RANDOM WALK on UNIPOTENT MATRIX GROUPS 1. Introduction

Home , Orthogonal matrix, Symmetric matrix, Unipotent

RANDOM WALK ON UNIPOTENT MATRIX GROUPS

PERSI DIACONIS AND BOB HOUGH

Abstract. We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure. As a second illustration, the method is used to study walks on the n ˆ n uni-upper triangular group with entries taken modulo p. The method allows sharp answers to the behavior of individual coordinates: coordinates immediately above the diagonal require order p2 steps for randomness, coordinates on the second diagonal require order p steps; coordinates on the kth diagonal require order 2 p k steps.

1. Introduction Let HpRq denote the real Heisenberg group 1 x z HpRq “ 0 1 y : x, y, z P R . $¨ ˛ , & 0 0 1 . 1 x z ˝ ‚ % - Abbreviate 0 1 y with rx, y, zs, identiﬁed with a vector in R3. ¨ 0 0 1 ˛ Consider simple˝ random‚ walk on G “ HpRq driven by Borel probability measure µ. For N ě 1, the law of this walk is the convolution power ˚N µ where, for measures ν, ξ on G, and for f P CcpGq,

xf, ν ˚ ξy “ fpghq dνpgq dξphq. żg,hPG Say that measure µ is non-lattice (aperiodic) if its support is not contained in a proper closed subgroup of G. For general non-lattice µ of compact support Breuillard [6] uses the representation theory of G to prove a local limit theorem for the law of µ˚N , asymptotically evaluating its density in translates of bounded Borel sets. However, in evaluating µ˚N on Borel sets translated on both the left and the 2010 Mathematics Subject Classification. Primary 60F05, 60B15, 20B25, 22E25, 60J10, 60E10, 60F25, 60G42. Key words and phrases. Random walk on a group, Heisenberg group, local limit theorem, unipotent group. We are grateful to Laurent Saloff-Coste, who provided us a detailed account of previous work. 1 2 PERSI DIACONIS AND BOB HOUGH right he makes a decay assumption on the Fourier transform of the abelianization of the measure µ, and raises the question of whether this is needed. We show that this condition is unnecessary. In doing so we give an alternative approach to the local limit theorem on G treating it as an extension of the classical local limit theorem on Rn, with the further advantage that our argument applies without significant change to arbitrary µ of compact support. We also obtain the optimal rate. The method of argument is related to, though simpler than, the analysis of quantitative equidistribution of polynomial orbits on G from [18]. Recall that the abelianization Gab “ G{rG, Gs of G is isomorphic to 2 R with projection p : G Ñ Gab given by pprx, y, zsq “ rx, ys. Assume that the probability measure µ satisfies the following conditions. (1) Compact support. (2) Centered. The projection p satisfies

ppgqdµpgq “ 0. żG (3) Lazy. For all open sets N with id P N, µpNq ą 0. (4) Full dimension. Let Γ “ xsupp µy be the closure of the subgroup of G generated by the support of µ. The quotient G{Γ is compact. Section 2 gives a characterization of closed subgroups Γ of G of full dimension. Note that a closed subgroup Γab of full dimension in the abelianization is isomorphic to one of R2, R ˆ Z, Z2. In the case of Z2, 2 there are v1, v2 P R such that Γab “ Zv1 ` Zv2; set V “ |v1 ^ v2| for the volume of the lattice, and define the parity function ε :Γ Ñ Z{2Z 2 by εpn1v1 `n2v2q ” n1n2 mod 2. In either case RˆZ or Z , write Γab,nl for the non-lattice component of Γab. Under the above conditions, the central limit theorem for µ is known, see the history at the end of the introduction. Let pdtqtą0 denote the 2 semigroup of dilations given by dtprx, y, zsq “ rtx, ty, t zs and denote the Gaussian semigroup pνtqtą0 defined by its generator (see [6], [10]) d A f “ fpgqdνtpgq dt ˇt“0 żgPG ˇ 1 1 “ zBˇfpidq ` xyB2 fpidq ` x2B2fpidq ` y2B2fpidq zˇ xy 2 x 2 y 2 2 2 2 2 2 where σx “ x “ g“rx,y,zsPG x dµpgq and similarly σy “ y , σxy “ xy, z. With ν “ ν , the central limit theorem for µ states that for f P C pGq, 1 ş c ˚N f, d ?1 µ Ñ xf, νy. N A E For g P G define the left and right translation operators Lg,Rg : L2pGq Ñ L2pGq,

Lgfphq “ fpghq,Rgfphq “ fphgq. RANDOM WALK ON UNIPOTENT MATRIX GROUPS 3

Our local limit theorem is as follows. Theorem 1. Let µ be a Borel probability measure of compact support on G “ HpRq, which is centered, lazy and full dimension. Let, as above, Γ “ xsupp µy be the closure of the group generated by the support of µ, let F be a fundamental domain for G{Γ having volume M ą 0 and set 1F χF “ M for its unit density. Let ν be the limiting Gaussian measure ˚N of d ?1 µ . Uniformly for g, h P G and for f P CcpGq, for N ě 1, N ˚N ? ´2 (1) LgRhf, χF ˚ µ “ LgRhf, d N ν ` oµ }f}1 N . If any@ of the following conditionsD @ holds D ` ˘ i. Γ is a discrete subgroup of HpRq 2 ii. Γ is not discrete, but Γab is a discrete subgroup of R and the Cram´ercondition holds

´i αx`βy` 2πλ z´ xy ` aεpx,yqV sup e p V p 2 2 qqdµpgq ă 1 α,βPR ˇżg“rx,y,zsPG ˇ aPt0,1u ˇ ˇ |λ|ą1 ˇ ˇ ˇ ˇ iii. Γab is not discrete, but the Cram´ercondition holds:

sup eíλ¨px,yqdµpgq ă 1 g“rx,y,zsPG λPΓab,nl, |λ|ą1 ˇż ˇ ˇ ˇ then uniformly for g, h Gˇ and f C G , ˇ { P ˇ P cp q ˇ 5 ˚N ? ´ 2 (2) LgRhf, χF ˚ µ “ LgRhf, d N ν ` Of,µ N . Remark.@The rate is best possibleD @ as may beD seen by projecting´ ¯ to the abelianization. A variety of other statements of the local theorem are also derived, see eqn. (9) in Section 3. Remark. For non-lattice µ, [6] obtains (1) with h “ id and for general h subject to Cramér’scondition. A condition somewhat weaker than Cramér’swould suffice to obtain (2) in the case of ii and iii. Remark. In the case that µ is discrete or has a density, [1, 2] obtains ´ 5 ˚N an error of O N 2 in approximating µ pgq, g P Γ with the corresponding heat´ kernel.¯ The estimates of Theorem 1 permit the following asymptotic evalua- tion of the probability of return to the identity in simple random walk on HpZq.

