Geometry of Random Cayley Graphs of Abelian Groups
aa r X i v : . [ m a t h . P R ] F e b Geometry of Random Cayley Graphs of Abelian Groups
Jonathan Hermon Sam Olesker-Taylor
Abstract
Consider the random Cayley graph of a finite Abelian group G with respect to k generatorschosen uniformly at random, with 1 ≪ log k ≪ log | G | . Draw a vertex U ∼ Unif( G ).We show that the graph distance dist( id , U ) from the identity to U concentrates at aparticular value M , which is the minimal radius of a ball in Z k of cardinality at least | G | ,under mild conditions. In other words, the distance from the identity for all but o ( | G | ) of theelements of G lies in the interval [ M − o ( M ) , M + o ( M )]. In the regime k & log | G | , we showthat the diameter of the graph is also asymptotically M . In the spirit of a conjecture of Aldousand Diaconis [1], this M depends only on k and | G | , not on the algebraic structure of G .Write d ( G ) for the minimal size of a generating subset of G . We prove that the order ofthe spectral gap is | G | − /k when k − d ( G ) ≍ k and | G | lies in a density-1 subset of N or when k − d ( G ) ≍ k . This extends, for Abelian groups, a celebrated result of Alon and Roichman [4].The aforementioned results all hold with high probability over the random Cayley graph. Keywords: typical distance, diameter, spectral gap, relaxation time, random Cayley graphs
MSC 2020 subject classifications:
Contents ≪ k ≪ log | G |
83 Typical Distance: k ≍ log | G |
124 Typical Distance: k ≫ log | G |
205 Diameter 226 Spectral Gap 237 Open Questions and Conjectures 28References 29
Jonathan Hermon Sam [email protected], math.ubc.ca/ ∼ jhermon/ [email protected], mathematicalsam.wordpress.comUniversity of British Columbia, Vancouver, Canada Department of Mathematical Sciences, University of Bath, UKSupported by EPSRC EP/L018896/1 and an NSERC Grant Supported by EPSRC Grants 1885554 and EP/N004566/1The vast majority of this work was undertaken while both authors were at the University of Cambridge Introduction and Statement of Results
We analyse geometric properties of a
Cayley graph of a finite group; the focus is on Abeliangroups. The generators of this graph are chosen independently and uniformly at random. Precisedefinitions are given in § G be a finite group, let k be an integer (allowed todepend on G ) and denote by G k the Cayley graph of G with respect to k independently anduniformly random generators. We consider values of k with 1 ≪ log k ≪ log | G | for which G k isconnected with high probability (abbreviated whp ), ie with probability tending to 1 as | G | grows.For an Abelian group G , write d ( G ) for the minimal size of a generating subset of G . · Typical Distance.
Draw U ∼ Unif( G ). We show that the law of the graph distance betweenthe identity and U concentrates. The leading order term in this typical distance depends onlyon k and | G | when 1 ≪ k ≪ log | G | / log log log | G | and k − d ( G ) ≍ k or k ≫ log | G | . · Diameter.
For k ≍ log | G | under mild conditions on the group and k ≫ log | G | for any Abeliangroup, we show that the diameter concentrates at the same value as the typical distance. · Spectral Gap.
For any 1 ≪ k . log | G | with k − d ( G ) ≍ k , we determine the order of thespectral gap of the random walk on the random Cayley graph.Introduced by Aldous and Diaconis [1], there has been a great deal of research into these randomCayley graphs. Motivation for this model and an overview of historical work is given in § Cayley graphs are either directed or undirected; we emphasise this by writing G + k and G − k ,respectively. When we write G k or G ± k , this means “either G − k or G + k ”, corresponding to the undir-ected, respectively directed, graphs with generators chosen independently and uniformly at random.Conditional on being simple, G + k is uniformly distributed over the set of all simple degree- k Cayley graphs. Up to a slightly adjusted definition of simple for undirected Cayley graphs, ourresults hold with G k replaced by a uniformly chosen simple Cayley graph of degree k ; see § G N ) N ∈ N of finite groups with | G N | → ∞ as N → ∞ . For ease ofpresentation, we write statements like “let G be a group” instead of “let ( G N ) N ∈ N be a sequence ofgroups”. Likewise, the quantities d ( G ) and, of course, k appearing in the statements all correspondto sequences, which need not be fixed (or bounded) unless we explicitly say otherwise. In the samevein, an event holds with high probability (abbreviated whp ) if its probability tends to 1.We use standard asymptotic notation: “ ≪ ” or “ o ( · )” means “of smaller order”; “ . ” or O ( · )”means “of order at most”; “ ≍ ” means “of the same order”; “ h ” means “asymptotically equivalent”. Our first result concerns typical distance in the random Cayley graph.
Definition A.
For a group G , k ∈ N and β ∈ (0 , , define the β -typical distance D G k ( β ) via B G k ( R ) := { x ∈ G (cid:12)(cid:12) dist G k ( id , x ) ≤ R } and D G k ( β ) := min (cid:8) R ≥ (cid:12)(cid:12) |B G k ( R ) | ≥ β | G | (cid:9) . Informally, we show that the mass (in terms of number of vertices) concentrates at a thin ‘slice’,or ‘shell’, consisting of vertices at a distance M ± o ( M ) from the origin, with M explicit.Investigating this typical distance for G k when k diverges with | G | was suggested to us byBenjamini [6]. Previous work concentrated on fixed k , ie independent of | G | ; see § G , write d ( G ) for the minimal size of a generating subset of G and m ∗ ( G ) := max (cid:8) min j ∈ [ d ] m j (cid:12)(cid:12) ⊕ d Z m j is a decomposition of G (cid:9) . Theorem A.
Let G be an Abelian group. The following convergences are in probability as | G | → ∞ .Consider ≪ k ≪ log | G | ; suppose that k − d ( G ) ≍ k and d ( G ) ≪ log | G | / log log | G | . Write D + := | G | /k / (2 e ) and D − := | G | /k /e . For all β ∈ (0 , , we have D ± G k ( β ) / D ± → P . Consider k h λ log | G | with λ ∈ (0 , ∞ ) ; suppose that d ( G ) ≤ log | G | / log log | G | and m ∗ ( G ) ≫ . There exists a constant α ± λ ∈ (0 , ∞ ) so that, for all β ∈ (0 , , we have D ± G k ( β ) / ( α ± λ k ) → P . Consider k ≫ log | G | with log k ≪ log | G | ; write ρ := log k/ log log | G | so that k = (log | G | ) ρ .(We allow ρ ≫ .) For all β ∈ (0 , , we have D ± G k ( β ) / (cid:0) ρρ − log k | G | (cid:1) → P . In all three cases, the implicit lower bound holds deterministically and for all Abelian groups.
Remark A.1.
We establish the concentration of typical distance via three distinct approaches, in § § §
4. Conceptually, all involve sizes of lattice balls and drawing elements uniformly fromballs. A precise statement for each approach is given, as is an outline of the proof. In summary,Theorem A is a direct consequence of Theorems 2.2, 3.2 and 4.2; see also Hypotheses A to C. △ Remark A.2.
For smaller k , namely 1 ≪ k ≪ p log | G | / log log log | G | , we can relax k − d ( G ) ≍ k to k − d ( G ) ≫
1. In order to generate the group, we certainly need k ≥ d ( G ), by definition. In manycases k − d ( G ) ≫ △ Remark A.3.
Interesting is how we prove this theorem. It is common in mixing time proofs to usegeometric properties of the graph, such as expansion or distance properties. We do the opposite: weuse mixing techniques to prove this geometric result. This is in the same spirit as [22]; see § △ We can extend our proof to consider the diameter , ie the maximal distances between pairs ofvertices in the graph, in the regime k & log | G | . For a graph H , denote by diam H its diameter.Our first diameter result gives concentration. A more refined statement is given in Theorem 5.1. Theorem B.
Let G be an Abelian group. The following convergences are in probability as | G | → ∞ .Consider k h λ log | G | with λ ∈ (0 , ∞ ) ; suppose that d ( G ) ≤ log | G | / log log | G | and m ∗ ( G ) ≫ . Let α ± λ ∈ (0 , ∞ ) be the constant from Theorem A. We have diam G ± k / ( α ± λ k ) → P . Consider k ≫ log | G | with log k ≪ log | G | ; write ρ := log k/ log log | G | so that k = (log | G | ) ρ .(We allow ρ ≫ .) We have diam G ± k / ( ρρ − log k | G | ) → P . The upper bound holds for all groups.In both cases, the implicit lower bound holds deterministically and for all Abelian groups.
Remark B.
For any Cayley graph H one has D H ( ) ≤ diam H ≤ D H ( )+1. (Note that ( x , ..., x ℓ )is a path in G ( z ) if and only if ( x ℓ , ..., x ) is a path in G ( z − ) for any generators z .) So the typicaldistance and diameter are always equivalent up to constants. Theorem B gives conditions underwhich they are asymptotically equivalent whp for random Cayley graphs.Combining Theorem A with [11, Theorem A] shows that t mix ( G k ) ≍ (diam G k ) /k whp when k − d ( G ) ≍ k & log | G | . One can also consider non-Abelian groups; see [12, Theorem E]. △ Our next diameter result shows, in a well-defined sense, that, amongst all groups, when k − log | G | ≍ k with log k ≪ log | G | , the group Z d gives rise to the largest typical diameter. Definition.
For two random sequences α := ( α N ) N ∈ N and β := ( β N ) N ∈ N of reals, we say that α ≤ β whp up to smaller order terms if there exist non-random sequences ( γ N ) N ∈ N and ( δ N ) N ∈ N of reals with δ N → as N → ∞ such that ( { α N ≤ (1 + δ N ) γ N } ) N ∈ N and ( { (1 − δ N ) γ N ≤ β N } ) N ∈ N both hold whp. We say that α h β whp if α ≤ β and β ≤ α whp up to smaller order terms. We now define the candidate radius which we show is an upper bound for diam G k whp.3 efinition C. Write R ( k, n ) for the minimal R ∈ N with (cid:0) kR (cid:1) ≥ n . We now state our second diameter result. A more refined statement is given in Theorem 5.2.
Theorem C.
Let G be an arbitrary group. Suppose that k − log | G | ≍ k and ≪ log k ≪ log | G | .Then diam G k ≤ R ( k, | G | ) up to smaller order terms whp; further, if H := Z d , then diam H k h R ( k, d ) = R ( k, | H | ) whp. (The limit is as | G | → ∞ .) This gives a quantitative sense in which Z d is the group giving rise to the largest diameter. Corollary C.
For all diverging d and n with n ≤ d and all groups G of size n , if k − log n ≍ k and log k ≪ log n , then diam G k ≤ diam H k where H := Z d up to smaller order terms whp. Wilson [31, Conjecture 7] conjectures an analogous statement for mixing times. When restrictedto nilpotent groups, we prove an extension of this conjecture in [11, Theorems C and D].
Our next result concerns the spectral gap and relaxation time of the random Cayley graph.
Definition D.
Consider a reversible Markov chain with (real) eigenvalues λ ≥ λ ≥ · · · ≥ λ n ≥ − of its transition matrix. The usual , respectively absolute , spectral gap is defined as γ := min i =1 { − λ i } = 1 − λ , respectively γ ∗ := min i =1 { − | λ i |} = 1 − max {| λ | , | λ n |} ; the usual , respectively absolute , relaxation time is defined as t rel := 1 /γ, respectively, t ∗ rel := 1 /γ ∗ . The spectral gap or relaxation time of a graph , is that of the simple random walk on the graph. It is classical that under reversibility in continuous-time the spectral gap asymptotically determ-ines the exponential rate of convergence to equilibrium, whereas in discrete-time it is determinedby the absolute spectral gap; see [19, §
12 and § z = [ z , ..., z k ] with z , ..., z k ∈ G ,write G + ( z ), respectively G − ( z ), for the undirected, respectively directed, Cayley graph with re-spect to generators z , ..., z k . We do not require k → ∞ as | G | → ∞ .A more refined statement than the one given below is given in Theorem 6.1. Theorem D.
There exists a positive constant c so that, for all Abelian groups G , all k and allmultisets of generators z of size k , we have t ∗ rel (cid:0) G − ( z ) (cid:1) ≥ t rel (cid:0) G − ( z ) (cid:1) ≥ c | G | /k . For all δ > , there exists a constant C δ > so that, for all Abelian groups G , if k ≥ (2+ δ ) d ( G ) , then P (cid:0) t ∗ rel ( G − k ) ≤ C δ | G | /k (cid:1) ≥ − C δ − k/C δ . Further, for all ε ∈ (0 , , there exists a density- (1 − ε ) subset A ⊆ N so that if | G | ∈ A then thecondition k ≥ (2 + δ ) d ( G ) can be relaxed to k ≥ (1 + δ ) d ( G ) ; the constants now also depend on ε . The method of proof for this result is rather different to our previous results and also somewhatdifferent to those used by others to study the spectral gap of random Cayley graphs; see § In this subsection, we give a fairly comprehensive account of previous work on distance metricson and spectral gap of random Cayley graphs; we compare our results with existing ones. We alsomention, where relevant, other results which we have proved in companion papers; see also § .3.1 Motivation: Random Cayley Graphs and Cutoff for Random Walks In their seminal paper, Aldous and Diaconis [1, 2] considered random walks on random
Cayleygraphs. Diaconis [9] gave the following (paraphrased) motivation.
Erd˝os, when considering classes of mathematical objects, often combinatorial or graphtheoretic, would often ask, “What does a typical object in this class ‘look like’ ?” If anobject is chosen uniformly at random, are there natural properties which hold whp?It is then natural to ask, “How does a typical random walk on a group behave?”
