of the conjectures (3) and (7) for the zeta function [19,20]. In the latter case, the integer moments of moments can be calculated using the shifted moment conjecture of [13,14]; see [10]. There has again been a good deal of progress in proving the conjecture corresponding to (3) using the analogue for the zeta function of the hierarchical structure exemplified by the branching random walk [1][2][3][4][5][6]21,22,24], and so we see our results for the branching random walk as being of interest in the number theoretical context as well.

Moments of Moments: the Branching Random Walk
Take a binary tree of depth n, and a choice of leaf l. Load to each branch in the tree an independent centred Gaussian random variable with variance 1 2 log 2. We write for the branching random walk from root to l where Y m (l) ∼ N (0, 1 2 log 2) are the branch weightings, see Fig. 1. Note that X n (l) ∼ N 0, n 2 log 2 (10) and that the distribution of X n (l) does not depend on the choice of leaf l (nor does the distribution of Y m (l) depend on the level m nor the leaf l), however including both labels will become useful later. Similarly, it will be important to record the points at which concurrent paths through the tree diverge.
Definition 1 Take two leaves l 1 , l 2 of a binary tree of depth n. The last common ancestor of l 1 , l 2 , denoted by lca(l 1 , l 2 ) is the furthest node from the root that has both l 1 and l 2 as descendants. The last common ancestor of k leaves is the furthest node from the root with all k leaves as descendants. Figure 2 shows an example involving three leaves on a tree of depth n = 4. It will be important for our purposes to keep track of the level of the last common ancestor. Hence, we also define the last common level lcl(l 1 , . . . , l k ) to be the level of lca(l 1 , . . . , l k ). For example, in Fig. 2, lcl(l 1 , l 2 , l 3 ) = 0 and lcl(l 1 , l 2 ) = 2. Fig. 1 An example of a random walk X 4 (l) = Y 1 (l) + · · · + Y 4 (l) on a binary tree of depth n = 4, from root to leaf l. The weightings Y j (l) are independent, centred Gaussian random variables with variance 1 2 log 2 lca(l 1 , l 2 , l 3 ) lca(l 1 , l 2 ) l 1 l 2 l 3

Fig. 2
A binary tree of depth 4 with three leaves l 1 , l 2 , l 3 highlighted. The last common ancestor of l 1 , l 2 is lca(l 1 , l 2 ). The last common ancestor of all three (and also lca(l 2 , l 3 ) and lca(l 1 , l 3 )) is the root node. The paths are differentiated by dashed and dotted lines As a process, {X n (l), l ∈ {1, . . . 2 n }} is log-correlated (see for example [1]). It is natural, therefore, to investigate the associated partition function 5 or moment generating function 1 2 n 2 n l=1 e 2β X n (l) = 1 2 n 2 n l=1 e 2β n m=1 Y m (l) (11) where, as in (9), Y m (l) ∼ N (0, 1 2 log 2) and are independent. In particular, we are interested in the moments of the partition function (11), where the expectation in (12) is with respect to the Gaussian random variables. These are the moments of moments for the branching random walk. They are the analogues of (6).

Results
By calculating the moments of moments (12), we are able to recover an asymptotic result of the form (13), albeit with different leading order coefficients. Explicitly, we prove the following.
For small values of k, one can calculate exact and explicit formulae for the moments of moments; we provide such examples for k = 1, . . . , 5 in Appendix A. Such moments were also considered by Derrida and Spohn [15] for general branching weightings and in continuous settings. In the discrete setting, they compute the first few low moments (which agree with the first few explicit examples computed in Appendix A) and establish a connection to the KPP equation in the continuous setting. Additionally, it is natural to ask if the leading order coefficient ρ(k, β) for kβ 2 < 1 in (14) could take the form f (β) k Γ (1 − kβ 2 ) for some function f , in line with Fyodorov-Bouchaud [17] or Remy-Zhu [28] formulae for related problems in the same regime (see also (13)). Such a statement does not appear to hold here.
Furthermore, we are able to establish that for integer values of the moment parameters the branching moments of moments are polynomials.
Thus, the branching moments of moments exhibit asymptotic behaviour identical to that of the random matrix moments of moments, once the identification N = 2 n is made.
The remainder of this section details the key ideas necessary for the proof of Theorem 1 and Corollary 1. Small cases of the moments of moments are explicitly calculated.