Corollary 2. Let µ0 be the measure on HpZq which assigns equal prob- 1 ability 5 to the each element of the generating set 1 ˘1 0 1 0 0 e, 0 1 0 , 0 1 ˘1 . $ ¨ ˛ ¨ ˛, & 0 0 1 0 0 1 . ˝ ‚ ˝ ‚ % - 4 PERSI DIACONIS AND BOB HOUGH

25 5 ˚N ´ 2 As N Ñ 8, µ0 peq “ 16N 2 ` O N . The basic idea which drives´ the proof¯ of Theorem 1 is that per- muting segments of generators in a typical word of the walk generates smoothness in the central coordinate of the product, while leaving the abelianized coordinates unchanged. This observation permits passing from a limit theorem to a local limit theorem by smoothing at a de- creasing sequence of scales. As a further application of this technique, answering a question of [13] we determine the mixing time of the central coordinate in a natural class of random walks on the group NnpZ{pZq of n ˆ n uni-upper triangular matrices with entries in Z{pZ. Theorem 3. Let n ě 2 and let µ be a probability measure on Zn´1 which satisﬁes the following conditions. (1) Bounded support. (2) Full support. xsupp µy “ Zn´1 (3) Lazy. µp0q ą 0 (4) Mean zero. xµ x 0 xPZn´1 p q “ (5) Trivial covariance. ř n´1 piq pjq x x µpxq “ In´1. ˜xP n´1 ¸ ÿZ i,j“1 Push forward µ to a probability measure µ˜ on NnpZq via, for all x P Zn´1, 1 xp1q 0 ¨ ¨ ¨ 0 . . p2q .. . ¨ 0 1 x . ˛ . . . . µ˜ ...... 0 “ µpxq. ˚ ‹ ˚ 0 1 xpn´1q ‹ ˚ ‹ ˚ 0 ¨ ¨ ¨ 0 1 ‹ ˚ ‹ ˝ ‚ Write Z : NnpZq Ñ Z for the upper right corner entry of a matrix of NnpZq. There exists C ą 0 such that, for all primes p, for N ě 1,

˚N 1 N µ˜ pZ ” x mod pq ´ ! exp ´C 2 . p n´1 x mod p ˇ ˇ ˜ p ¸ ÿ ˇ ˇ ˇ ˇ 2 Remark. Informally,ˇ the top right cornerˇ entry mixes in time O p n´1 . This is tight, since archimedean considerations show that the´L1 dis-¯ 2 tance to uniform is " 1 if the number of steps of the walk is ! p n´1 . Remark. Although we have considered only the top right corner entry in UnpZ{pZq, this result determines the mixing time of each entry above the diagonal by iteratively projecting to the subgroups determined by the top left or bottom right m ˆ m sub-matrices. RANDOM WALK ON UNIPOTENT MATRIX GROUPS 5

Remark. The argument presented here appears to be suﬃcient to give a joint local limit theorem for random walk on Un, n ą 3 and other nilpotent groups analogous to Theorem 1 for random walk on U3, but we have not checked this carefully.

History Random walk on groups is a mature subject with myriad projections into probability, analysis and applications. Useful overviews with ex- tensive references are in [5], [24]. Central limit theorems for random walk on Lie groups were first proved by [27] with [26] carrying out the details for the Heisenberg group. Best possible results under a second moment condition for nilpotent Lie groups are in [23]. A general local limit theorem for the Heisenberg group appears in [6], which contains a useful historical review. There similar conditions to those of our Theorem 1 are made, but the argument treats only the non-lattice case and needs a stronger condition on the characteristic function of the measure projected to the abelianization. Remarkable local limit theorems are in [1, 2]. The setting is groups of polynomial growth, and so “essentially” nilpotent Lie groups via Gromov’s Theorem. The first paper gives quite complete results assuming that the generating measure has a density. The second paper gives results for measures supported on a lattice. The arguments in [2] have been adapted in [7] to give a local limit theorem for non-lattice measures supported on finitely many points. Just as for the classical abelian case, many variations have been stud- ied. Central limit theorems for walks satisfying a Lindeberg condition on general Lie groups are proved in [25], see also references therein. Large deviations for walks on nilpotent groups are proved in [3]. Cen- tral limit theorems on covering graphs with nilpotent automorphism groups are treated in [20, 21]. This allows walks on Cayley graphs with some edges and vertices added and deleted. Brownian motion and heat kernel estimates are also relevant, see [19, 17]. Random walk on finite nilpotent groups are a more recent object of study. Diaconis and Saloff-Coste [15, 14, 13] show that for simple symmetric random walk on Z{nZ, order n2 steps are necessary and sufficient for convergence to uniform. The first paper uses Nash inequalities, the second lifts to random walk on the free nilpotent group and applies central limit theorems of Hebisch, Saloff-Coste and finally Harnack inequalities to transfer back to the finite setting. The third paper uses geometric ideas of moderate growth to show that for groups of polynomial growth, diameter-squared steps are necessary and sufficient to reach uniformity. This paper raises the question of the behavior of the individual coordinates on UnpZ{pZq which is finally answered in Theorem 3. A direct non-commuting Fourier approach to HpZ{pZq is carried out in [8], where it is shown that order p log p steps suffice to 6 PERSI DIACONIS AND BOB HOUGH make the central coordinate random, improved here to order p steps, which is best possible. For a review of the HpZq results, see [12]. Fi- nally there have been quite a number of papers studying the walk on UnpZ{pZq when both p and n grow. We refer to [22], which contains a careful review and definitive results.