This lead Aldous and Diaconis [1, 2] to consider the set of all Cayley graphs of a given group G with k generators. Drawing such a Cayley graph uniformly at random corresponds to choosing generators Z , ..., Z k ∼ iid Unif( G ), conditional on giving rise to a simple graph; see § Aldous and Diaconis [1, 2] made the following (informal) conjecture: regardless of the particulargroup G , provided k ≫ log | G | , the random walk on the random Cayley graph exhibits cutoff whpat a time which depends only on k and | G | . This was established for Abelian groups by Douand Hildebrand [10, 16]; in [12], we provide a counter-example using unit upper triangular matrixgroups. For more details, see our companion articles [11, 12] where we study cutoff extensively.The point of the Aldous–Diaconis conjecture is that certain statistics should be “independent ofthe algebraic structure of the group”, ie only depend on G through | G | . The current article showshow very related statements to those above hold when “cutoff” is replaced by “typical distance”.Namely, we give conditions under which the typical distances concentrates on a value that dependsonly on k and | G | ; see Theorems 2.2, 3.2 and 4.2. Previous work on distance metrics (detailed below) had concentrated on the case where thenumber of generators k is a fixed number. The results establish (non-degenerate) limiting laws.This restricts the (sequences of) groups which can be studied; eg, in order for it to be even possibleto generate the group—never mind having independent, uniform generators do so whp—one needs d ( G ) ≤ k ≍
1. We discuss generation of groups further in [13, § § k → ∞ as | G | → ∞ and we establish concen-tration of the observables. This allows us to consider a much wider range of groups, in particularwith d ( G ) diverging with | G | . This line of enquiry was suggested to us by Benjamini [6]Amir and Gurel-Gurevich [5] studied the diameter of the random Cayley graph of cyclic groupsof prime order. They prove (for fixed k ) that the diameter is order | G | /k ; see [5, Theorems 1and 2]. They conjecture that the diameter divided by | G | /k converges in distribution to somenon-degenerate distribution as | G | → ∞ ; see [5, Conjecture 3].Marklof and Str¨ombergsson [24] consider, as a consequence of a quite general framework, thediameter of the random Cayley graph of Z n with respect to a fixed number k of random generat-ors, for a random n , without any primality assumption. They derive distributional limits for thediameter, the average distance (defined with respect to various L p metrics) and the girth. Theydetermine limit distributions for each of these, and in some cases derive explicit formulas.Shapira and Zuck [30] build on the framework of Marklof and Str¨ombergsson [24], again forfixed k ; they are able to consider non-random n , as well as Abelian groups of arbitrary (fixed)rank, instead of only cyclic groups. In particular, they verify the conjecture of Amir and Gurel-Gurevich [5, Conjecture 3]; they additionally work with average distance and girth.Lubetzky and Peres [22] derive an analogous typical distance result for n -vertex, d -regularRamanujan graphs: whp all but o ( n ) of the vertices lie at a distance log d − n ± O (log log n ); theyestablish this by proving cutoff for the non-backtracking random walk at time log d − n .Related work on the diameter of random Cayley graphs, including concentration of certainmeasures, can be found in [20, 29]. 5he Aldous–Diaconis conjecture for mixing can be transferred naturally to typical distance: themass should concentrate at a distance M , where M can be written as a function only of k and | G | ;ie there is concentration of mass at a distance independent of the algebraic structure of the group.In [12, Theorem E] we consider typical distance analogously to this paper; there the underlyinggroup is a non-Abelian matrix group. In contrast with the Abelian groups in Theorem A, the M for these non-Abelian groups cannot be written as a function only of k and | G | . Hough [17, Theorem 1.1] showed that, for any prime p , the relaxation time of the randomwalk on any Cayley graph of Z p with respect to an arbitrary set of k generators is order at least | Z p | /k = p /k , provided that k ≤ log p/ log log p . Using a different approach, we extend Hough’sresult, removing the restrictions on p and k and considering general Abelian groups; see Theorem D.This extends, in the Abelian set-up, a celebrated result of Alon and Roichman [4, Corollary 1],which asserts that, for any finite group G , the random Cayley graph with at least C ε log | G | randomgenerators is whp an ε -expander, provided C ε is a sufficiently large (in terms of ε ). (A graph isan ε -expander if its isoperimetric constant is bounded below by ε ; up to a reparametrisation,this is equivalent to the spectral gap of the graph being bounded below by ε .) There has beena considerable line of work building upon this general result of Alon and Roichman. (Pak [26]proves a similar result.) Their proof was simplified and extended, independently, by Loh andSchulman [21] and Landau and Russell [18]; both were able to replace log | G | by log D ( G ), where D ( G ) is the sum of the dimensions of the irreducible representations of the group G ; for Abeliangroups D ( G ) = | G | . A ‘derandomised’ argument for Alon–Roichman is given by Chen, Moore andRussell [7]. Both [7, 18] use some Chernoff-type bounds on operator valued random variables.Christofides and Markstr¨om [8] improve these further by using matrix martingales and proving aHoeffding-type bound on operator valued random variables. They also improved the quantificationfor C ε , showing that one may take C ε := 1 + c ε with c ε → ε →
0; this means that, whp, thegraph is an ε -expander whenever k ≥ (1 + c ε ) log D ( G ) and c ε → ε →
0. They also generaliseAlon–Roichman to random coset graphs. The proofs use tail bounds on the (random) eigenvalues.Alon and Roichman [4, Theorem 2] also specifically consider Abelian groups. There they do acalculation directly in terms of the eigenvalues, rather than using a probabilistic tail bound.In [11, Theorem E], we analyse for a nilpotent group G the spectral gap of G k in the regime k ≍ log | G | : we show that G k is an expander whp under a certain natural condition on k . In thespecial case of Abelian groups, this becomes k − d ( G ) ≍ k ; the general condition is k − d ( G ) ≍ k where G is the direct product of the quotients in the lower central series of G . Hence in this set-upit extends Theorem D by removing the restriction that | G | lies in a large-density subset of N .There are some fairly standard ways in which one can get bounds on the (usual) spectral gapof a Markov chain. The first is to look at the mixing time. For c > ε ∈ (0 , π c min ], we have t mix ( ε ) ≍ t rel log(1 /ε ) , where n is the size of the state space of the (reversible) Markov chain, π min is the minimal valueof the invariant distribution of the Markov chain and c is a constant; see, eg, [19, Theorem 20.6and Lemma 20.11]. Thus, if one can bound the mixing time at level π c min then one can bound therelaxation time. This method is used by Alon and Roichman [4] and Pak [26]; we use it in [11].Another method is to obtain a tail estimate on the value of a random eigenvalue; one can thenuse the union bound to say that all (non-unitary) eigenvalues are at most some fixed value, whichin turn lower bounds the spectral gap (ie upper bounds the relaxation time).All these references consider the regime k ≍ log | G | ; our results also apply when 1 ≪ k ≪ log | G | .From a technical perspective, in order to obtain failure probability via a large deviation boundfor a random eigenvector of O (1 / | G | ), one needs k & log | G | . The purpose of this is to carry outa union bound over the | G | eigenvalues; see, eg, [8]. Likewise, arguments that bound the 1 / | G | c mixing time, for some constant c , in terms of some generator getting picked once (cf [28]) cannotwork unless k & log | G | . As such, to consider 1 ≪ k ≪ log | G | , a different approach is needed. Westill use a union bound, but instead of asking for an error probability O (1 / | G | ) for each eigenvalue,we group together eigenvalues according to a certain gcd and bound the error for each group.6 .4 Additional Remarks Consider a finite group G . Let Z be a multisubset of G . We consider geometric properties,namely through distance metrics and the spectral gap, of the Cayley graph of (
G, Z ); we call Z the generators . The undirected , respectively directed , Cayley graph of G generated by Z , denoted G − ( Z ), respectively G + ( Z ), is the multigraph whose vertex set is G and whose edge multiset is (cid:2) { g, g · z } | g ∈ G, z ∈ Z (cid:3) , respectively (cid:2) ( g, g · z ) | g ∈ G, z ∈ Z (cid:3) . We focus attention on the random
Cayley graph defined by choosing Z , ..., Z k ∼ iid Unif( G );when this is the case, denote G + k := G + ( Z ) and G − k := G − ( Z ). While we do not assume that theCayley graph is connected (ie, Z may not generate G ), in the Abelian set-up the random Cayleygraph G k is connected whp whenever k − d ( G ) ≫
1; see [13, Lemma 8.1].The graph depends on the choice of Z . Sometimes it is convenient to emphasise this; we use asubscript, writing P G ( z ) ( · ) if the graph is generated by the group G and multiset z . Analogously, P G k ( · ) stands for the random law P G ( Z ) ( · ) where Z = [ Z , ..., Z k ] with Z , ..., Z k ∼ iid Unif( G ). The directed Cayley graph G + ( z ) is simple if and only if no generator is picked twice, ie z i = z j for all i = j . The undirected Cayley graph G − ( z ) is simple if in addition no generator is the inverseof any other, ie z i = z − j for all i, j ∈ [ k ]. In particular, this means that no generator is of order 2,as any s ∈ G of order 2 satisfies s = s − —this gives a multiedge between g and gs for each g ∈ G .Abusing terminology, we relax the definition of simple Cayley graphs to allow order 2 generators,ie remove the condition z i = z − i for all i .Given a group G and an integer k , we are drawing the generators Z , ..., Z k independently anduniformly at random. It is not difficult to see that the probability of drawing a given multisetdepends only on the number of repetitions in that multiset. Thus, conditional on being simple, G k is uniformly distributed on all simple degree- k Cayley graphs. Since k ≪ p | G | , the probability ofsimplicity tends to 1 as | G | → ∞ . So when we say that our results hold “whp (over Z )”, we couldequivalently say that the result holds “for almost all degree- k simple Cayley graphs of G ”.Our asymptotic evaluation does not depend on the particular choice of Z , so the statistics inquestion depend very weakly on the particular choice of generators for almost all choices. In manycases, the statistics depend only on G via | G | and d ( G ). This is a strong sense of ‘universality’. This paper is one part of an extensive project on random Cayley graphs. There are three mainarticles [11, 12, 14] (including the current one [14]), a technical report [13] and a supplementarydocument [15].
Each main article is readable independently.
The main objective of the project is to establish cutoff for the random walk and determiningwhether this can be written in a way that, up to subleading order terms, depends only on k and | G | ; we also study universal mixing bounds, valid for all, or large classes of, groups. Separately,we study the distance of a uniformly chosen element from the identity, ie typical distance, and thediameter; the main objective is to show that these distances concentrate and to determine whetherthe value at which these distances concentrate depends only on k and | G | .[11] Cutoff phenomenon (and Aldous–Diaconis conjecture) for general Abelian groups; also, fornilpotent groups, expander graphs and comparison of mixing times with Abelian groups.[14] Typical distance, diameter and spectral gap for general Abelian groups.[12] Cutoff phenomenon and typical distance for upper triangular matrix groups.[13] Additional results on cutoff and typical distance for general Abelian groups.[15] Deferred technical results mainly regarding random walk on Z and the volume of lattice balls.7 .4.4 Acknowledgements This whole random random Cayley graphs project has benefited greatly from advice, discussionsand suggestions of many of our peers and colleagues. We thank a few of them specifically here. · Justin Salez for reading this paper in detail and giving many helpful and insightful commentsas well as stimulating discussions ranging across the entire random Cayley graphs project. · Itai Benjamini for discussions on typical distance. · Evita Nestoridi and Persi Diaconis for general discussions, consultation and advice. ≪ k ≪ log | G | This section focusses on concentration of distances from the identity in the random Cayleygraph of an Abelian group when 1 ≪ k ≪ log | G | . (Subsequent sections deal with k & log | G | .)The main result of the section is Theorem 2.2.The outline of this section is as follows: · § · § · § · § · § To start the section, we recall the typical distance statistic.
Definition 2.1.
Let H be a graph and fix a vertex ∈ H . For r ∈ N , write B H ( r ) for the r -ball inthe graph H , ie B H ( r ) := { h ∈ H | d H (0 , h ) ≤ r } , where d H is the graph distance in H . Define D H ( β ) := min (cid:8) r ≥ (cid:12)(cid:12) |B H ( r ) | ≥ β | H | (cid:9) for β ∈ (0 , . When considering sequences ( k N , G N ) N ∈ N of integers and Abelian groups, abbreviate D N ( β ) := D G N ([ Z ,...,Z kN ]) ( β ) where Z , ..., Z k N ∼ iid Unif( G N ) . Finally, considering such sequences, we define the candidate radius for the typical distance: D + N := k N | G N | /k N / (2 e ) and D − N := k N | G N | /k N /e for each N ∈ N . As always, if we write D N , then this is either D + N or D − N according to context. We show that, whp over the graph (ie choice of Z ), this statistic concentrates. The result willbe valid for all Abelian groups, under some conditions on k in terms of G . Further, the value atwhich the typical distance concentrates, which will be D ± above, depends only on k and | G | . Thisis in agreement with the spirit of the Aldous–Diaconis conjecture. Hypothesis A.
The sequence ( k N , G N ) N ∈ N satisfies Hypothesis A if the following hold: lim inf N →∞ | G N | = ∞ , lim sup N →∞ k N / log | G N | = 0 , lim inf N →∞ ( k N − d ( G N )) = ∞ and k N − d ( G N ) − k N ≥ k N log | G N | + 2 d ( G N ) log log k N log | G N | for all N ∈ N . We study 1 ≪ k ≪ log | G | here. In Remark 2.3 below, we give some sufficient conditionsfor Hypothesis A to hold. Throughout the proofs, we drop the subscript- N from the notation, egwriting k or n = | G | , considering sequences implicitly. Write D k ( β ) for the β -typical distance of G k .We now state the main theorem of this section.8 heorem 2.2. Let ( k N ) N ∈ N be a sequence of positive integers and ( G N ) N ∈ N a sequence of finite,Abelian groups; for each N ∈ N , define Z ( N ) := [ Z , ..., Z k N ] by drawing Z , ..., Z k N ∼ iid Unif( G N ) .Suppose that ( k N , G N ) N ∈ N satisfies Hypothesis A. Then, for all β ∈ (0 , , we have D ± N ( β ) / D ± N → P (in probability) as N → ∞ . Moreover, the implicit lower bound holds deterministically, ie for all choices of generators, and forall Abelian groups, ie Hypothesis A need not be satisfied—we just need lim sup N k N / log | G N | = 0 . Remark 2.3.