Structure of Proof
Establishing the statement of Theorem 1 in the simplest instance, k = 1, follows from a moment generating function calculation. Recall that in the random matrix case, MoM U (N ) (1, β) has an exact (finite N ) expression: for Re(β) > − 1 2 , see [23]. For β ∈ N, the right hand side of (15) simplifies to 0≤i, j≤β−1 As N → ∞, where c(1, β) is the ratio of Barnes G-functions appearing in the first regime in (13). The asymptotic behaviour (17) for integer β follows from (16); for general β it was determined by Keating and Snaith [23]. As is consistent with (13), for k = 1 there is no phase transition as β varies. The equivalent case of k = 1 for the branching moments of moments (see (12)) requires calculating the following moment In terms of the binary tree, this can be interpreted as 'loading' the root with one particle. Consequently, each summand is the contribution from that particle passing through the tree and ending at leaf l. Since the Y j (l) are independent between each level of the binary tree, we have . By making the identification N = 2 n , the branching moments of moments exhibit the same asymptotic growth (although with a different leading order coefficient, and no lower order terms) as (17).
When k ≥ 2, one has the additional difficulty of the paths X n (l j ) no longer being independent typically. In order to introduce the key ideas of the proof for general k, it is instructive also to calculate explicitly the case for k = 2. This case is the first where a phase change can be seen as β varies, and the calculation demonstrates how to handle the dependence between paths. For ease of notation, henceforth we write for the branching moments of moments in (12) Additionally, since the case for β = 0 is trivial, henceforth we assume β = 0. Thus, take β = 0 and consider (20) for k = 2, where λ := lcl(l 1 , l 2 ). As up to level λ the paths are identical, and thereafter independent, we may rewrite (22) as This follows because 2 4βλ is the contribution from the joined paths, and 2 λ is the number of choices of lca(l 1 , l 2 ) given lcl(l 1 , l 2 ) = λ. At this point observe that the expectation on the right hand side of (23) is the same as calculated for the first moment of moments, except on a tree of depth n − λ − 1 (and with an additional step prior to the new root node). Hence we proceed inductively, Thus, the general method for proving Theorem 1, and hence Corollary 1, will follow via strong induction. In order to demonstrate the three different asymptotic regimes, we examine (25) for different values of β. If 2β 2 > 1 then as n → ∞. Instead, if 2β 2 < 1, then as n → ∞ Finally, if 2β 2 = 1, then using (24) we have Hence, as n → ∞, at 2β 2 = 1, In the next section, we prove Theorem 1 and Corollary 1 using the techniques presented in this section. In particular, we make liberal use of the iterative properties of the binary tree underpinning (20).

Proof Details
We proceed by strong induction. Recall that we write for β ∈ R and k ∈ N In Sect. 2.2 we established the base cases of MoM n (1, β), MoM n (2, β). As the case β = 0 is trivial, here and henceforth β = 0. Now assume for all j < k, and k ≥ 2, that where ρ( j, β), σ ( j, β), τ ( j, β) are the positive leading order coefficients (depending on the moment parameters j and β) of MoM n ( j, β) in each of the three regimes 7 .
Throughout we write Σ for a sum without the diagonal term. We now consider the kth case, where in the last two lines λ := lcl(l 1 , . . . , l k ). At the initial separation on level λ, j particles will split in one direction, and k − j in the other for j ∈ {1, . . . , k − 1}. Thereafter, one is essentially the dealing with two subtrees of depth n − λ − 1, with j particles on one and k − j on the other. Note also that there are 2 λ choices for lca(l 1 , . . . , l k ) given lcl(l 1 , . . . , l k ) = λ, and that since only off-diagonal terms appear in the sum, λ ∈ {0, . . . , n − 1}. Let Y ∼ N (0, 1 2 log 2), then MoM n (k, β) To complete the proof of Theorem 1, we determine the asymptotic behaviour of (40) by separately considering the ranges kβ 2 < 1, kβ 2 = 1, and kβ 2 > 1. It transpires that we will need to further partition the case kβ 2 > 1, for more details see Sect. 3.3.