Notation and conventions Vectors from Rd, d ě 1 are written in plain text w, their coordinates with superscripts wpiq, and sequences of vectors with an underline w. The sum of a sequence of vectors w is indicated w. wt denotes the transpose of w. We have been cavalier in our use of transpose; in- terpretation of vectors as rows or columns should be clear from the context. We frequently identify matrix elements in the group Un with vectors from Euclidean space, and have attempted to indicate the way in which the vectors should be interpretted. As a rule of thumb, when the group law is written multiplicatively, the product is in the group Un, and when additively, in Euclidean space. The arguments presented use permutation group actions on sequences of vectors. Given integer N ě 1, denote SN the symmetric group on rNs “ Z X r1,Ns, which acts on length N sequence of vectors by per- muting the indices:

SN Q σ : pw1, ..., wN q ÞÑ pwσp1q, ..., wσpNqq.

d C2 is the two-element group. For d ě 1, identify C2 with the d- d d dimensional hypercube t0, 1u . 1d is the element of C2 corresponding d to the sequence of all 1’s on the hypercube. C2 acts on sequences of vectors of length 2d with the jth factor determining the relative order of the ﬁrst and second blocks of 2j´1 elements. To illustrate the action 2 of C2 on x “ x1x2x3x4:

p0, 0q ¨ x “ x1x2x3x4

p1, 0q ¨ x “ x2x1x3x4

p0, 1q ¨ x “ x3x4x1x2

p1, 1q ¨ x “ x3x4x2x1.

d The 2-norm on R is indicated } ¨ } and } ¨ }pR{Zqd denotes distance d to the nearest integer lattice point. Given ξ P R , eξp¨q denotes the d 2πiξ¨x character of R , eξpxq “ e . Given m ą 1 and a mod m, ea,mp¨q 2πiax denotes the character of Z{mZ, ea,mpxq “ e m . d Use δx to indicate the Dirac delta measure at x P R . Given f P d CcpR q and measure µ, xf, µy denotes the bilinear pairing

xf, µy “ fpxqdµpxq. d żR RANDOM WALK ON UNIPOTENT MATRIX GROUPS 7

Denote the Fourier transform of function f, resp. the characteristic function of measure µ by, for ξ P Rd, ˆ fpξq “ e´ξpxqfpxqdx, µˆpξq “ e´ξpxqdµpxq. d d żR żR d For x P R , Txf denotes function f translated by x, ˆ Txfpyq “ fpy ´ xq, Txfpξq “ e´ξpxqfpξq and for real t ą 0, f denotes f dilated by t, t y 1 x f pxq “ f , f pξq “ fˆptξq . t t t t ´ ¯ For smooth f, the Plancherel identity isp

xf, µy “ fˆpξqµˆpξqdξ. d żR For r P R and σ ą 0, ηpr, σq denotes the one-dimensional Gaussian distribution with mean r and variance σ2, with density and characteristic function

px´rq2 exp ´ 2 2σ 2 2 2 ηpr, σqpxq “ ? , ηpr, σqpξq “ e ξprq exp ´2π σ ξ . ´ 2πσ ¯ ´ ` ˘ A centered (mean zero) normal distribution{ η in dimension d is speciﬁed by its covariance matrix

d σ2 “ xpmqxpnqηpxq d ˆżR ˙m,n“1 and has density and characteristic function

t 2 ´1 exp ´ x pσ q x 2 2 t 2 ηp0, σqpxq “ 1 , ηp0, σqpξq “ exp ´2π ξ σ ξ . ´d 2 ¯ p2πq 2 pdet σ q 2 ` ˘ All of our arguments concern the repeated{ convolution µ˚N of a ﬁxed measure µ on the upper triangular matrices. Asymptotic statements are with respect to N as the large parameter. The Vinogradov notation A ! B, resp. A " B, means A “ OpBq, resp. B “ OpAq. A — B means A ! B and B ! A.

2. Background to Theorem 1 This section collects together several background statements regarding the Heisenberg group, its Gaussian semigroups of probability measures and statements of elementary probability which are needed in the course of the argument. 8 PERSI DIACONIS AND BOB HOUGH

Write A “ r1, 0, 0s, B “ r0, 1, 0s, C “ r0, 0, 1s. The following com- mutators are useful, rA, Bs “ ABA´1B´1 “ r0, 0, 1s “ C, rA´1,B´1s “ A´1B´1AB “ r0, 0, 1s “ C, rA, B´1s “ AB´1A´1B “ r0, 0, ´1s “ C´1, rA´1,Bs “ A´1BAB´1 “ r0, 0, ´1s “ C´1.

A convenient representation for rx, y, zs P HpRq is CzByAx. Using the commutator rules above, the multiplication rule for w P HpRqN becomes N p1q p2q p3q p1q p2q p3q (3) wi , wi , wi “ w , w , w ` Hpwq i“1 ź ” ı “ ‰ where ¨ and H act on sequences of vectors from Rd (d ě 1, resp. d ě 2) via p1q p2q (4) w “ wi,Hpwq “ wi wj . i iăj ÿ ÿ It is also convenient to deﬁne 1 1 N (5) H˚pwq “ Hpwq ´ wp1qwp2q ` wp1qwp2q 2 2 i i i“1 1 ÿ “ wp1qwp2q ´ wp2qwp1q , 2 i j i j 1ďiăjďN ÿ ´ ¯ 1 and for w “ rx, y, zs,w ˜ “ x, y, z ´ 2 xy , so that the multiplication rule may be written “ ‰ N 1 (6) w “ w˜ ` 0, 0, wp1qwp2q ` H˚pwq . i 2 i“1 ź „  Let S “ supp µ. Recall that Γ “ xSy is the closure of the group generated by S. Its abelianization, Γab “ Γ{rΓ, Γs is equal to ppΓq where p is the projection p : G Ñ Gab. Let Γ0 be the semigroup generated by S. We record the following descriptions of Γ and Γ0.