Write n := | G | . Any of the following conditions imply Hypothesis A:1 ≪ k . p log n/ log log log n and k − d ≫ ≪ k . p log n and k − d ≫ log log k ;1 ≪ k ≪ log n/ log log log n and k − d ≥ δk for some suitable δ = o (1); d ≪ log n/ log log log n and k − d ≍ k. △ As remarked after the summarised statement (in Remark A.3), when considering properties ofthe random walk on a graph, such as the mixing time, geometric properties of the graph are oftenderived and used. In a reversal of this, we use knowledge about the mixing properties of a suitablerandom variable to derive a geometric result. We explain this in a little more detail now.For the lower bound, for any Cayley graph G of an Abelian group of degree k , (trivially) wehave |B G ( R ) | ≤ | B k ( R ) | , where B k ( R ) is the k -dimensional lattice ball of radius R . If | B k ( R ) | ≪ n, then immediately |B G ( R ) | ≪ n, and so D G ( β ) ≥ R for all β ∈ (0 , n .Consider now the upper bound. We fix some target radius kL and draw W , ..., W k ∼ iid Geom(1 /L ) in the directed case. For the undirected case, we add to each W i a uniform sign.It is well-known that the law of W := ( W , ..., W k ) given k W k = R is uniform on the L sphere ofradius R . Since the k W k = P k | W i | is an iid sum, it concentrates around its mean, ie kL , when kL ≫
1. So this is roughly like drawing uniformly from a sphere of radius kL , except that we havethe added benefit that the coordinates W , ..., W k are (unconditionally) independent.We can then interpret W i as the number of times which generator i is used in getting from theidentity to W · Z . We show that W · Z is well-mixed whp when kL is slightly larger than the targetradius. Now, if the law of W · Z is mixed in TV and k W k ≤ kL (1 + δ ) whp, then the law of W · Z conditional on k W k ≤ kL (1 + δ ) is also mixed in TV. Thus, using the concentration of k W k , wededuce that a proportion 1 − o (1) of vertices x ∈ G can be written as x = w · Z for some w with k w k h kL ; this gives a path of length at most kL from the identity to x .We show this mixing estimate via a (modified) L argument, where W is conditioned to be‘typical’, namely we define a set W and condition that W ∈ W . The most important part isto bound the probability that two independent copies of W are equal conditional on both beingin W ; this must be o (1 /n ). Since k W k concentrates and W is uniform on the sphere of thisradius, we need to choose L so that the sphere of radius kL has volume slightly more than n . Inhigh dimensions—here we consider balls in k ≫ W uniformly from a ball of radius kL . However,the lack of independence between the coordinate causes difficulties, in particular in Lemma 2.13below. We thus use this vector of geometrics as a proxy for the uniform distribution, but with thekey property that the coordinates are independent. Z k We desire an R ± so that | B ± k ( R ± ) | ≈ n , where B ± k ( R ) is the lattice ball of radius R , ie B − k ( R ) := (cid:8) w ∈ Z k (cid:12)(cid:12) k w k ≤ R (cid:9) and B + k ( R ) := (cid:8) w ∈ Z k + (cid:12)(cid:12) k w k ≤ R (cid:9) . efinition 2.4. Set ω := max { (log k ) , k/n / (2 k ) } . Note that ≪ ω ≪ k ≪ log n . Define R ± := inf (cid:8) R ∈ N (cid:12)(cid:12) | B k ( R ) | ≥ ne ω (cid:9) . The following lemma controls the size of balls. Its proof is given in [15, § E]; see in particular[15, Lemmas E.2a and E.3a] where the index q corresponds to a type of L q lattice balls; take q := 1to recover the usual L lattice balls here. Recall D ± from Definition 2.1. Lemma 2.5.
Assume that ≪ k ≪ log n . For all ξ ∈ (0 , , we have |R − D | / D ≪ and (cid:12)(cid:12) B k (cid:0) D (1 − ξ ) (cid:1)(cid:12)(cid:12) ≪ n. From the results in § Proof of Lower Bound in Theorem 2.2.
Let ξ ∈ (0 ,
1) and set R := R (1 − ξ ). Since the un-derlying group is Abelian, applying Lemma 2.5, we have |B k ( R ) | ≤ | B k ( R ) | ≪ n. Hence, for all β ∈ (0 ,
1) and all Z , we have D k ( β ) ≥ R = R (1 − ξ ), asymptotically in n . The argument given here is in a similar vein to that of [11, § ε > L := (1 + 3 ε ) R /k .Draw W = ( W i ) k ∼ Geom(1 /L ) ⊗ k ; later, we condition on k W k ≤ Lk . Here the geometricrandom variables have support { , , ... } . Define χ := ( χ i ) k as follows: in the undirected case, χ i ∼ iid Unif( {± } ); in the directed case, χ i := 1 for all i . Set S := ( χW ) · Z where χW := ( χ i W i ) k .Define W ′ and χ ′ as independent copies of W and χ , respectively; set S ′ := ( χ ′ W ′ ) · Z .In [11, § W was ‘typical’ in aprecise sense. There we were interested in the law of the random walk; the introduction of typicalitywas a tool to study this, for establishing mixing bounds for the random walk. Here, somewhat inreverse, we can choose which random variable we study. Definition 2.6.
Abbreviate L := L (1 − log k/ √ k ) . Define W := (cid:8) w ∈ Z k + (cid:12)(cid:12) L + 1 ≤ k w k /k ≤ L, max i w i ≤ L log k (cid:9) . When W and W ′ are independent copies, write typ := { W, W ′ ∈ W} . Lemma 2.7 (Typicality) . We have P ( W ∈ W ) ≍ and hence P ( typ ) ≍ . Proof.
We consider the three parts of typicality separately: · the lower bound on k W k holds with probability 1 − o (1) by Chebyshev’s inequality; · the upper bound on k W k holds with probability bounded away from 0 by Berry–Esseen; · the upper bound on max i W i holds with probability 1 − o (1) by the union bound.We control the L distance between S conditional on W ∈ W and the uniform distribution. Proposition 2.8.
Suppose that Hypothesis A is satisfied. Then E (cid:0)(cid:13)(cid:13) P G k (cid:0) S ∈ · | W ∈ W (cid:1) − Unif( G ) (cid:13)(cid:13) (cid:1) = o (1) , where we recall that P G k ( · ) is the random law corresponding to the random Cayley graph G k . We now have all the ingredients to prove the upper bound on typical distance.10 roof of Upper Bound in Theorem 2.2.
Let W have the law of W conditional on W ∈ W . ByProposition 2.8, the L distance between S := W · Z and Unif( G ) is o (1) whp. Thus the support S of S is a proportion 1 − o (1) of the vertices whp. In particular, there is a path of length at most Lk from id to all vertices in S whp, as k W k ≤ Lk by definition of typicality. Hence D k ( β ) ≤ Lk = (1+3 ε ) R whp. Applying Lemma 2.5 then gives ( D k ( β ) − D ) / D ≤ ε whp.The remainder of this subsection is devoted to proving Proposition 2.8. We have E (cid:0)(cid:13)(cid:13) P G k (cid:0) S ∈ · | W ∈ W (cid:1) − Unif( G ) (cid:13)(cid:13) (cid:1) = n P (cid:0) S = S ′ | typ (cid:1) − , recalling that χ ′ and W ′ are independent copies of χ and W , respectively, and S ′ := ( χ ′ W ′ ) · Z .First we control the probability that χW = χ ′ W ′ ; in this case we necessarily have S = S ′ . Lemma 2.9.
We have P ( χW = χ ′ W ′ | typ ) = o (1 /n ) . Proof.
Recall that L := L (1 − log k/ √ k ). Consider the directed case first, ie χ = 1 = χ ′ . Then P (cid:0) W = W ′ , typ (cid:1) ≤ P w : k w k ≥ k ( L +1) P (cid:0) W = w = W ′ (cid:1) = P w : k w k ≥ k ( L +1) P (cid:0) W ′ = w (cid:1) Q k P (cid:0) W i = w i (cid:1) = P w : k w k ≥ k ( L +1) P (cid:0) W ′ = w (cid:1) Q k L − (1 − L − ) w i − = P w : k w k ≥ k ( L +1) P (cid:0) W ′ = w (cid:1) · L − k (1 − L − ) k w k − k ≤ L − k (1 − L − ) kL = (cid:0) L − (1 − L − ) L (1 − √ log k/k ) (cid:1) k ≤ ( eL ) − k exp (cid:0)p k log k (cid:1) ≤ n − e − εk/ , with the final inequality using the fact that L + ≥ (1+2 ε ) n /k /e , using Lemma 2.5. In the undirectedcase, we also need χ = χ ′ , which happens with probability 2 − k , and is independent of ( W, W ′ ).Hence the same inequality holds with the event { W = W ′ } replaced by { χW = χ ′ W ′ } , recallingthat L − h L + . Finally, P ( typ ) ≍ . Thus Bayes’s rule combined with the above calculation gives P (cid:0) χW = χ ′ W ′ | typ (cid:1) ≤ n − e − εk ≪ /n. The following lemma describing the distribution of v · Z for a given v ∈ Z k is crucial. Lemma 2.10.
For all v ∈ Z k with gcd( v , ..., v k , n ) = γ , we have v · Z ∼ Unif( γG ) . We thus now need to control | γG | . Lemma 2.11.
For all Abelian groups G and all γ ∈ N , we have | G | / | γG | ≤ γ d ( G ) . These two lemmas were used in [11, § V := χW − χ ′ W ′ and g := gcd( V , ..., V k , n ) . Corollary 2.12.
We have n P (cid:0) V · Z = 0 , V = 0 | typ (cid:1) . E (cid:0) g d ( V = 0) | typ (cid:1) . Proof.
The conditioning does not affect Z . The corollary follows from Lemmas 2.10 and 2.11. Lemma 2.13.
Given Hypothesis A, we have E ( g d ( V = 0) | typ ) = 1 + o (1) . roof. Each coordinate of V is unimodal and symmetric about 0. From this we can deduce that P (cid:0) V ∈ γ Z | V = 0 (cid:1) ≤ /γ, as in [11, Lemma 2.14]. The probability of V = 0 is roughly 1 / (2 L ) ≍ n − /k ; in particular, it is atmost 3 n − /k . The coordinates are independent. Since P ( typ ) ≍ , we thus have P (cid:0) g = γ | typ (cid:1) . (cid:0) /γ + 3 /n /k (cid:1) k . By typicality, g ≤ L log k ≤ n /k log k . Hence, summing over γ , we obtain E (cid:0) g d ( V = 0) | (cid:1) . P n /k log kγ =1 γ d (cid:0) /γ + 3 /n /k (cid:1) k . We handle almost exactly the same sum in [11, Corollary 2.15]. Hypothesis A here is designedprecisely to control this sum; it is identical to [11, Hypothesis A]. There the 3 /n /k part is replacedwith 2 /n /k , but exactly the same arguments apply showing that the sum is 1 + o (1).Proposition 2.8 now follows immediately from Lemmas 2.9 and 2.13 and Corollary 2.12. Proof of Proposition 2.8.
By Lemmas 2.9 and 2.13 and Corollary 2.12, we have n P (cid:0) S = S ′ | typ (cid:1) ≤ n P (cid:0) V = 0 | typ (cid:1) + n P (cid:0) V · Z = 0 , V = 0 | typ (cid:1) ≤ n P (cid:0) V = 0 | typ (cid:1) + E (cid:0) g d ( V = 0) | typ (cid:1) = 1 + o (1) . k ≍ log | G | This section focusses on concentration of distances from the identity in the random Cayley graphof an Abelian group when k ≍ log | G | . (The previous section dealt with 1 ≪ k ≪ log | G | and thenext deal with k ≫ log | G | .) The main result of the section is Theorem 3.2; see also Hypothesis B.The outline of this section is as follows: · § · § · § · § · § · § · § L -type graph distances to L q -type. To start the section, we recall the typical distance statistic.
Definition 3.1.
Let H be a graph and fix a vertex ∈ H . For r ∈ N , write B H ( r ) for the r -ball inthe graph H , ie B H ( r ) := { h ∈ H | d H (0 , h ) ≤ r } , where d H is the graph distance in H . Define D H ( β ) := min (cid:8) r ≥ (cid:12)(cid:12) |B H ( r ) | ≥ β | H | (cid:9) for β ∈ (0 , . When considering sequences ( k N , G N ) N ∈ N of integers and Abelian groups, abbreviate D N ( β ) := D G N ([ Z ,...,Z kN ]) ( β ) where Z , ..., Z k N ∼ iid Unif( G N ) . As always, if we write D N , then this is either D + N or D − N according to context. We show that, whp over the graph (ie choice of Z ), this statistic concentrates. Here we consider k h λ log | G | for any λ ∈ (0 , ∞ ). The result holds for a large class of Abelian groups. Further, forthese groups, the typical distance concentrates at α λ k where α λ ∈ (0 , ∞ ) is a constant; so thisdepends only on k and | G | . This is in agreement with the spirit of the Aldous–Diaconis conjecture.12ecall that any Abelian group can be decomposed as ⊕ d Z m j for some d, m , ..., m d ∈ N . Foran Abelian group G , we define the dimension and minimal side-length , respectively, as follows: d ( G ) := min (cid:8) d ∈ N (cid:12)(cid:12) ⊕ d Z m j is a decomposition of G (cid:9) ; m ∗ ( G ) := max (cid:8) min j ∈ [ d ] m j (cid:12)(cid:12) ⊕ d Z m j is a decomposition of G (cid:9) . It can be shown that there is a decomposition which is optimal for both these statistics: there exist d, m , ..., m d ∈ N so that ⊕ d Z m j is a decomposition of G with d = d ( G ) and min j ∈ [ d ] m j = m ∗ ( G ).From now on, we assume that we are always using such an optimal decomposition.There are some conditions which the Abelian groups must satisfy. Hypothesis B.