Range: |ˇ| > 1 √ k
In this range, kβ 2 > 1 so we expect MoM n (k, β) to grow like 2 (k 2 β 2 −k+1)n . As previously mentioned, it will be necessary to further partition the span of values. The three divisions 8 are:

Range: 1
. . , k}. We first record a useful rewriting of (40) due to the symmetric nature of the summands. If k is odd then Instead if k is even, then In either case, MoM n ( j, β) is paired with MoM n (k − j, β). Hence, we first consider the case of 2 < m ≤ k 2 and split the sums at m in order to apply (33). Then, (56) becomes Instead if k is even and 2 < m ≤ k 2 MoM n (k, β) = 2 k 2 β 2 −k+1 If k 2 < m ≤ k and k odd, If k 2 < m ≤ k and k even, Now, applying (33) to (58), for odd k and 2 < m ≤ k Hence, in order to show that MoM n (k, β) grows like 2 (k 2 β 2 −k+1)n , we need to establish both that 2 j(β 2 ( j+1−2k)+1)n is subleading, for j = 1, . . . , m − 1, as well as 2 since lβ 2 > 1 for l = m, . . . , k. Hence as required. Now take j = m, . . . , k−1 2 . By assumption, lβ 2 > 1 for l ≥ m, and m ≤ k−1 2 . We therefore have If k is even, but still 2 < m ≤ k 2 then entirely similarly to the odd k case we find thus, the same arguments hold for the first two sums of (68) as for the odd k case. We are done provided additionally that the contribution from 2 (1− k 2 β 2 2 )n is subleading. As kβ 2 > 1 and k > 2, we therefore have k 2 β 2 > 2, and so Moving to the case where k−1 2 < m ≤ k and k odd, and applying (33) to (60) we find Since kβ 2 > 1 by assumption, we have that the terms in the second sum of (70) do grow asymptotically like 2 (k 2 β 2 −k+1)n . To confirm that this is also true for the terms in the first sum, we check that j(β 2 ( j + 1 − 2k) + 1 < 0 for j = 1, . . . , k − m and k−1 2 < m ≤ k. This is true by the same arguments as above (see (64) and (65)).
To conclude we consider the case of even k and k 2 < m ≤ k, we find here that thus we employ the arguments of (70). This concludes the proof for |β| ∈ 1

√ m
Assume that |β| = 1 √ m for some m ∈ {2, . . . , k − 1} (the cases m = 1, k were dealt with in Sects. 3.3.2 and 3.2 respectively). We revisit the techniques used in Sect. 3.3.1. Beginning with the odd k case, we separate (56) around the mth term to find Since the sums over j = 1, . . . , m − 1 and j = m + 1, . . . , k−1 2 can be handled using the arguments of Sect. 3.3.1, we only need to determine how the middle term (i.e. j = m) grows asymptotically. Using (33) we examine Computing the sum over λ, we have Combining (77) and (78) gives Thus, the result follows once it is established that 2(m − k) + 1 is negative. By assumption m ∈ {2, . . . , k − 1}, so 2(k − m) > 1 and thus we conclude. The case for even k follows from precisely the same reasoning, except in the case where m = k 2 . Assume k ≥ 4 is even 9 and m = k 2 , then by (57) we have As above, the sum has already been handled by the argument in Sect. 3.3.1, thus we only consider the penultimate term. Since k ≥ 4, when we apply (33), we may replace By assumption, k 2 β 2 = 1, so k 2 β 2 − k + 1 simplifies to k + 1. Thus, the result follows if 2 2n is subleading to 2 (k+1)n . This follows since k ≥ 4.

Proof of Corollary 1
Assume k, β ∈ N. When k = 1, 2, the result follows from the computation in Sect. 2.2, see (18), (19) and (25). There it is shown that thus both are polynomials in 2 n of degree k 2 β 2 − k + 1. Proceeding inductively, assume that for j < k, and that α j r j are the appropriate coefficients for the polynomial MoM n ( j, β). Then, using (40), MoM n (k, β) Thus, MoM n (k, β) is a sum of polynomials in 2 n . Since we determined in Sect. 3.3 that when kβ 2 > 1, MoM n (k, β) grows like 2 (k 2 β 2 −k+1)n , we hence have shown that MoM n (k, β) is a polynomial in 2 n of this degree. This completes the proof of Corollary 1.
The second equality follows by splitting the paths before and after the last common level λ := lcl(l 1 , l 2 , l 3 ). Note that at the splitting point, there are 2 λ choices for lca(l 1 , l 2 , l 3 ) given λ, and hence 3 1 choices for which paths are paired, and which is the single path. The third equality follows by substituting in (25). Hence, calculating the sum over λ, we find MoM n (3, β) Thus, if 3β 2 > 1 then If instead 3β 2 < 1 then Finally, if 3β 2 = 1 then using (94) we find Thus, as n → ∞ Hence, overall as n → ∞,   Observe that in all cases computed, in the limit n → ∞, the leading order coefficient of MoM n (k, β) fails to be analytic in β. Figures 3a, b, 4a, b plot the leading order coefficients of MoM n (k, β) for k = 2, 3, 4, 5 as β varies.