Proposition 4. Let Γ ď HpRq be a closed subgroup of full dimension. The structure of the abelianization Γab “ Γ{rΓ, Γs and of Γ falls into one of the following cases. (1) 2 Γab “ R , Γ “ trγ, rs : γ P Γab, r P Ru 2 (2) There exist non-zero orthogonal v1, v2 P R , such that

Γab “ tnv1 ` rv2 : n P Z, r P Ru, Γ “ trγ, rs : γ P Γab, r P Ru RANDOM WALK ON UNIPOTENT MATRIX GROUPS 9

2 (3) There exist non-zero v1, v2 P R , linearly independent over R, such that

Γab “ tn1v1 ` n2v2 : n1, n2 P Zu. In this case, Γ falls into one of two further cases (a) Γ “ trγ, rs : γ P Γab, r P Ru (b) There exists λ P Rą0 and f :Γab Ñ R such that

Γ “ trγ, λpfpγq ` nqs : γ P Γab, n P Zu . Proof of Proposition 4. The full dimension condition guarantees that 2 Γab is a two dimensional closed subgroup of R , and the three possi- bilities given are all such closed subgroups. Likewise, the center of Γ is a non-trivial subgroup of R, hence either R or λ ¨ Z for some real λ ą 0. It follows that the fiber over γ P Γab is a translate of the center. Let v1, v2 be two linearly independent elements of the abelianization, and choose g1 “ rv1, z1s, g2 “ rv2, z2s in Γ. The commutator ´1 ´1 rg1, g2s “ g1g2g1 g2 is bilinear in v1, v2, is non-zero, and lies in the center. It follows that if one of v1, v2 may be scaled by a continuous parameter in the abelianization then the center is R. In the case in which Γ is discrete, a more detailed description of the fibers over the abelianization is available. In this case Γab is a lattice in R2. Let V be the covolume of this lattice 2 (7) V “ vol R {Γab and let ε :Γab Ñ t0, 1u be the parity` function˘ defined by choosing a basis pv1, v2q for Γab and setting

(8) @n1, n2 P Z, εpn1v1 ` n2v2q ” n1n2 mod 2. It is straightforward to check that this deﬁnition is independent of the basis chosen. Lemma 5. Let

|S| A “ a P Z : aggab “ 0 . # gPS + ÿ V The central ﬁber tx P R : r0, 0, xs P Γu is, for some L “ m ą 0, m P Z x y ε V V ¨ ` a z ´ g g ` g : a P A “ L ¨ . Z g g 2 2 Z #gPS + ÿ ˆ ˙ Proof. Recall (3) the multiplication rule for w P SN with wp1q “ wp2q “ 0, N p3q 1 w “ id, w˜ ` H˚pwq ;z ˜ “ z ´ xy. i 2 i“1 ź ” ı 10 PERSI DIACONIS AND BOB HOUGH

Choose a basis pv1, v2q for Γab, and write each wi as wi “ siv1 ` tiv2. ˚ V Then H pwq is some integral multiple of 2 , and the integral multiple is odd if and only if the number of wi with odd parity is odd:

v ^ v H˚pwq “ 1 2 2 s t ` s t . 2 i j i i ˜ iăj i ¸ ÿ ÿ

Lemma 6. Γ0 “ Γ.

Proof. Write Γ0,ab “ ppΓ0q where p denotes projection to the abelianization Gab. We first prove Γ0,ab “ Γab. Let u P Γ0,ab. We claim that for some r ă 0, ru P Γ0,ab. To see this, first choose v P Γ0,ab such that xu, vy ă 0. If v ‰ ru then choose w P Γ0,ab such that xu, wy ă 0 and xprojuK v, projuK wy ă 0 (this is guaranteed since otherwise all of Γ0,ab is contained in a single half-plane determined by v). Form positive 1 integer combinations of u, v, w to obtain u “ ru with r ă 0 in Γ0,ab. Note that this guarantees that Γ0,ab X tsu : s P Ru is a group. As u was arbitrary Γ0,ab is a group, hence equal to Γab. 1 1 1 Let 0 ă ă 4 be a fixed small parameter and choose x, x , y, y in Γ0 such that 1 1 ppxq, ppx q, ppyq, ppy q « e1, é1, e2, é2 where the approximation means to within distance . Take a word w in T “ tid, x, x1, y, y1u of length 4n with product approximating 1 1 the identity in Γab to within , which is such that each of x, x , y, y appear ą p1 ´ Opqqn times in w. The abelianization of the product is independent of the ordering of w, but if the letters in w appear in order y, x, y1, x1 then the central element is ă ´p1 ` Opqqn2, while if they appear in order y1, x, y, x1 then the central element is ą p1 ` Opqqn2. Moving from an ordering of the first type to an ordering of the second by swapping generators one at a time changes the central element by Op1q at each step. Let Ó 0 to deduce that Γ0 contains positive and negative central elements, and hence that Γ0 is a group, equal to Γ. More quantitative structural statements are as follows.

Lemma 7. Let µ be a measure on HpRq, with abelianization µab not supported on a lattice of R2. If the Cram´ercondition holds for the 1 measure µab then it holds also for the measure on µ on R obtained by ˚ pushing forward µab b µab by H pw1, w2q.

Proof. Let ξ P R, |ξ| ě 1 and ﬁx w2 P supppµabq, bounded away from ˚ w1^w2 1 0. Write H pw1, w2q “ 2 “ 2 w1 ¨ wˆ2. The claim follows since ˚ e´ξ pH pw1, w2qq dµabpw1q is bounded away from 1 uniformly in ξ and w2. ˇş ˇ ˇ ˇ RANDOM WALK ON UNIPOTENT MATRIX GROUPS 11

Lemma 8. Let µ be a measure on R2 of compact support, with support generating a subgroup of R2 of full dimension. If µ is lattice supported, assume that the co-volume of the lattice is at least 1. There is a constant 1 1 c “ cpµq ą 0 such that, uniformly in 0 ă ξ ă 2 , for N “ Npξq “ ξ , Y ] ˚ ˚N ˚N e´ξ pH pw1, w2qq dµ pw1qdµ pw2q ď 1 ´ cpµq. 2 2 ˇżR ˆR ˇ ˇ ˇ Proof. Standardˇ application of the functional centralˇ limit theorem im- ˇ 1 ˚ ˚N ˚N ˇ plies that N H pw1, w2qdµ pw1qdµ pw2q converges to a non-zero density on R as N Ñ 8. Normalize Haar measure on HpRq to be given in coordinates by dg “ dxdydz. The density of a Gaussian measure ν on HpRq can be understood as the rescaled limit of the density of a random walk with independent Gaussian inputs in the abelianization. Consider the distribution on the Heisenberg group given by ν2 “ rηp0, 1q, ηp0, 1q, 0s, which has projection to the abelianization given by a two dimensional symmetric standard normal distribution, and with trivial central ﬁber. ˚N The rescaled distribution d ?1 ν2 converges to a Gaussian measure ν0 N on HpRq as N Ñ 8. Note that we have not included a covariance term, which can be accommodated with a linear change of coordinates. Also, we do not consider? randomness in the central coordinate as it would scale only as N, whereas the central coordinate has distribution at scale N. Let α P R2 and ξ P R. Write the modiﬁed characteristic function of xy ν0 as (recallz ˜ “ z ´ 2 )