The sequence ( k N , G N ) N ∈ N satisfies Hypothesis B if lim N →∞ k N = ∞ , lim N →∞ k N / log | G N | ∈ (0 , ∞ ) , lim inf N →∞ m ∗ ( G N ) = ∞ and d ( G N ) ≤ log | G N | / log log | G N | for all N ∈ N . We are now ready to state the main theorem of this section.
Theorem 3.2.
Let ( k N ) N ∈ N be a sequence of positive integers and ( G N ) N ∈ N a sequence of finite,Abelian groups; for each N ∈ N , define Z ( N ) := [ Z , ..., Z k N ] by drawing Z , ..., Z k N ∼ iid Unif( G N ) .Suppose that ( k N , G N ) N ∈ N satisfies Hypothesis B. Let λ := lim sup N k N / log | G N | . Then thereexists a constant α ± λ ∈ (0 , ∞ ) so that, for all β ∈ (0 , , we have D ± N ( β ) / ( α ± λ k N ) → P (in probability) as N → ∞ . Moreover, the implicit lower bound holds deterministically, ie for all choices of generators, and forall Abelian groups, ie Hypothesis B need not be satisfied—we just need lim N k N / log | G N | ∈ (0 , ∞ ) . For ease of presentation, in the proof we drop the N -subscripts. Remark 3.3. In § L -type graph distances to L q -type.An analogous concentration of typical distance is given. See Hypothesis B ′ and Theorem 3.11. △ The outline here is very similar to that from before; see § d -thpower of a gcd. Issues arose when k became too large while k − d is fairly small; see the proof ofLemma 2.13. This arose from the fact that we used the estimate P (cid:0) V ∈ γ Z (cid:1) ≤ P (cid:0) V ∈ γ Z | V = 0 (cid:1) + P (cid:0) V = 0 (cid:1) ≤ /γ + 3 /n /k . Once this was raised to the power k , the second term became an issue. We alleviate this by defining I := (cid:8) i ∈ [ k ] | V i = 0 (cid:9) and studying P ( V i ∈ γ Z | i ∈ I ); the problematic term 3 /n /k then does not exist as we consideronly non-zero coordinates of V . If G = ⊕ d Z m j , then we are actually interested in V i mod m j foreach j . Recall that m ∗ = min j m j . ‘Typically’, one has | V i | ≤ m ∗ . We suppose initially that m ∗ islarge enough so that max i | V i | < m ∗ whp. Thus looking at V i = 0 or V i ≡ m j is no different.For large | I | , the gcd analysis goes through similarly to before. When | I | is small, eg smallerthan d , it is more difficult to control; in this case, we use a fairly naive bound on the gcd, butcontrol carefully the probability of realising such an I . The case I = ∅ , which corresponds to V = 0, is handled by taking the lattice ball to be of large enough volume.Previously we used a vector of geometrics as a proxy for a uniform distribution on a ball. Herewe are able to let W be uniform on a ball. The coordinates are no longer independent, which makesthe gcd analysis is slightly complicated. However, since we only consider i with V i = 0, this can behandled; see Lemma 3.9. This uniformity simplifies some other calculations somewhat.13 .3 Estimates on Sizes of Balls in Z k We wish to determine the size of balls B k ( R ) when k ≍ log n . In particular, we are interestedin the growth when the volume is around n . Definition 3.4.
Define M ±∗ ( k, N ) to be the minimal integer M satisfying | B ± k ( M ) | ≥ N . Lemma 3.5.
For all λ ∈ (0 , ∞ ) , there exists a function ω ≫ and a constant α ± so that, for all ε ∈ (0 , , if k h λ log n , then M ±∗ := M ±∗ ( k, ne ω ) satisfies M ±∗ h α ± k h α ± λ log n and (cid:12)(cid:12) B ± k (cid:0) α ± k (1 − ε ) (cid:1)(cid:12)(cid:12) ≪ n. This will follow easily from the following auxiliary lemma controlling the size of lattice balls.
Lemma 3.6.
There exists a strictly increasing, continuous function c ± : (0 , ∞ ) → (0 , ∞ ) so that,for all a ∈ (0 , ∞ ) , we have (cid:12)(cid:12) B ± k ( ak ) (cid:12)(cid:12) = exp (cid:0) k (cid:0) c ± ( a ) + o (1) (cid:1)(cid:1) . Proof.
The directed case follows immediately from Stirling’s approximation and the fact that (cid:12)(cid:12) B + k ( ak ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:8) b ∈ Z k + (cid:12)(cid:12) P k b i ≤ ak (cid:9)(cid:12)(cid:12) = (cid:0) ⌊ ak ⌋ + kk (cid:1) = (cid:0) ⌊ ( a +1) k ⌋ k (cid:1) Consider now the undirected case. Omit all floor and ceiling signs. By considering the numberof coordinates which equal 0, we obtain (cid:12)(cid:12) B − k ( ak ) (cid:12)(cid:12) = P ki =0 A i where A i := A i ( k, a ) := (cid:0) ki (cid:1) k − i (cid:0) k − i + akak (cid:1) . Choose i ∗ := i ∗ ( k, a ) that maximises A i . Then A i ∗ ≤ | B − k ( ak ) | ≤ ( k + 1) A i ∗ . Observe that A i +1 A i = ( k − i ) i + 1)( k (1 + a ) − i ) , and hence one can determine i ∗ as a function of k and a , conclude that i ∗ ( a, k ) /k converges as k → ∞ and thus determine c + ( a ) in terms of the last limit. We omit the details. Knowing thislimit allows us to plug this into the definition of A i and use Stirling’s approximation to get A i ∗ = exp (cid:0) k (cid:0) c − ( a ) + o (1) (cid:1)(cid:1) , for some strictly increasing function c − : (0 , ∞ ) → (0 , ∞ ). Since k +1 = e o ( k ) , the claim follows.From this lemma, Lemma 3.5 follows easily. Proof of Lemma 3.5.
Set α := c − (1 /λ ). The upper bound is an immediate consequence of thecontinuity of c . The lower bound follows from the exponential growth rate. From the results in § Proof of Lower Bound in Theorem 3.2.
Let ξ ∈ (0 ,
1) and set R := α ± λ k (1 − ξ ). Since the un-derlying group is Abelian, applying Lemma 3.5, we have |B ± k ( R ) | ≤ | B ± k ( R ) | ≪ n. Hence, for all β ∈ (0 ,
1) and all Z , we have D ± k ( β ) ≥ R = α ± λ k (1 − ξ ), asymptotically in n . m ∗ ( G ) ≫ k Define M ±∗ , ω and α ± as in Definition 3.4 and Lemma 3.5. In this subsection we draw W ± ∼ Unif( B ± k ( M ±∗ )), ie uniform on a ball of radius M ±∗ . We show that W ± · Z is well-mixed on G , andhence its support contains almost all the vertices.14 roposition 3.7. Suppose that Hypothesis B is satisfied. Then E (cid:0)(cid:13)(cid:13) P G k (cid:0) W ± · Z ∈ · (cid:1) − π G (cid:13)(cid:13) (cid:1) = o (1) , Given this proposition, the upper bound in Theorem 3.2 follows easily.
Proof of Upper Bound in Theorem 3.2 Given Proposition 3.7. If k P G k ( W ± · Z ∈ · ) − π G k ≤ ε, then the support S of W ± · Z satisfies π G ( S c ) ≤ ε . Combined with Lemma 3.5 and Proposition 3.7,the upper bound in Theorem 3.2 follows.The remainder of this subsection is devoted to proving Proposition 3.7. We tend to dropthe ± -superscript from the notation, only writing + or − if there is ambiguity. Let W, W ′ ∼ iid Unif( B k ( M ∗ )) and let V := W − W ′ . The standard L calculation gives E (cid:0)(cid:13)(cid:13) P G k (cid:0) W · Z ∈ · (cid:1) − π G (cid:13)(cid:13) (cid:1) = E (cid:0) n P (cid:0) V · Z = 0 | Z (cid:1) − (cid:1) = n P (cid:0) V · Z = 0 (cid:1) − . First, it is immediate that P ( V = 0) = P ( W = W ′ ) = | B k ( M ∗ ) | − ≤ n − e − ω ≪ n − . Nowconsider V = 0. As in § g j := gcd (cid:0) V , ..., V k , m j (cid:1) for each j ∈ [ d ]; set g := gcd (cid:0) V , ..., V k , n (cid:1) . The following lemma is equivalent to Lemma 2.10, rephrased slightly.
Lemma 3.8.
Conditional on V , we have V · Z ∼ Unif( ⊕ d g j Z m j ) . For the remainder of this subsection, we assume that the minimal side-length m ∗ := m ∗ ( G )satisfied m ∗ ≫ k ≍ M ∗ . In the next subsection, we remove this assumption: we extend the proofto m ∗ ≫
1, as in Hypothesis B. Given this, we have max i ∈ [ k ] | V i | < max j ∈ [ d ] m j . Hence I := (cid:8) i ∈ [ k ] (cid:12)(cid:12) V i m j ∀ j ∈ [ d ] (cid:9) = (cid:8) i ∈ [ k ] (cid:12)(cid:12) W i = W ′ i (cid:9) . To analyse the expected gcd, we breakdown according to the value of I . Lemma 3.9.
There exists a constant C so that, for all I ⊆ [ k ] with I = ∅ , we have n P (cid:0) V · Z = 0 | I = I (cid:1) ≤ E (cid:0) g d | I = I (cid:1) ≤ ( C d (2 M ∗ ) d −| I | +2 when | I | ≤ d + 1 , · ( ) d −| I | when | I | ≥ d + 2 . Lemma 3.10.
For all I ⊆ [ k ] with | I | ≪ k , we have P ( I = I ) ≤ e − ω n − o (1) . If I = ∅ , then the o (1) term may be taken to be 0. Given these two lemmas, we have all the ingredients required to prove Proposition 3.7, fromwhich we deduced the main theorem (Theorem 3.2). We defer the proofs of Lemmas 3.9 and 3.10until after the proof of Proposition 3.7, which we give now.
Proof of Proposition 3.7.
Here k h λ log n , M := M ∗ h αk h αλ log n and d ≤ log n/ log log n .As noted previously, the standard L calculation gives E (cid:0)(cid:13)(cid:13) P G k (cid:0) W · Z ∈ · (cid:1) − π G (cid:13)(cid:13) (cid:1) = E (cid:0) n P (cid:0) V · Z = 0 | Z (cid:1) − (cid:1) = n P (cid:0) V · Z = 0 (cid:1) − n P I ⊆ [ k ] P (cid:0) V · Z = 0 , I = I (cid:1) − . Consider I = ∅ . Then V · Z = 0 (for all Z ). By Lemma 3.10, we have P ( I = ∅ ) ≤ n − e − ω . Thus n P (cid:0) V · Z = 0 , I = ∅ (cid:1) ≤ e − ω = o (1) . Consider I ⊆ [ k ] with 1 ≤ | I | ≤ d + 1. There are at most ( d + 1) (cid:0) kd +1 (cid:1) ≤ k d +2 such sets I . Sincelog k = log log n + log λ + o (1), we have k d +2 ≤ n / . Applying Lemmas 3.9 and 3.10 gives n P (cid:0) V · Z = 0 , I = I (cid:1) ≤ C d (3 αλ log n ) d +2 −| I | · n − o (1) ≤ k − d − n − / , d ≪ k ≍ log n and (so) 2 d = n o (1) . We now sum over all I with 1 ≤ | I | ≤ d + 1: n P ≤| I |≤ d +1 P (cid:0) V · Z = 0 , I = I (cid:1) ≤ n − / = o (1) . Consider I ⊆ [ k ] with d + 2 ≤ | I | ≤ L := log n/ log log n ; then L − d ≫
1. Similarly to above,there are at most L (cid:0) kL (cid:1) ≤ k L +1 such sets I . Applying Lemmas 3.9 and 3.10 gives n P (cid:0) V · Z = 0 , I = I (cid:1) ≤ n − o (1) ≤ k − L − n − / , noting that k L ≤ n / o (1) . We now sum over all I with d + 2 ≤ | I | ≤ L : n P d +2 ≤| I |≤ L P (cid:0) V · Z = 0 , I = I (cid:1) ≤ n − / = o (1) . Finally consider I ⊆ [ k ] with | I | ≥ L . Sum over these using Lemma 3.9: n P L ≤| I |≤ k P (cid:0) V · Z = 0 , I = I (cid:1) ≤ · ( ) d − L = 1 + o (1) . Combining these four parts into a single sum, we deduce the result.It remains to prove the auxiliary Lemmas 3.9 and 3.10.
Proof of Lemma 3.9.
The first inequality is an immediate consequence of Lemma 3.8.Note that g ≤ M ∗ since max i | V i | ≤ M ∗ . For α, β ∈ Z , write α ≀ β if α divides β . Thus E (cid:0) g d (cid:12)(cid:12) I = I (cid:1) ≤ P Mγ =1 γ d P (cid:0) γ ≀ V i ∀ i ∈ I | I = I (cid:1) For a set I ⊆ [ k ], write W I := ( W i ) i ∈ I and W \ I := W [ k ] \ I . Consider conditioning on I = I . Let W \ I and W ′\ I be given; since I = I , we have W \ I = W ′\ I . Let U have the distribution of W I given W \ I and define U ′ analogously. Write D i := D i ( γ ) := { γ ≀ ( U i − U ′ i ) } . Then P (cid:0) γ ≀ V i ∀ i ∈ I (cid:12)(cid:12) I = I, k W \ I k (cid:1) = P (cid:0) D i ∀ i ∈ I (cid:1) . By exchangeability, it suffices to consider the case I = { , ..., ℓ } . We then have P (cid:0) D i ∀ i ∈ I (cid:1) = P (cid:0) D ℓ (cid:1) P (cid:0) D ℓ − (cid:12)(cid:12) D ℓ (cid:1) · · · P (cid:0) D (cid:12)(cid:12) D , ..., D ℓ (cid:1) = Q ℓi =1 P (cid:0) D i (cid:12)(cid:12) D i +1 , ..., D ℓ (cid:1) . For i ∈ [ k ], define M i := M ∗ −k W \{ ,...,i } k and M ′ i analogously. Let i ∈ [ ℓ − u i +1 , ..., u ℓ )and ( u ′ i +1 , ..., u ′ ℓ ) be two vectors in the support of ( U i +1 , ..., U ℓ ). Then,conditional on ( U i +1 , ..., U ℓ ) = ( u i +1 , ..., u ℓ ) and ( U ′ i +1 , ..., U ′ ℓ ) = ( u ′ i +1 , ..., u ′ ℓ )we have ( U , ..., U i ) ∼ Unif (cid:0) B i ( R ) (cid:1) and ( U ′ , ..., U ′ i ) ∼ Unif (cid:0) B i ( R ′ ) (cid:1) for some R, R ′ ∈ R . (Recall that the subscript in B k denotes the dimension of the ball.)In the case of undirected balls, the law of U i − U ′ i given this conditioning is symmetric andunimodal on Z \ { } ; see [27, Theorem 2.2]. It follows, as in the proof of Lemma 2.13, that P (cid:0) D − i (cid:12)(cid:12) D − i +1 , ..., D − ℓ (cid:1) ≤ /γ. Further, this holds not just conditional on D − i +1 ∩ · · · ∩ D − ℓ , but conditional on any choice of( U i +1 , ..., U ℓ ) and ( U ′ i +1 , ..., U ′ ℓ ) which satisfy D − i +1 ∩· · ·∩ D − ℓ . By the same reasoning, P ( D − ℓ ) ≤ /γ .Hence, for undirected balls, P (cid:0) D − i ∀ i ∈ I (cid:1) = P (cid:0) γ ≀ V − i ∀ i ∈ I (cid:12)(cid:12) I = I (cid:1) ≤ γ −| I | . (The − -superscript emphasises that this is for undirected balls.)We now turn our attention to directed balls. In this case, U i and U ′ i are both unimodal, butwith potentially different modes, if R = R ′ . Instead of direct computation, we compare withthe undirected case. Specifically, if U i and U ′ i have the same sign in the undirected case, then | V i | = | U i − U ′ i | has the same law as in the directed case. The choice of sign is independent of16verything else; the two have the same sign with probability . Hence, by conditioning on thespecific values of ( U i +1 , ..., U ℓ ) and ( U ′ i +1 , ..., U ′ ℓ ), we obtain1 /γ ≥ P (cid:0) D − i (cid:12)(cid:12) D − i +1 , ..., D − ℓ (cid:1) ≥ P (cid:0) D + i (cid:12)(cid:12) D + i +1 , ..., D + ℓ (cid:1) . For γ = 2, note that the probabilities are actually the same: this is because x − y is even if andonly if | x | − | y | is even, since x and − x have the same parity.From this we deduce, for both the undirected and directed cases, that E (cid:0) g d | I = I (cid:1) ≤ d −| I | + P Mγ =3 γ d (2 /γ ) | I | = 1 + 2 d −| I | + 2 d P Mγ =3 ( γ/ d −| I | . A case-by-case analysis, according to d − | I | , completes the proof. Proof of Lemma 3.10.
Recall from Definition 3.4 that | B k ( M ∗ ) | ≥ ne ω . Thus P (cid:0) I = ∅ (cid:1) = P (cid:0) W = W ′ (cid:1) = (cid:12)(cid:12) B k ( M ∗ ) (cid:12)(cid:12) − ≤ n − e − ω . Using the law of W I given W \ I determined in the previous proof, we have P (cid:0) W \ I = W ′\ I (cid:1) = P ( W = W ′ ) P ( W = W ′ | W \ I = W ′\ I ) = | B k ( M ∗ ) | − E ( | B | I | ( M ∗ − k W \ I k ) | − ) ≤ | B | I | ( M ∗ ) || B k ( M ∗ ) | . It is a standard balls-in-bins combinatorial identity that (cid:12)(cid:12) B + ℓ ( R ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:8) b ∈ Z ℓ + (cid:12)(cid:12) P ℓ b i ≤ R (cid:9)(cid:12)(cid:12) = (cid:0) ⌊ R ⌋ + ℓℓ (cid:1) . For the undirected case, we can choose a sign for each coordinate. Hence we see that (cid:12)(cid:12) B + ℓ ( R ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) B − ℓ ( R ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:8) b ∈ Z ℓ (cid:12)(cid:12) P ℓ | b i | ≤ R (cid:9)(cid:12)(cid:12) ≤ ℓ (cid:0) ⌊ R ⌋ + ℓℓ (cid:1) . Abbreviate M := M ∗ and ℓ := | I | . It suffices to consider I with ℓ ≤ ck , for an arbitrarily smallpositive constant c . From Lemma 3.5, we have M ≤ αk . So (cid:12)(cid:12) B ± ℓ ( M ) (cid:12)(cid:12) ≤ ℓ (cid:0) ⌊ M ⌋ + ℓℓ (cid:1) ≤ (cid:0) e (2 αk/ℓ + 1) (cid:1) ℓ ≤ (8 eαk/ℓ ) ℓ , with the last inequality requiring 2 αk/ℓ ≥
1, which holds if c is sufficiently small, as ℓ ≤ ck . Now,for c sufficiently small, the map ℓ (8 eαk/ℓ ) ℓ is increasing on [1 , ck ]. Hence (cid:12)(cid:12) B ± ℓ ( M ) (cid:12)(cid:12) ≤ (8 eαk/ℓ ) ℓ ≤ (8 eα/c ) ck ≤ (8 eα/c ) cλ log n . By taking c sufficiently small, we can upper bound this by an arbitrarily small power of n . m ∗ ( G ) ≫ For the upper bound, we have been assuming that the minimal side length m ∗ ( G ) satisfies m ∗ ( G ) ≫ log | G | . (Recall that the lower bound had no conditions on m ∗ ( G ).) We now describehow to relax this condition to m ∗ ( G ) ≫
1. We could go even further, with statements like “onlya small number of j in G = ⊕ d Z m j have m j ≍ m ∗ ( G ) ≫ L and L ∞ balls. To distinguish these we use a superscript: · B ℓ ( R ) will be the L ball in ℓ dimensions of radius R ; · B ∞ ℓ ( R ) will be the L ∞ ball in ℓ dimensions of radius R .For a set I ⊆ [ k ], recall that we write W I := ( W i ) i ∈ I and W \ I := ( W i ) i/ ∈ I .We describe the adaptations for undirected graphs. The adaptations for directed graphs arecompletely analogous: simply replace appearances of Z k with Z k + and | W i | with W i .17 utline of Proof. The idea behind the proof is intuitive. Since R ≍ k , by symmetry we have E ( | W i | ) ≤ R/k ≍ i . Thus ‘almost all’ the coordinates should be smaller than any divergingfunction (these coordinates are good ). Further, the contribution to the radius k W k due to the bad coordinates should be small, ie o ( k ). Roughly this allows us to replace k with k = k (1 − o (1)) and R with R = R (1 − o (1)). Choosing R := α k/ log n k · (1 + 2 ε ) for ε > R ≥ α k/ log n k · (1 + ε ) and hence | B k ( R ) | ≫ n. This was the key element in the proof previously; the remainder of the proof is as before.We now proceed formally and rigorously.
Relaxing Minimal Side-Length Condition.
Let ε > λ := lim k/ log n . Set R := α λ k (1 + 2 ε )and draw W ∼ Unif( B k, ( R )). Let ν satisfy 1 ≪ ν ≪ m ∗ ( G ). For w ∈ Z k , define J ( w ) := (cid:8) i ∈ [ k ] (cid:12)(cid:12) | w i | ≤ ν (cid:9) . Call these coordinates good . By Markov’s inequality, clearly | [ k ] \ J ( W ) | . /ν = o (1) whp.As always, we look at two independent realisations W and W ′ . We then wish to look atcoordinates i ∈ [ k ] which are good for both W and W ′ , ie in J := J ( W ) ∩ J ( W ′ ). We needto make sure that the contribution to the radius from the (abnormally large) bad coordinates isnot too large. For δ > w ∈ Z k , write L δ ( w ) for the collection of the ⌈ δk ⌉ -largest (in absolutevalue) coordinates of w . We then define typicality in the following way: for δ, δ ′ >
0, set W := (cid:8) w ∈ Z k (cid:12)(cid:12) k w k ≤ R, (cid:12)(cid:12) [ k ] \ J ( w ) (cid:12)(cid:12) ≤ δk, k w L δ ( w ) k ≤ δ ′ k (cid:9) . In particular now, if w, w ′ ∈ W , then k w J ( w ) ∩J ( w ′ ) k ≥ k − δ ′ k . It is not difficult to see that wecan choose δ, δ ′ = o (1) with P ( W ∈ W ) = 1 − o (1); we give justification at the end of the proof.Consider now W, W ′ ∼ iid Unif( B k, ( R )). We have the following conditional law: W J , W ′ J ∼ iid Unif (cid:0) B k ( R ) ∩ B ∞ k ( ν ) (cid:1) conditional on W \ J = w \ J = W ′\ J and J = J where J = J ( W ) ∩ J ( W ′ ) , k := |J | and R := R − k w \ J k . Write typ := { W, W ′ ∈ W} . On the event typ , given J = J and ( W J , W ′ J ), we have k ≥ k (1 − δ ) = k (cid:0) − o (1) (cid:1) and R ≥ R (1 − δ ′ ) = R (cid:0) − o (1) (cid:1) . In particular, we may choose η > ε ) so that R ≥ α λ (1 − η ) k (1 − η )(1 + ε ) and k ≥ k (1 − η ) , and hence | B k ( R ) | ≫ n. Since typicality holds with probability 1 − o (1), we have (cid:12)(cid:12) B k ( R ) ∩ B ∞ k ( ν ) (cid:12)(cid:12) ≫ n. The remainder of the proof follows similarly to before. Formally, we define W and W ′ as follows: W i := W i and W ′ i := W ′ i for i ∈ J ; W i := 0 and W ′ i := 0 for i / ∈ J . Since this is a projection, { W I = W ′ I } ⊆ { W I = W I } for any I ⊆ [ k ]. Now instead of decomposingaccording to the value (or size) of I := { i ∈ [ k ] | W i = 0 } , we use the set I := I ∩ J . The fact that (cid:12)(cid:12) B k ( R ) ∩ B ∞ k ( ν ) (cid:12)(cid:12) ≫ n allows all the previous estimates for I to follow through for I here.The last change to mention is the gcd calculations of Lemma 3.9. The only property of thedistribution of ( W, W ′ ) required was that each coordinate (while not independent) is unimodal andsymmetric about 0, even conditional on W I = W ′ I and W ′ I = w ′ I for some I ⊆ [ k ] and w I , w ′ I ∈ Z | I | .For ( W , W ′ ), this property still holds. Hence the identical argument applies here too.It remains to argue that P ( W ∈ W ) = 1 − o (1) for some δ, δ ′ = o (1). First, as noted above, P ( | [ k ] \ J ( W ) | > δk ) = o (1) by Markov’s inequality and the fact that E ( | W | ) ≍
1. The fact that P ( k W L δ ( W ) k > δk ) = o (1) follows by a union bound over all (cid:0) k ⌈ δk ⌉ (cid:1) possible values of the set L δ ( W )and applying Bernstein’s inequality; take δ ′ := Cδ log(1 /δ ) for a sufficiently large constant C .18 emark. We believe that the typical distance should concentrate if k ≍ log | G | and k − d ≫ m ∗ ( G ). However, without any such condition, we do havereason to believe that the value at which this concentration happens should depend on more thanjust k and | G | —the algebraic structure of G should be important. This exact phenomenon occurswhen studying the mixing time of the random walk on the Cayley graph. See [11, Theorem A], inparticular contrasting the case k ≍ log | G | ≍ d ( G ) with 1 ≪ k . log | G | and d ( G ) ≪ log | G | . △ L q -Type Graph Distances Graph distances in Cayley graphs have some special properties. Consider a collection z =[ z , ..., z k ] of generators and distances in the Cayley graph G ( z ). For a path ρ in G ( z ), for each i ∈ [ k ], write ρ i, + for the number of times z i is used, ρ i, − for the number of times z − i is used (ifin the undirected case, otherwise ρ i, − := 0) and ρ i := ρ i, + − ρ i, − . The path connects the identitywith ρ · z . Then the length, in the usual graph distance, of ρ is k ρ k := P k ( ρ i, + + ρ i, − ).For any q ∈ [1 , ∞ ), define the L q graph distance of ρ by k ρ k qq := P k ( ρ qi, + + ρ qi, − ) . For the L ∞ graph distance , define k ρ k ∞ := max i { ρ i, + + ρ i, − } . (The usual graph distance is given by q = 1.)For Abelian groups, clearly for any q ∈ [1 , ∞ ) an L q geodesic , ie a path of minimal L q weight,will only use either z i or z − i , not both (since the terms in the product can be reordered), ie ρ i, + ρ i, − = 0 for all i . Thus k ρ k qq = P k | ρ i | q . Similarly, any L ∞ geodesic ρ can be adjusted into anew path ρ ′ with ρ · z = ρ ′ · z and k ρ k ∞ = k ρ ′ k ∞ satisfying ρ ′ i, + ρ ′ i, − = 0 for all i .We define the L q typical distance D G ( z ) ,q ( · ) analogously to D G ( z ) ( · ), ie the q = 1 case. Hypothesis B ′ . The sequence ( k N , G N ) N ∈ N and q ∈ [1 , ∞ ] jointly satisfy Hypothesis B ′ if thefollowing conditions hold (defining k / ∞ := 1 for k ∈ N ): lim N →∞ k N = ∞ , lim N →∞ k N / log | G N | = 0 and lim N →∞ k /qN | G N | /k N /m ∗ ( G N ) = 0; if q ∈ (1 , ∞ ) then additionally k N ≤ log | G N | / log log | G N | for all N ∈ N ;lim sup N →∞ d N k N < ( for undirected graphs , for directed graphs . Finally we set up a little more notation. Make the following definitions: C − q := 2 Γ(1 /q + 1)( qe ) /q , C + q := C − q and D ± q ( k, n ) := k /q n /k /C ± q , where the case q = ∞ is to be interpreted as the limit q → ∞ ; eg, C −∞ = 2 and D + ∞ ( k, n ) = n /k . When these are sequences ( k N , | G N | ) N ∈ N , for N ∈ N and q ∈ [1 , ∞ ], write D ± N,q := D ± q ( k N , | G N | ).Similarly, for a sequence ( G N ) N ∈ N of finite groups with corresponding multisubsets ( Z ( N ) ) N ∈ N of sizes ( k N ) N ∈ N , for N ∈ N , β ∈ [0 ,
1] and q ∈ [1 , ∞ ], define D ± N,q := D G ± N ( Z ( N ) ) ( β ) . Using an extension of the methodology from this section ( § L q latticeballs, we can prove the following theorem. We have already considered q = 1 and k ≍ log | G | . Theorem 3.11.