Ipα, ξq “ e´αpgabqe´ξpz˜qdν0pgq żg“rx,y,zsPG and ˚ x H pxq bN Ipα, ξ; Nq “ e´α ? e´ξ dν2,ab pxq . 2 N N N żpR q ˆ ˙ ˆ ˙ In view of the multiplication rule (6), for }α}, |ξ| “ Op1q lim Ipα, ξ; Nq Ñ Ipα, ξq. NÑ8 The following rate of convergence is given in Appendix A. Theorem 9. For all α P R2, ξ P R such that p1 ` }α}2qp1 ` ξ2q ă N, p1`}α}2qp1`ξ2q 1 ` O N I pα, ξ; Nq “ . 2´π}α}2 ¯ exp ξ coth πξ cosh πξ In particular, ´ ¯ 2π}α}2 exp ´ ξ coth πξ Ipα, ξq “ . ´cosh πξ ¯ 12 PERSI DIACONIS AND BOB HOUGH

Remark. While Ipα, ξq characterizes the Gaussian measure, it does not behave well under convolution. We make the following convention regarding rare events. Say that N a sequence of measurable events tAN uNě1 such that AN Ă S occurs with high probability (w.h.p.) if the complements satisfy the decay estimate, bN c ´C @ C ě 0, µ pAN q “ OC N as N Ñ 8. The sequence of complements is said` to˘ be negligible.A bN sequence of events tAN u which is negligible for µ is also negligible bN when µ is conditioned on a non-negligible sequence of events tBN u. Recall the classical local limit theorem for sums of independent random variables on Rn, see e.g. [16]. Theorem 10 (Local limit theorem for Rn). Let µ be a lazy probability measure of mean zero, covariance matrix n σ2 “ xpiqxpjqdµpxq , n ˆżR ˙i,j“1 compact support and such that Γ “ xsupp µy satisﬁes Rn{Γ is compact. Let be a fundamental domain for n Γ, and let χ 1F , where F R { F “ volpF q n χF “ δ0 in the case that Γ “ R . Denote by ηp0, σq the standard 2 8 n normal distribution with covariance matrix σ . Let f P Cc pR q, f ě 0 n and, given x P R , write Txfpyq “ fpy ´ xq for the translated function. n 1 For N ě 1, uniformly in x P R , for all 0 ă ă 3 , ? ˚N Txf ˚ χF , µ “p1 ` op1qq Txf, η 0, Nσ

A 1 ´´ ¯E @ D ` O exp ´N 3 .

If the Cram´ercondition is satisﬁed,´ ´ ¯¯

sup e´λpxqdµpxq ă 1 λP n Rn Rnl ˇż ˇ |λ|ą1 ˇ ˇ y ˇ ˇ ˇ n ˇ 8 n then, for all N ě 1, uniformly in x P R , for all f P Cc pR q, f ě 0 1 and for all 0 ă ă 3 , 3 ? ˚N ´ 1 |x| T f ˚ χ , µ “ 1 ` O N 2 ` T f, η 0, Nσ x F N 2 x ˆ ˆ ˙˙ A ´ ¯E @ D 1 ´ ` O exp Ń 3 . ´ ´ ¯¯ Proof sketch. Suppose first that xsupp µy “ Rn. It may be assumed 1 2 that f has been smoothed at scale exp Ń 3 and that |x| ! N 3 . ´ ¯ RANDOM WALK ON UNIPOTENT MATRIX GROUPS 13

First assume the Cram´ercondition. Write

˚N ˆ N Txf, µ “ fpξqe´ξpxqµˆpξq dξ. n żR @ D ´1 ´ε ´ 1 ´ε The integral is split into the ranges |ξ| ď N 3 , N 3 ă |ξ| ď ε and ε ă |ξ|. On the ﬁrst range write

´ 1 N 2 t 2 3 @ξ P C, |ξ| ! N 3 , µˆpξq “ exp ´2π Nξ σ ξ 1 ` O N|ξ| , Make the linear change of coordinates` ˘ ` ` ˘˘ ? iσ´1x ξ :“ Nσξ ´ ? , N and shift the integral to be on the real plane. Bounding

´2 ˆ σ x sup f ξ0 ` i !}f}1 n N ξ0PR ˇ ˆ ˙ˇ ˇ ˇ ˇ ˇ gives ˇ ˇ ? 1 ´ Txf, η 0, Nσ ` O }f}1 exp ´N 3 A ´ }f} ¯E ´´xtpσ2q´1x´ 1 ¯¯ |x|3 ? 1 ? ` O n exp ` 2 . det σ2N 2 2 N N ˆ ˆ ˙ ˆ ˙˙ ´ 1 ´ε In the intermediate range N 3 ă |ξ| ď ε, ξ P R, bound simply 1 ´ |µˆpξq| " ξtσ2ξ, which gives another error bounded by

1 ´ O }f}1 exp ´N 3 . ´ ´ ¯¯ On the remaining range invoke Cram´er’scondition to obtain a bound, for some ą 0,

O pexp pŃqq , which is smaller than the bound claimed. In the case where Craméris not assumed, approximate f in L1 from above and below by band-limited functions f` and f´ and repeat the argument. In the case where xsupp µy has a lattice component of dimension d, d dimensions of the integral are replaced by integrals over a torus, and compactness of the torus replaces the Cramércondition on this component. 14 PERSI DIACONIS AND BOB HOUGH

3. Proof of Theorem 1 The argument differs slightly according as the abelianized walk is lattice or non-lattice. We initially discuss only the case in which the walk takes place within a discrete subgroup of HpRq, and then describe the necessary modifications needed to handle the other cases. Let v1, v2 be a basis for Γab and choose representatives rv1, z1s, rv2, z2s P Γ. Recall (7), V “ |v1 ^ v2| and the notation from Lemma 5, for 1 rx, y, zs P G,z ˜ “ z ´ 2 xy. Given an element v “ n1v1 ` n2v2 of Γab, the fiber above v in Γ is, as a collection of third coordinates, the coset n n V n z˜ ` n z˜ ` 1 2 vp1qvp2q ` ¨ . 1 1 2 2 2 1 2 m Z 3 Given n “ rn1, n2, n3s P Z , set n n V g “ n v ` n v , n z˜ ` n z˜ ` 1 2 vp1qvp2q ` n . n 1 1 2 2 1 1 2 2 2 1 2 3 m „  In order to prove Theorem 1 in the lattice case, it thus suffices to prove the following estimate.

3 Proposition 11. For each n “ pn1, n2, n3q P Z , 2 ˚N V dν ´ 5 1 2 (9) ppn1, n2, n3q :“ µ ptgnuq “ 2 ¨ d ? gn ` O N . mN dg N ´ ¯ ´ ¯ N Introduce two collections of words: WN “ S and N n WN “ w P WN : wi P r0, 0,V ¨ Zs ¨ gn . # i“1 + ź pεq N Recall the parity function ε from (8) and define w “ i“1 εpwi,abq. The condition defining W n is a condition on the pair pw˜, wpεqq as a N ř vector in a lattice walk in R3 ˆZ{2Z; the first two coordinates are fixed, and, within the fiber over these coordinates, the last two variables are fixed in an index m coset of the lattice. In particular, the lattice case 4 of the local limit theorem on R gives that (write n1v1 ` n2v2 “ v) V p1 ` o p1qq vtpσ2q´1v µbN pW n q “ exp ´ N 2πmN det σ2 2N ˆ ˙ 1 ´ ` Oaexp Ń 3 , ´ ´ ¯¯ where σ2 denotes the covariance matrix of the walk projected orthogo- nally onto the first two coordinates. In particular, it may be assumed that }v}! N log N, n a condition which guarantees thataWN is non-negligible. RANDOM WALK ON UNIPOTENT MATRIX GROUPS 15

n It suﬃces to determine the conditional probability Prob ptgnu|WN q. bN In what follows, µ is abbreviated UN and the conditional measure bN n n µ p¨|WN q is abbreviated UN p¨q . Recalling the multiplication rule N p3q 1 wp1q, wp2q, wp3q “ wp1q, wp2q, w˜ ` wp1qwp2q ` H˚pwq i i i 2 i“1 ź ” ı „  V set zn “ n1z˜1 ` n2z˜2 ` n3 m . Also, write 1 t ρptq “ χ ´ 1 , 1 ptq, ρV ptq “ ρ . r 2 2 q V V ˆ ˙ 1 p1q p2q Shifting the left and right integrand by 2 w w , n (10) ptgnuq “ V ρV ˚ δz , ρV ˚ E n δ p3q , UN n UN w˜ `H˚pwq where A ” ıE xf, gy “ fg. żR 3.1. Reduction to central limit theorem. In frequency space,

n p3q n ˚ (11) UN ptgnuq “ V e´ξ pznq E eξ H pwq ` w˜ dξ. 1 UN R{ Z ż V ” ´ ¯ı The following two lemmas reduce to a quantitative central limit theorem by truncating frequency space to the scale of the distribution. Lemma 12. For any A ą 0 there is C “ CpAq ą 0 such that if C log N }V ξ}R{Z ě N ,

˚ p3q ´A E n eξ H pwq ` w˜ ď N . UN ˇ ” ´ ¯ıˇ Proof. Choose k “ˇ kpξq according to the ruleˇ ˇ ˇ 1 1 1 kpξq “ 1, |ξ| ą , , |ξ| ď . 10V 2V |ξ| 10 Z ^ 1 N N 1 Let N “ 2k . The group Gk “ C2 acts on strings of length N with jth factor exchanging the order of the substrings of length k ending at X \ n p2j ´ 1qk and 2jk. The action preserves WN , since only the value of H˚pwq is altered, and it changes by a multiple of V . Given string w, writew ˆ for the string of length 2N 1 with jth entry given by k

wˆj “ wpj´1qk`i. i“1 ÿ Write N 1 ˚ 1 2 2 ˚ H pwq “ Hk pwq ` Hk pwq,Hk pwq “ H pwˆ2j´1, wˆ2jq . j“1 ÿ 16 PERSI DIACONIS AND BOB HOUGH

1 Hk is invariant under Gk. One has

˚ p3q E n eξ H pwq ` w˜ UN

” ´ ˚¯ı p3q “ E n Eτ G eξ H pτ ¨ wq ` w˜ UN P k ” ” ´ ¯ıı

1 p3q 2 “ E n eξ H pwq ` w˜ Eτ G eξ H pτ ¨ wq , UN k P k k and, therefore, ” ´ ¯ “ ` ˘‰ı

˚ p3q 2 E n eξ H pwq ` w˜ ď E n Eτ G eξ H pτ ¨ wq UN UN P k k ˇ ” ´ ¯ıˇ Op1q 2 ˇ ˇ ď N “Eˇ UN E“τPG`k eξ Hk p˘‰τ ¨ˇ‰wq . ˇ ˇ ˇ ˇ One checks “ˇ “ ` ˘‰ˇ‰ ˇ ˇ N 1 2 ˚ EτPGk eξ Hk pτ ¨ wq “ cos p2πξH pwˆ2j´1, wˆ2jqq . j“1 “ ` ˘‰ ź Let δ “ δpµq ą 0 be a small parameter. Let Ekpjq be the event }ξ ¨ H˚pwˆ , wˆ q} ě δ. Choosing w according to , for j ă N 1 2j´1 2j R{Z UN the events Ekpjq are i.i.d. Furthermore, if δ is sufficiently small then Lemma 8 implies that Ekpjq occurs with positive probability uniformly in k. Let C ą 0 be a small constant, and let Ebad be the event 1 j Ekpjq ă CN . If C is sufficiently small then 1 ř UN pEbadq ď exp pĆN q c 1 while on Ebad there is C ą 0 such that 2 1 1 EτPGk eξ Hk pτ ¨ wq ď exp pĆ N q , completing the estimate.“ ` ˘‰ The above lemma permits truncation of the integral in ξ at |ξ| ! log N N . Next the conditioning is removed by fixing the abelianized variables in frequency space. n Recall mL “ V . The conditions for w P WN are, for some x mod m,

p3q mL w “ n v ` n v , w˜ ` wpεq P xL ` mL . ab 1 1 2 2 2 Z This may be imposed as