Let ( k N ) N ∈ N be a sequence of positive integers and ( G N ) N ∈ N a sequence of finite,Abelian groups; for each N ∈ N , define Z ( N ) := [ Z , ..., Z k N ] by drawing Z , ..., Z k N ∼ iid Unif( G N ) .Suppose that ( k N , G N ) N ∈ N satisfies Hypothesis B ′ . Then, for all β ∈ (0 , , we have D ± N,q ( β ) / D ± N,q → P (in probability) as N → ∞ . Moreover, the implicit lower bound holds for all choices of generators and for all Abelian groups,only requiring the conditions in Hypothesis B ′ which depend only on ( k N , | G N | ) N ∈ N and q . The arguments used to prove this theorem really are analogous to those used in this section( § L q and in dimension1 ≪ k ≪ log n , rather than L and k ≍ log n . Other than this, the remainder of the analysis, inparticular the reduction to a gcd and the consideration of the set I of non-zero coordinates of W ,is exactly the same. (Now W is uniform on an L q ball of appropriate radius.) We do not give thedetails here; they can be found in [13, § Typical Distance: k ≫ log | G | This section focusses on concentration of distances from the identity in the random Cayleygraph of an Abelian group when k ≫ log | G | . (The previous sections dealt with 1 ≪ k . log | G | .)The main result of the section is Theorem 4.2.The outline of this section is as follows: · § · § · § · § · § To start the section, we recall the typical distance statistic.
Definition 4.1.
Let H be a graph and fix a vertex ∈ H . For r ∈ N , write B H ( r ) for the r -ball inthe graph H , ie B H ( r ) := { h ∈ H | d H (0 , h ) ≤ r } , where d H is the graph distance in H . Define D H ( β ) := min (cid:8) r ≥ (cid:12)(cid:12) |B H ( r ) | ≥ β | H | (cid:9) for β ∈ (0 , . When considering sequences ( k N , G N ) N ∈ N of integers and Abelian groups, abbreviate D N ( β ) := D G N ([ Z ,...,Z kN ]) ( β ) where Z , ..., Z k N ∼ iid Unif( G N ) . Finally, considering such sequences, we define the candidate radius for the typical distance: D N := ρ N ρ N − log | G N | / log k N where ρ N := log k N / log log | G N | for each N ∈ N . As always, if we write D N , then this is either D + N or D − N according to context. Up to subleadingorder, the typical distance will be the same for the undirected graphs as for the directed graphs. We show that, whp over the graph (ie choice of Z ), this statistic concentrates. Here we consider k ≫ log | G | . The result holds for all Abelian groups; in fact, the implicit upper bound is valid forall groups. Further, the typical distance concentrates at a distances which depends only on k and | G | . This is in agreement with the spirit of the Aldous–Diaconis conjecture. Hypothesis C.
The sequence ( k N , n N ) N ∈ N satisfies Hypothesis C if lim inf N →∞ k N log n N = ∞ and lim inf N →∞ log k N log n N = 0 . Theorem 4.2.
Let ( k N ) N ∈ N be a sequence of positive integers and ( G N ) N ∈ N a sequence of finite,Abelian groups; for each N ∈ N , define Z ( N ) := [ Z , ..., Z k N ] by drawing Z , ..., Z k N ∼ iid Unif( G N ) .Suppose that ( k N , | G N | ) N ∈ N satisfies Hypothesis C. Then, for all β ∈ (0 , , we have D ± N ( β ) / D N → P (in probability) as N → ∞ . Moreover, the implicit lower bound holds deterministically, ie for all choices of generators, and theimplicit upper bound holds for all groups, not just Abelian groups.
As always, for ease of presentation, in the proof we drop the N -subscripts. When k ≫ log | G | , one can see that the typical distance statistic D must satisfy D ≪ k . Bysymmetry, the expected number of times a generator is used when drawing from a ball B k ( R ) is o (1). The number of ways that precisely R can be chosen is (cid:0) kR (cid:1) . We choose R with (cid:0) kR (cid:1) ≈ | G | .20 .3 Estimates on Sizes of Balls in Z k We consider balls and spheres in the L and L ∞ senses: write B k, ( · ), respectively S k, ( · ), forthe L ball, respectively sphere, in Z k ; write B k, ∞ (1) for the L ∞ unit ball in Z k . Lemma 4.3.
For all R ≥ , we have | B ± k, ( R ) | ≤ R (cid:0) ⌊ R ⌋ + k ⌊ R ⌋ (cid:1) and (cid:12)(cid:12) S ± k, ( R ) ∩ B ± k, ∞ (1) (cid:12)(cid:12) ≥ (cid:0) k ⌊ R ⌋ (cid:1) . Furthermore, if R ≪ k , then R (cid:0) ⌊ R ⌋ + k ⌊ R ⌋ (cid:1) = exp (cid:0) R log( k/R ) · (cid:0) o (1) (cid:1)(cid:1) = (cid:0) k ⌊ R ⌋ (cid:1) In particular, if k = (log n ) ρ ≫ log n and ε > is a constant, then (cid:12)(cid:12) S ± k, (cid:0) (1 + ε ) ρρ − log k n (cid:1) ∩ B k, ∞ (1) (cid:12)(cid:12) ≫ n. Proof.
In the first display, the upper bound is proved in [15, Lemma E.2a]; the lower bound is theusual formula for the number of subsets of [ k ] of size R . The second display is a simple application ofStirling’s approximation and asymptotics of the binary entropy function. The final display followsby combining the previous two and performing a simple calculation. From the results in § Proof of Lower Bound in Theorem 4.2.
Let ξ ∈ (0 ,
1) and set R := D (1 − ξ ). Since the underlyinggroup is Abelian, applying Lemma 4.3, a simple calculation gives |B k ( R ) | ≤ | B k, ( R ) | ≤ exp (cid:0) D log( k/ D ) · (1 − ξ ) (cid:1) ≪ n. Hence, for all β ∈ (0 ,
1) and all Z , we have D k ( β ) ≥ R = D (1 − ξ ), asymptotically in n . Lemma 4.3 gives a quantitative sense in which | B k, ( R ) | ≈ (cid:12)(cid:12) S k, ( R ) ∩ B k, ∞ (1) (cid:12)(cid:12) ≥ (cid:0) k ⌊ R ⌋ (cid:1) ; inform-ally, this means that we do not really lose any volume by restricting to the sphere and requiringthat each generator is used at most once. We show the upper bound for arbitrary groups. Proof of Upper Bound in Theorem 4.2.
Let ξ > R := D (1 + ξ ). Draw W, W ′ ∼ iid Unif( S k, ( R ) ∩ B k, ∞ (1)). Define S := Z W · · · Z W k k and S ′ similarly. We show that S is well-mixedwhp (this time in the L sense) to deduce the upper bound. By the standard L calculation, E (cid:0) k P G k (cid:0) S ∈ · (cid:1) − π G k (cid:1) = n P (cid:0) S ′ = S ′ (cid:1) − . If W = W ′ , then there exists an i ∈ [ k ] so that W i = 1 and W ′ i = 0 or vice versa. By the uniformityand independence of the generators, S ′ S − ∼ Unif( G ) for all (not just Abelian) groups. Thus n P (cid:0) S = S ′ (cid:1) − ≤ n P (cid:0) W = W ′ (cid:1) = n (cid:12)(cid:12) S k, ( R ) ∩ B k, ∞ (1) (cid:12)(cid:12) − ≪ , using Lemma 4.3 for the final relation. This completes the proof. Remark.
This upper bound, ie on typical distance with k ≫ log | G | , can be easily deduced frommixing results proved in the ’90s. Specifically, it was shown by Dou and Hildebrand [10, Theorem 1]that the mixing time for the usual random walk is upper bounded by ρρ − log k | G | for any group;Roichman [28, Theorems 1 and 2] subsequently gave a simpler proof, using an argument not thatdissimilar from our proof above. The lower bound does not follow from mixing results, though.There are a few reasons for including the proof above. Foremost is that we use the sameargument in § k with k − log | G | ≍ k , not just k ≫ log | G | .Additionally, we need to do most of the work for the lower bound anyway, and it demonstrateshow easily our method adapts to this new regime. △ Diameter
In this section we consider the diameter of the random Cayley graph. Our analysis is separatedinto two distinct sections. § k & log | G | , and that the value at which itconcentrates is the same as for typical distance. § k with k − log | G | ≍ k , that the group giving rise to the largest diameter(amongst all groups) is Z d . k & log | G | Recall that in Theorem 3.2 we showed, in the regime k ≍ log n and under some assumptions,that, up to subleading order terms, the typical distance concentrates at αk , for some constant α .The next theorem shows, in the same set-up, that the diameter does the same. The argument usesthe typical distance result as a ‘black box’, then extending from this to diameter. Theorem 5.1.
Let ( k N ) N ∈ N be a sequence of positive integers and ( G N ) N ∈ N a sequence of finite,Abelian groups; for each N ∈ N , define Z ( N ) := [ Z , ..., Z k N ] by drawing Z , ..., Z k N ∼ iid Unif( G N ) .Suppose that ( k N , G N ) N ∈ N satisfies either Hypotheses B or C. For λ ∈ (0 , ∞ ) , let α ± λ ∈ (0 , ∞ ) be the constant from Theorem 3.2; for each N ∈ N , write ρ N := log k N / log log | G N | , so that k N = (log | G N | ) ρ N . Then the following convergences in probability hold: diam G N ( Z ( N ) ) / (cid:0) α ± λ k N (cid:1) → P when lim N k N / log | G N | = λ ∈ (0 , ∞ );diam G N ( Z ( N ) ) / (cid:0) ρ N ρ N − log k N | G N | (cid:1) → P when lim N k N / log | G N | = ∞ . Moreover, the implicit lower bound on the diameter holds deterministically, ie for all choices ofgenerators, and for all Abelian groups, and, when k ≫ log | G | , the implicit upper bound holds forall groups, not just Abelian groups. Remark.
While we only state and prove the result for k & log | G | , the argument can be extendedto allow k ≪ log | G | , provided log | G | /k diverges sufficiently slowly. This requires a little more care;we do not explore the details here. As always, we drop the N -subscripts in the proof, eg writing diam G k or | G | . Proof of Theorem 5.1.
Clearly diam G k = D k (1) ≥ D k ( β ) for all β ∈ [0 , k h λ log | G | for some λ ∈ (0 , ∞ ). Let ε ≪
1, vanishing slowlyand specified later. Define α := α ± λ as in Theorem 3.2. Let A := [ Z , ..., Z (1 − ε ) k ] be the first (1 − ε ) k generators and B := [ Z (1 − ε ) k +1 , ..., Z k ] be the remaining εk . By transitivity, it suffices to considerdistances from the identity. The idea is to take L steps using A and then one more using B , where L is the minimal radius of a ball in the | A | -dimensional lattice of volume at least ne ω , for someslowly diverging ω . Write M := αk . By Lemma 3.5, we have L/M h − ε h . The key point isthat when k ≍ log | G | replacing k with (1 − ε ) k changes the typical distance by a factor 1+ o ε → (1).By Theorem 3.2, whp, A is typical in the sense that the proportion of elements of the groupwhich can be reached via a word of length at most L , using only the generators from A , is at least1 − e − ν , for some ν ≫
1, independent of ε .Condition on A , and that it is typical; write P for the probability measure induced by thisconditioning. Denote by H the set of elements which can be reached in the above sense. (This isthe vertex set of the ball of radius L in G ( A ).) Fix x ∈ G . Note that if b ∼ Unif( G ), then P (cid:0) x ∈ b + H (cid:1) ≥ − e − ν where b + H := { b + h | h ∈ H } . Furthermore, if b, b ′ ∼ Unif( G ) are independent then the events { x ∈ b + H } and { x ∈ b ′ + H } are P -independent; this is because we have conditioned on A , and so H is a deterministic set.Using the εk generators from B , informally we get εk Bernoulli trials to get to x using b + H for b ∈ B , and each trial has success probability 1 − o (1). Formally, write R for the set of elements22eachable from the identity via a word of length at most L + 1 (ie the ‘range’); let b ′ be an arbitraryelement of B , so b ′ ∼ Unif( G ). (Recall that the conditioning makes H non-random.) Then P (cid:0) x / ∈ R (cid:1) ≤ P (cid:0) x / ∈ B + H (cid:1) = P (cid:0) x / ∈ b + H ∀ b ∈ B (cid:1) = P (cid:0) x / ∈ b ′ + H (cid:1) | B | ≤ e − νεk . Since ν → ∞ , we may choose ε → νε → ∞ . Then, since k ≍ log n , we have P (cid:0) R = G (cid:1) = P (cid:0) ∃ x ∈ G st x / ∈ R (cid:1) ≤ n P (cid:0) x / ∈ R (cid:1) ≤ ne − νεk = o (1) . Averaging over A establishes an upper bound of diam G k ≤ L + 1 whp, and L ≤ M (1 + ε ).Finally consider Hypothesis C, so k ≫ log | G | . Exactly the same argument holds here, usingthe typical distance to first get to almost all the elements and then one more step. Recall fromTheorem 4.2 that the upper bound is valid for arbitrary groups. k − log | G | ≍ k In this subsection we show that the group Z d gives rise to the random Cayley graph with thelargest diameter when k − log | G | ≍ k whp, up to smaller order terms.Recall that R ( k, n ) is the minimal R ∈ N with (cid:0) kR (cid:1) ≥ n . Theorem 5.2.