1 1 p3q mL (12) 1 pw P W n q “ e ´x ` w˜ ` wpεq N m a,m L 2 a mod m ÿ ˆ ˆ ˙˙ t ´1 ˆ eα ´rn1, n2s ` rv1, v2s wab dα. 2 żpR{Zq 1 t t ´1 ` ˘ Let pα q “ α rv1, v2s . RANDOM WALK ON UNIPOTENT MATRIX GROUPS 17

C log N Lemma 13. Let A, ą 0 and 0 ď }V ξ}R{Z ď N where C is as in Lemma 12. For all N suﬃciently large, if either a ı 0 mod m or 1 ´ 2 }α}R2{Z2 ě N , then

p3q w˜ 1 pεq ˚ p3q Á 1 EUN ea,m ` w eα pwabq eξ H pwq ` w˜ ď N . « ˜ Lm 2 ¸ ff ´ ¯ 1 1´ε Proof. Let ε “ 2 and let N “ tN u. Let w0 be w truncated at 1 N and let wt be the remainder of w so that w is the? concatenation w0 ‘ wt. Treat wt as fixed and assume that }wt} ď N log N, which holds w.h.p. Write

˚ ˚ ˚ ˚ H pwq “ H pw0q ` H pw0, wtq ` H pwtq.

Let Ekpxq denote the degree k Taylor expansion of e1pxq about 0. Let ˚ ˚ k “ kpA, q be suﬃciently large so that Ek pξH pw0qq ´ eξpH pw0qq ď 1 2N A holds w.h.p. The error in making this approximation may be ˚ bounded by taking a moment higher than k of ξH pw0q, and is negligible. It thus suﬃces to estimate instead

p3q w˜0 m pεq ˚ p3q 1 EUN1 ea,m ` w0 eα w0,ab eξ H pw0, wtq ` w˜0 « ˜ mL 2 ¸ ` ˘ ´ ¯ ˚ ˆEk pξH pw0qq . ﬀ

Expand Ek into PolypNq terms, each depending on boundedly many indices from w0. Expectation over the remaining terms factors as a product which is exponentially small in a power of N, hence negligible.

3.2. Quantitative Gaussian approximation. In the range }α} ď 1 log N ´ 2 N , |ξ| ! N , expectation with respect to µ is replaced with expectation taken over a measure with projection to the abelianization given by a Gaussian of the same covariance matrix as µab. The modi- ﬁed characteristic function in the Gaussian case is evaluated precisely in Theorem 9, which ﬁnishes the proof.

2 Lemma 14. Write σ for the covariance matrix of µab. Let

σα1 “ α2, δ “ det σ. 18 PERSI DIACONIS AND BOB HOUGH

1 C log N ´ 2 There exists c ą 0 such that, for }α} ď N and |ξ| ď N ,

˚ p3q 1 EUN eα pwabq eξ H pwq ` w˜

” 1´ 2 ¯ı ´1 “ eξ Nz˜ I N 2 α , Nδξ; N ` OpN q ` ˘ ´ ¯ (13) ` min O }α}p1 ` N}α}2qp1 ` N 3|ξ|3q „ ` 1˘ ˆ exp ć }α}2 min N, , N|ξ|2 ˆ ˆ ˆ ˙˙˙ 3 1 2 2 (14) O N}α} ` N 2 |ξ|p1 ` N |ξ| q exp pćN|ξ|q . ´ ¯  p3q Proof. Without loss, let z˜ “ 0. Then eξpw˜ q may be removed by Taylor expansion at the end of the argument, the details are omitted. Let ν be a centered Gaussian on R2 with covariance matrix equal to that of µab. Since the expectation no longer depends upon the third bN coordinate, write µab in place of UN . For 0 ď n ď N consider the bn bNń measure µn “ µab b ν in which the first n coordinates are i.i.d. with measure µab and last N ´ n coordinates are i.i.d. ν. Write ˚ 1 En “ Eµn reα pwabq eξ pH pwqqs .

1 2 Since E0 “ I N 2 α , Nδξ; N (the expectation is real) it suﬃces to bound the diﬀerence´ ¯ N

EN ´ E0 “ Ej ´ Ej´1. j“1 ÿ To bound Ej ´ Ej´1, write Ej as an N-fold integral, and move inte- grination with respect to the jth coordinate inside. This inner expectation is the complex conjugate of the characteristic function of µab at frequency t ξ α pwq “ α1 ` p´1qδpiăjqwp2q, p´1qδpiąjqwp1q . j 2 i i «i‰j i‰j ﬀ ÿ ÿ By Taylor expansion, (15) 4 Eµab eαj pwq pwjq “ Eν eαj pwq pwjq 1 ` T pαjpwqq ` Op}αjpwq} q where T“ p¨q is a degree‰ 3 polynomial.“ ‰ Since ` ˘ 4 ´2 Eµj }αjpwq} ! N it suﬃces to bound the error“ that results‰ from T . The error from Tj is bounded in one of two ways depending upon the relative sizes of }α} and |ξ|. First, by an argument analogous to the argument in the case of Lemma 12 in which the order of blocks RANDOM WALK ON UNIPOTENT MATRIX GROUPS 19

1 of length k „ |ξ| are swapped in a fixed string, we obtain an error, 5 3 3 summed over all j, of O exp pćN|ξ|q pN 2 |ξ| ` N}α} q , which is the 1 bound claimed in (14).´ To make this argument, form G¯k by omitting 1 from Gk the factor which moves index j. Then Tj is invariant under Gk. Averaged with respect to UN , the expected size of |Tj| can be separated ˚ from the expected size of E 1 re pH pτ ¨ wqqs by Cauchy-Schwarz. τPGk ξ To obtain decay in }α}, iterate the estimate (15) to find ´1 EN ` OpN q

˚ “ EνbN eα1 pwabq eξ pH pwqq 1 ` T pαjpwqq . « ˜ j ¸ﬀ ÿ The degree 3 polynomial T “ j Tjpαjpwqq consists in monic monomials of which ř i. OpNq are constant in w and cubic in α ii. OpN 2q are linear in ξw and quadratic in α iii. OpN 3q are quadratic in ξw and linear in α. Of these OpN 2q have a repeated factor from w iv. OpN 4q are cubic in ξw. Of these, OpN 3q have a repeated factor from w. Given a typical monomial M of T , write ωpMq for the number of variables from w which are odd degree in M. Consider expectation ˚ EM “ EνbN rMeα1 pwabq eξ pH pwqqs .