Let ( k N ) N ∈ N be a sequence of positive integers and ( G N ) N ∈ N a sequence of finitegroups; for each N ∈ N , define Z ( N ) := [ Z , ..., Z k N ] by drawing Z , ..., Z k N ∼ iid Unif( G N ) .Suppose that lim inf N ( k N − log | G N | ) /k N > and lim sup N log k N / log | G N | = 0 . Then lim sup N →∞ diam G N ( Z ( N ) ) / R ( k N , | G N | ) ≤ in probability . Proof.
From Lemma 4.3 and Theorem 5.1, when k ≫ log | G | , the diameter concentrates at R ( k, | G | ) when the underlying group is Abelian, and this is an upper bound for all groups.Thus it remains to consider k with k − log | G | ≍ k and k ≍ log | G | . All that was requiredfor the upper bound on typical distance when k ≫ log | G | was that P ( W = W ′ ) ≪ / | G | where W, W ′ ∼ iid Unif( S k, ( D ) ∩ B k, ∞ (1)) with D := D (1 + ξ ), where D was the candidate typicaldistance radius and ξ > ξ > R := R ( k, | G | )(1 + ξ ). Before proceeding, let us determine someestimates on R . Let h : (0 , → (0 ,
1) : p
7→ − p log p − (1 − p ) log(1 − p ) denote the binary entropyfunction (in nats). It is standard that Stirling’s approximation, like in Lemma 4.3, gives (cid:0) kr (cid:1) = exp (cid:0) k h ( r/k ) · (cid:0) o (1) (cid:1)(cid:1) . Thus if k − log | G | ≍ k , then we see that R ( k, | G | ) ≍ k . Further, the fact that the derivative of h is continuous and strictly positive on (0 , ) gives (cid:0) kR (cid:1) ≫ | G | ; hence P ( W = W ′ ) ≪ / | G | .This shows that the typical distance D k ( β ) ≤ R ( k, | G | ) whp up to smaller order terms for allconstants β ∈ (0 , § R ( k, | G | ) ≍ k . In this section, we calculate the spectral gap; see Theorem D. We first prove it for k ≥ d ( G ).In § k − d ( G ) ≍ k and then to k − d ( G ) ≍ k for a density-(1 − ε )subset of values for | G | . The lower bound holds deterministically, without any conditions. For an Abelian group G , we write d ( G ) for the minimal size of a generating set. It is convenientto phrase the statement in terms of the relaxation time , which is the inverse of the spectral gap.23 heorem 6.1 (Spectral Gap) . First, there exists an absolute constant c > so that, for all Abeliangroups G and all multisets z of generators of size k , we have t ∗ rel (cid:0) G − ( z ) (cid:1) ≥ t rel (cid:0) G − ( z ) (cid:1) ≥ c | G | /k . (6.1a) Second, for all δ > , there exist constants c δ , C δ > so that, for all Abelian groups G , if k ≥ (2 + δ ) d ( G ) and Z , ..., Z k ∼ iid Unif( G ) , then P (cid:0) t ∗ rel ( G − k ) ≤ C δ | G | /k (cid:1) ≥ − C δ − k/c δ . (6.1b) Furthermore, for all ε ∈ (0 , , there exists a subset A ⊆ N of density at least − ε so that if | G | ∈ A then then condition k ≥ (2 + δ ) d ( G ) can be relaxed to k ≥ (1 + δ ) d ( G ) and (6.1b) stillholds; the constant C δ now also depends on ε , ie becomes C δ,ε , but c δ need not be adjusted. We prove this for the non-absolute spectral gap, ie min λ =1 { − λ } , where the minimum is overeigenvalues; the same proof also works for the absolute spectral gap, ie min λ =1 { − | λ |} . In this subsection, we establish the lower bound in Theorem 6.1.
Proof of Lower Bound in Theorem 6.1.
Write n := | G | . Abbreviate simple random walk by SRW .We may assume that k ≤ log ( n ), as otherwise (6.1a) indeed holds for some c >
0. Let L := ⌊ (( n ) /k − ⌋ . By our assumption on k , we have L ≥
1. Consider the set A := (cid:8) w · Z | w ∈ Z k and | w i | ≤ L ∀ i = 1 , ..., k (cid:9) ⊆ G. (6.2)Clearly | A | ≤ (2 L + 1) k ≤ n . Let t ≥
0, and let ( Y s ) s ≥ be a continuous-time rate-1 SRW on Z .Writing τ A c := inf { s ≥ | S s / ∈ A } for the exit time of A by the SRW S , observe that P (cid:0) τ A c > t (cid:1) ≤ P (cid:0) max s ∈ [0 ,t/k ] | Y s | ≤ L (cid:1) k , (6.3)where 0 ∈ A is the identity of the group. It follows from Lemma 6.3 below that P (cid:0) max s ∈ [0 ,t/k ] | Y s | ≤ L (cid:1) ≥ exp (cid:0) − π ( t/k ) / ( L + 1) (cid:1) . Substituting this into (6.3) we get P (cid:0) τ A c > t (cid:1) ≥ exp (cid:0) − tπ / ( L + 1) (cid:1) . (6.4)The minimal Dirichlet eigenvalue of a set A is defined to be the minimal eigenvalue of minusthe generator of the walk killed upon exiting A ; we denote it by λ A . For connected A , we show inLemma 6.4 below that, for all a ∈ A , we have − t log P a (cid:0) τ A c > t (cid:1) → λ A as t → ∞ . From this and (6.4), it then follows that λ A ≤ λ where λ := π / ( L + 1) ≤ π / (cid:0) ( n ) /k + 1 (cid:1) . Since | A | ≤ n , applying [3, Corollary 3.34], we get t rel ≥ (1 − n | A | ) /λ ≥ / (2 λ ) . This concludes the proof of the lower bound in Theorem 6.1, namely (6.1a).24 .3 Upper Bound on Relaxation Time
In this subsection, we establish the upper bound in Theorem 6.1, namely (6.1b). We prove itfor the usual spectral gap t rel ; the same proof applies to bound the absolute spectral gap t ∗ rel . Inparticular, we bound the probability that 1 − λ is small; a completely analogous calculation canbe used to bound the probability that 1 + λ n is small. We only present the former calculation.For ease of presentation, we assume first that k ≥ d ( G ). In § Proof of Upper Bound in Theorem 6.1.
Decompose G as ⊕ d Z m j . An orthogonal basis of eigen-vectors for P , the transition matrix of the corresponding discrete-time walk, is given by( f x | x ∈ G ) where f x ( y ) := cos (cid:0) π P di =1 x i y i /m i (cid:1) , with corresponding eigenvalues given by (cid:0) λ x | x ∈ G (cid:1) where λ x = k P ki =1 cos (cid:0) π (¯ x · Z i ) (cid:1) , where ¯ x j = x j /m j for all j = 1 , ..., d and ¯ x · Z i = P dj =1 x j Z ji /m j is the standard inner-product on R d , where Z ji is the j -th coordinate of the i -th generator Z i ; herewe identify ¯ x and Z j with elements of R d in a natural manner.Observe that λ = 1. Our goal is to bound min x ∈ G \{ } { − λ x } from below. For α ∈ R , let { α } be the unique number in ( − , ] so that α − { α } ∈ Z . It follows from Lemma 6.5 below that1 − λ x ≥ π k P ki =1 { ¯ x · Z i } . (6.5)For each x ∈ G , we make the following definitions: g j := g j ( x ) := gcd( x j , m j ) for each j ≥ s ∗ := s ∗ ( x ) := max (cid:8) m j /g j | j ∈ { , ..., d } (cid:9) ; A ( s ) := (cid:8) x ∈ G | s ∗ ( x ) = s (cid:9) for each s ≥ φ ( j ) := (cid:12)(cid:12)(cid:8) j ′ ∈ { , ..., j } | gcd( j, j ′ ) = 1 (cid:9)(cid:12)(cid:12) for each j ≥ . From this, we claim that we are able to deduce, for s ≥
2, that | A ( s ) | ≤ (cid:0)P sj =1 φ ( j ) (cid:1) d ≤ (cid:0) P sj =2 ( j − (cid:1) d ≤ (cid:0) s (cid:1) d . (6.6)Indeed, φ ( j ) ≤ j − j ≥
2, and observe thatif r divides m , then (cid:12)(cid:12)(cid:8) a ∈ { , ..., m } (cid:12)(cid:12) gcd( a, m ) = r (cid:9)(cid:12)(cid:12) = φ ( m/r );hence, summing over the set of possible values for m j /g j , which by definition of A ( s ) is { , ..., s } ,we have | A ( s ) | /d ≤ P sj =1 φ ( j ). We are then able to deduce the upper bound, ie (6.1b), fromProposition 6.2, which we state precisely below. Indeed, first write p ( s ) := max x : s ∗ ( x )= s P (cid:0) − λ x ≤ c n − /k (cid:1) . By (6.5) along with Proposition 6.2 and Lemma 6.5 (stated below), letting c ′ := c · π , we have X x ∈ G \{ } P (cid:0) − λ x ≤ c ′ n − /k (cid:1) ≤ n max s>C n /k p ( s ) + X ≤ s ≤ C n /k | A ( s ) | p ( s ) ≤ − k + 2 − d P s ≥ s d (2 s ) − k/ . − k , where we have used k ≥ d and the fact that s ∗ ( x ) > x = 0.Modulo the proofs of the quoted results, ie Proposition 6.2 and Lemmas 6.3 to 6.5, this concludesthe proof of the upper bound in Theorem 6.1, namely (6.1b).It remains to state and prove the quoted results, ie Proposition 6.2 and Lemmas 6.3 to 6.5.25 roposition 6.2. There exist absolute constants c ∈ (0 , and C such that P (cid:0) k P ki =1 { ¯ x · Z i } ≤ c n − /k (cid:1) ≤ ( s ∗ ( x ) − k/ where s ∗ ( x ) ≤ C n /k , − k /n where s ∗ ( x ) > C n /k . (6.7a)(6.7b) Proof.
Fix x ∈ G . First consider the case that s := s ∗ ( x ) > C n /k , ie (6.7b). Let j := j ( x ) bea coordinate satisfying s = m j /g j . Denote m := m j ( x ) and g := g j ( x ) . Observe that x j Z ji ∼ iid Unif { g, g, ..., m } for each i . Hence, for each i , we have U i := ¯ x j Z ji ∼ Unif { /s, /s, ..., } . (6.8)By averaging over ( a i ) ki =1 , where a i := { P ℓ ∈{ ,...,d }\{ j } x ℓ Z ℓi /m ℓ } , recalling that { α } is theunique number in ( − , ] so that α − { α } ∈ Z , it suffices to show thatmax b ,...,b k ∈ [ − / , / P (cid:0) k P ki =1 { U i + b i } ≤ c n − /k (cid:1) ≤ − k /n. (6.9)Replacing c with 4 c we may assume that b i ∈ s Z . Indeed, if | b i − ℓ/s | ≤ / (2 s ) , ie | b i − ℓ/s | = min (cid:8) | b i − α | | α ∈ s Z (cid:9) , then { U i + ℓ/s } ≤ { U i + b i } . Henceif k P kj =1 { U i + b i } ≤ c n − /k then k P kj =1 { U i + ℓ/s } ≤ c n − /k . In this case, { U i + b i } has the same law as { U i } . Hence it suffices to prove (6.9) for b = · · · = b k = 0.We now split [0 , ] into M := ⌈ n /k ⌉ consecutive intervals of equal length J , ..., J M , where J := [0 , M ] and J ℓ := ( ℓ − M , ℓ M ] for ℓ >
1. Let Y i := ℓ − |{ U i }| ∈ J ℓ . Clearly, Y i /M ≤ Y i /M ≤ { U i } . It thus suffices to show that P (cid:0) k P ki =1 Y i ≤ (cid:1) ≤ − k /n. This last claim follows by a simple counting argument: there are M k total assignments of the Y i -s,but at most L ( k ) := (cid:0) ⌈ k/ ⌉ k − (cid:1) ≤ k assignments satisfy k P ki =1 Y i ≤ , since L ( k ) /M k ≤ − k n − .We now prove the case s := s ∗ ( x ) ≤ C n /k , ie (6.7a). By the same reasoning as for (6.9), itsuffices to show that max b ,...,b k ∈ [ − / , / P (cid:0) k P ki =1 { U i + b i } ≤ c n − /k (cid:1) ≤ s − k/ . (6.10)Regardless of b i , there is at most one a := a ( b i ) ∈ { /s, /s, ..., } such that { a + b i } < (2 s ) − ,and hence by (6.8), for all i , we have P (cid:0) { U i + b i } < (2 s ) − (cid:1) ≤ /s. If there is no such value a ( b i ), then set a ( b i ) := − { U i + b i } ≥ (2 s ) − for at least q := k · c s n − /k of the i -s, ie if (cid:12)(cid:12)(cid:8) i ∈ { , ..., k } | U i = a ( b i ) (cid:9)(cid:12)(cid:12) ≥ q, then k P ki =1 { U i + b i } ≥ c n − /k , as desired. As s ≤ C n /k , by taking c sufficiently small interms of C , we can make q/k sufficiently small so that the following holds: P (cid:0)(cid:12)(cid:12)(cid:8) i ∈ { , ..., k } | U i = a ( b i ) (cid:9)(cid:12)(cid:12) < q (cid:1) . (cid:0) kq (cid:1) s q − k . s − k/ . We now state the auxiliary lemmas referenced above, ie Lemmas 6.3 to 6.5. These are technicalresults; their proofs are given in [15, § D].
Lemma 6.3.
Let ℓ ∈ N and τ := inf { s ≥ | | Y s | = ℓ } , where ( Y s ) s ≥ is a continuous-time rate-1simple random walk on Z . Let θ := π/ℓ and λ := 1 − cos θ . Then, for all s ≥ , we have P (cid:0) τ > s (cid:1) ≥ e − λs ≥ exp (cid:0) − s ( π/ℓ ) (cid:1) . P and a set A , let λ A be the minimal Dirichlet eigenvalue , defined tobe the minimal eigenvalue of minus the generator of the chain killed upon exiting A , ie of I A − P A where ( I A − P A )( x, y ) := (cid:0) x, y ∈ A (cid:1)(cid:0) ( x = y ) − P ( x, y ) (cid:1) . Also, for a set A , write τ A c for the (first) exit time of this set by the chain. Lemma 6.4.
Consider a rate-1, continuous-time, reversible Markov chain with transition matrix P . Let A be a connected set, and let λ A and τ A c be as above. Then, for all a ∈ A , we have − t log P a (cid:0) τ A c > t (cid:1) → λ A as t → ∞ . Lemma 6.5.
For θ ∈ [ − , ] , we have πθ ) ≥ − cos(2 πθ ) ≥ ( πθ ) . Remark.
Our proof gives an explicit form for c in (6.1a). If k ≪ log n , then we get t rel ≥ π − | G | /k · (cid:0) o (1) (cid:1) . Indeed, in this case, in the definition of the set A in (6.2), we can take L := ⌊ ( εn ) /k ⌋ for any ε >
0, making | A | / | G | arbitrary small. One can improve the constant by replacing A with (cid:8) w · Z (cid:12)(cid:12) w ∈ Z k and P ki =1 | w i | ≤ L ( k, n ) (cid:9) , where L ( k, n ) is the maximal integer satisfying |{ w ∈ Z k | P ki =1 | w i | ≤ L ( k, n ) }| ≤ n. △ k In this subsection, we explain how to relax the conditions on k . First we can relax from k ≥ d ( G ) to k − d ( G ) ≍ k , valid for every group size n = | G | .We now give conditions under which this can be relaxed to k − d ( G ) ≍ k . If G = Z dp for aprime p , then one can relax this further to k − d & d , and even allow k − d ( G ) ≪ d ( G ), provided p diverges. (In this case, the term 2 − k has to be replaced by another term which tends to zeroat a slower rate as k → ∞ .) This follows from the fact that now we only need to consider (6.6)above with s := p and we can replace (6.6) with | A ( p ) | = p d −
1. So the condition k − d ( G ) ≍ k issufficient when G = Z dp with p prime.We now show that if | G | is ‘typical’ (in a precise sense), then the same condition is sufficient.In the proof above, in (6.6), we used the crude bound | A ( s ) | ≤ (cid:0)P i ∈ [ s ] φ ( i ) (cid:1) d ≤ (cid:0) s (cid:1) d . Instead, recalling that we write i ≀ n to mean that i divides n , we can use the improved bound | A ( s ) | ≤ (cid:0)P i ∈ [ s ] i ( i ≀ n ) (cid:1) d . In [15, Lemma F.7], we show that, for all ε >
0, there exists a constant C ′ ε and a density-(1 − ε )set B ε ⊆ N such that, for all n ∈ B ε and all 2 ≤ s ≤ n , we have P i ∈ [ s ] i ( i ≀ n ) ≤ C ′ ε s (log s ) . Using this to derive an improved bound on | A ( s ) | , and adjusting some of the constants in the proofin an appropriate manner, an inspection of the proof reveals that, for all n ∈ B ε and all δ > C ε,δ so that, for all Abelian groups of size n , if k ≥ (1 + δ ) d , then P (cid:0) t rel ( G k ) ≥ C ε,δ n /k (cid:1) ≤ e − k/C ε,δ . Open Questions and Conjectures
We close the paper with some questions which are left open.
1: Typical Distance and Diameter for All Abelian Groups
In our typical distance theorem, there were some conditions on the group. We allowed any groupwith d ( G ) ≪ log | G | / log log k if 1 ≪ k ≪ log | G | , but once d ( G ) became larger than this or k be-came order log | G | , we had to impose conditions. We conjecture that these are artefacts of the proof. Conjecture 1.
Let G be an Abelian group. Suppose that ≪ k . log | G | and k − d ( G ) ≫ . Then the typical distance statistic concentrates at a value which dependsonly on k and G , not the particular realisation of the generators. Further, if k ≪ log | G | and k − d ( G ) ≍ k , then it concentrates at a value which depends only on k and | G | . The claim when 1 ≪ k ≪ log | G | and k − d ( G ) ≍ k is a natural extension of Theorem 2.2.Further, if k ≪ p log | G | / log log log | G | , then k − d ( G ) ≫ k − d ( G ) ≫
1, for larger k , we still expect concentration of typical distance for allAbelian groups, but now the value will likely depend on the specific group. Compare this with theoccurrence of cutoff for the random walk on the random Cayley graph established in [11].There are two levels on which concentration occurs: first, for a fixed graph G ( z ), one drawsa U ∼ Unif( G ) and looks for concentration of dist( id , U ) at some value, say f ( z ); second, onedraws Z uniformly and looks for concentration of f ( Z ). The second is the meat of Conjecture 1.Indeed, our lower bound on typical distance holds for all Abelian groups and all Cayley graphswith k generators, thus necessarily P G ( z ) (dist( id , U ) & k | G | /k ) = 1 − o (1) for all such Cayleygraphs G ( z ). Additionally, our spectral gap estimate (Theorem D) says that the gap is order | G | − /k if k − d ( G ) ≍ k (or when k − d ( G ) ≍ k and | G | is ‘typical’) whp over uniform Z .Since u dist( id , u ) is a 1-Lipschitz function, by Poincar´e’s inequality Var G ( z ) (dist( id , U )) ≤ t rel ( G ( z )). For all multisets z of size k satisfying the aforementioned spectral gap estimate fromTheorem D (which holds whp for G k ), using our deterministic lower bound on the typical distance,we see that dist G ( z ) ( id , U ) concentrates at some value f ( z ), which may depend on z , by Chebyshev’sinequality. We conjecture that in fact f ( Z ) concentrates at some value D .It is easy to see that the typical distance and diameter are always the same up to constants. Weconjecture that the diameter of G k concentrates whp whenever 1 ≪ k . log | G | and k − d ( G ) ≫
2: Isoperimetry for Random Cayley Graphs
The isoperimetric , or
Cheeger , constant of a finite d -regular graph G = ( V, E ) is defined asΦ ∗ := d min ≤| S |≤ | V | Φ( S ) where Φ( S ) := | S | (cid:12)(cid:12)(cid:8) { a, b } ∈ E (cid:12)(cid:12) a ∈ S, b ∈ S c (cid:9)(cid:12)(cid:12) . More generally, the isoperimetric constant is defined for Markov chains; see [19, § P , it is easy to see that the original chain P , the time-reversal P ∗ and theadditive symmetrisation ( P + P ∗ ) all have the same isoperimetric profile. Thus the isoperimetricconstant for a directed Cayley graphs is the same as that for the undirected version.The following conjecture asserts that the Cheeger constant is, up to a constant factor, the sameas that of the standard Cayley graph of Z kL where L is such that n ≍ L k . Conjecture 2.
There exists a constant c so that, for all ε ∈ (0 , , there exist constants n ε and M ε so that, for every finite group G of size at least n ε , when k ≥ M ε , we have P (cid:0) Φ ∗ ( G k ) ≤ c | G | − /k (cid:1) ≤ ε, where Φ ∗ ( G k ) is the Cheeger constant of a random Cayley graph with k generators.
28y [23, Theorem 6.29], which regards expansion of general Cayley graphs, along with out upperbound on typical distance (and hence on diameter), we can prove this conjecture up to a factor k .By the well-known discrete analogue of Cheeger’s inequality, discovered independently by mul-tiple authors—see, for example, [19, Theorem 13.10]—we have γ ≤ Φ ∗ ≤ √ γ . Determining thecorrect order of Φ ∗ in our model remains an open problem. We conjecture that the correct orderof Φ ∗ is given by √ γ , ie order | G | − /k , using Theorem D for the order of the spectral gap.The celebrated Alon–Roichman theorem states that the Cayley graph of any finite group G is a (1 − ε )-expander (ie Φ ∗ ≥ − ε ) whp when k ≥ C ε log | G | , for some constant C ε ; the bestknown upper bound on C ε is O (1 /ε ). Naor [25, Theorem 1.2] refines this for Abelian groups: heshowed that one can in fact bound | Φ( S ) − | ≤ ε p log | S | / log | G | for all S with 1 ≤ | S | ≤ | V | simultaneously, when k/ log n ≥ C/ε , for a constant C . References [1] D. Aldous and P. Diaconis (1985). Shuffling Cards and Stopping Times.
Technical Report 231, Depart-ment of Statistics, Stanford University . Available online[2] D. Aldous and P. Diaconis (1986). Shuffling Cards and Stopping Times.
Amer. Math. Monthly . .5(333–348) MR841111 DOI[3] D. Aldous and J. A. Fill (2002). Reversible Markov Chains and Random Walks on Graphs . UnfinishedMonograph
Available at stat.berkeley.edu/~aldous/RWG/book.html [4] N. Alon and Y. Roichman (1994). Random Cayley Graphs and Expanders.
Random Structures Algorithms . .2 (271–284) MR1262979 DOI[5] G. Amir and O. Gurel-Gurevich (2010). The Diameter of a Random Cayley Graph of Z q . Groups Complex.Cryptol. .1 (59–65) MR2672553 DOI[6] I. Benjamini (2018). Private Communication.[7] S. Chen, C. Moore and A. Russell (2013). Small-Bias Sets for Nonabelian Groups: Derandomizations ofthe Alon–Roichman Theorem. Approximation, Randomization, and Combinatorial Optimization , LectureNotes in Comput. Sci. Springer, Heidelberg (436–451) MR3126546 DOI[8] D. Christofides and K. Markstr¨om (2008). Expansion Properties of Random Cayley Graphs and VertexTransitive Graphs via Matrix Martingales.
Random Structures Algorithms . .1 (88–100) MR2371053DOI[9] P. Diaconis (2019). Private Communication.[10] C. Dou and M. Hildebrand (1996). Enumeration and Random Random Walks on Finite Groups. Ann.Probab. .2 (987–1000) MR1404540 DOI[11] J. Hermon and S. Olesker-Taylor (2021). Cutoff for Almost All Random Walks on Abelian Groups. Available on arXiv [12] J. Hermon and S. Olesker-Taylor (2021). Cutoff for Random Walks on Upper Triangular Matrices.
Available on arXiv [13] J. Hermon and S. Olesker-Taylor (2021). Further Results and Discussions on Random Cayley Graphs.
Available on arXiv [14] J. Hermon and S. Olesker-Taylor (2021). Geometry of Random Cayley Graphs of Abelian Groups.
Avail-able on arXiv [15] J. Hermon and S. Olesker-Taylor (2021). Supplementary Material for Random Cayley Graphs Project.
Available on arXiv [16] M. Hildebrand (1994). Random Walks Supported on Random Points of Z /n Z . Probab. Theory RelatedFields . .2 (191–203) MR1296428 DOI[17] R. Hough (2017). Mixing and Cut-Off in Cycle Walks. Electron. J. Probab. (Paper No. 90, 49 pp.)MR3718718 DOI[18] Z. Landau and A. Russell (2004). Random Cayley Graphs Are Expanders: A Simple Proof of the Alon–Roichman Theorem. Electron. J. Combin. .1 (Research Paper 62, 6 pp.) MR2097328 DOI[19] D. A. Levin, Y. Peres and E. L. Wilmer (2017). Markov Chains and Mixing Times . Second ed., AmericanMathematical Society, Providence, RI, USA MR3726904 DOI
20] M. E. Lladser, P. Potoˇcnik, J. ˇSir´aˇn and M. C. Wilson (2012). Random Cayley Digraphs of Diameter 2and Given Degree.
Discrete Math. Theor. Comput. Sci. .2 (83–90) MR2992954[21] P.-S. Loh and L. J. Schulman (2004). Improved Expansion of Random Cayley Graphs. Discrete Math.Theor. Comput. Sci. .2 (523–528) MR2180056[22] E. Lubetzky and Y. Peres (2016). Cutoff on All Ramanujan Graphs. Geom. Funct. Anal. .4 (1190–1216) MR3558308 DOI[23] R. Lyons and Y. Peres (2016). Probability on Trees and Networks . Cambridge University Press, NewYork MR3616205 DOI[24] J. Marklof and A. Str¨ombergsson (2013). Diameters of Random Circulant Graphs.
Combinatorica . .4(429–466) MR3133777 DOI[25] A. Naor (2012). On the Banach-Space-Valued Azuma Inequality and Small-Set Isoperimetry of Alon–Roichman Graphs. Combin. Probab. Comput. .4 (623–634) MR2942733 DOI[26] I. Pak (1999). Random Cayley Graphs with O (log | G | ) Generators Are Expanders.
Algorithms—ESA ’99(Prague) , Lecture Notes in Comput. Sci. Springer, Berlin (521–526) MR1729149 DOI[27] S. Purkayastha (1998). Simple Proofs of Two Results on Convolutions of Unimodal Distributions.
Stat-istics & Probability Letters . .2 (97–100) MR1652520 DOI[28] Y. Roichman (1996). On Random Random Walks. Ann. Probab. .2 (1001–1011) MR1404541 DOI[29] N. T. Sardari (2019). Diameter of Ramanujan Graphs and Random Cayley Graphs. Combinatorica . .2(427–446) MR3962908 DOI[30] U. Shapira and R. Zuck (2019). Asymptotic Metric Behavior of Random Cayley Graphs of Finite AbelianGroups. Combinatorica . .5 (1133–1148) MR4039604 DOI[31] D. B. Wilson (1997). Random Random Walks on Z d . Probab. Theory Related Fields . .4 (441–457)MR1465637 DOI.4 (441–457)MR1465637 DOI