1 C Let C ą 0 be a small constant, and let N “ min N, |ξ| . Let w0 be the initial string of w of length N 1 and assume that´ this¯ includes any variables from M; the general case may be handled by a trivial modiﬁcation. Write w “ w0 ‘ wt so that wt contains the remaining variables. Treat wt as ﬁxed and average over w0. After performing a rotation of R2 simultaneously in each coordinate one may assume that α1 is proportional to r1, 1st. Write ˚ ˚ ˚ ˚ H pwq “ H pw0q ` H pw0, wtq ` H pwtq. t 1 ξ p2q p1q ˚ Writeα ˆ “ α ` 2 wt , ´wt . Expand eξpH pw0qq in Taylor series to degree L :“ tN”2u. The errorı in doing so is bounded by taking the next even moment, and is negligible, see the argument below. It remains to bound L ` p2π|ξ|q ˚ ` E bN1 Me pw qH pw q `! ν αˆ 0 0 `“0 ÿ ˇ “ ‰ˇ L ˇ ` ˇ p2π|ξ|q p1q p1q p2q p2q ď E bN1 Meαˆpw qw ¨ ¨ ¨ w w ¨ ¨ ¨ w . `! ν 0 m1 m` n1 n` l“0 m,nPrN 1s` ÿ ÿ ˇ “ ‰ˇ ˇ ˇ 20 PERSI DIACONIS AND BOB HOUGH

Perform expectation over all variables not appearing among m, n,M. This obtains a factor of, for some c ą 0, exp pć}αˆ}2N 1q. In the remaining variables, Taylor expand the exponential eαˆ. The dominant term then comes from the least even term in each variable. Summing over m, n, the dominant contribution comes with each mi, nj appearing once and not overlapping M. This obtains a bound of L pOp1q}αˆ}2|ξ|pN 1q2q` ! exp ć}αˆ}2N 1 }αˆ}ωpMq `! `“0 ` ˘ ÿ ! exp ´}αˆ}2pćN 1 ` Op|ξ|pN 1q2q }αˆ}ωpMq. If the constant C` is chosen sufficiently small,˘ this is bounded by, for some c ą 0 C exp ć}αˆ}2 min N, }αˆ}ωpMq. |ξ| ˆ ˆ ˙˙ The resulting bound is acceptable unless }αˆ} ă c}α} for a small constant c. In this case one obtains }ξwt}"}α} which is an event which }α}2 occurs with probability ! exp ć Nξ2 , which is again satisfactory. The bound claimed in (13) follows´ on¯ considering the description of monomials M given above. Proof of Theorem 1, lattice case. Combining Lemmas 12, 13, and 14 1 obtains, for any A ą 0, 0 ă ă 4 , for some c ą 0 Á ppn1, n2, n3q ` OApN q V “ e rn , n st e n z˜ ` n z˜ ` n L ´ Nz˜ m ´α 1 2 ´ξ 1 1 2 2 3 ¡ 1 }α}ďN ´ 2 ` ˘ ` ˘ log N |ξ|ď N ? ˆ I Nα2, Nδξ; N ` O pEq dαdξ, where the error” ´ term E satisfies¯ the estimateı of Lemma 14. Over the ´ 5 range of integration this integrates to OpN 2 q. Again on the range of integration, ? ? I Nα2, Nδξ; N “ I Nα2, Nδξ 1 ` OpN ´1q , see Theorem´ 9. Making a change¯ ´ of variables and¯ ` extending the˘ integral to R3 obtains ´ 5 ppn1, n2, n3q ` O N 2 2 ´ ¯ V 2 ´ 1 n1v1 ` n2v2 2 ? “ 2 2 e´α pσ q mδ N 3 N żR ˆ „ ˙ 1 n z˜ ` n z˜ ` n L ˆ e 1 1 2 2 3 ´ z˜ I pα, ξq dαdξ. ´ξ δ N ˆ ˆ ˙˙ The right hand side is the Gaussian density of the limit theorem. RANDOM WALK ON UNIPOTENT MATRIX GROUPS 21

Proof of Corollary 2. The required probability is pp0, 0, 0q. One has 2 4 25 m “ V “ 1 and δ “ 25 . The value 16 is obtained since 1 Ipα, ξqdαdξ “ . 3 4 żR 4. Non-lattice measures The most straightforward modification is to the case that the abelianized measure is discrete but the central fiber is not. In this case the point gn is replaced with a point gn1,n2,z “ rn1v1 ` n2v2, zs where z is a real parameter. The function V ρV of (10) is replaced with a smooth function f of compact support. The integral over ξ in (11) becomes an integral over R instead of a torus. This integral is split into the k 1 product of an integral and a sum by setting ξ “ ξ0 ` V with |ξ0| ď 2V and k P Z. Lemma 12 handles the case that |ξ0| is large. When Cramér is assumed, k ‰ 0 is analogous to a ‰ 0 in Lemma 13. If Craméris not assumed then f is approximated above and below in L1 with band- limited functions so that only finitely many k need be considered, but in this case a rate is not obtained. The remainder of the argument goes through as before. In the case where the abelianized walk has a continuous factor the details are only slightly more involved. In this case the fibered distribution is also dense in R. When both abelianized coordinates are continuous test against functions of type xy fprx, y, zsq “ F x ´ x , y ´ y , z ´ ´ Ax ´ By ´ z 0 0 2 0 where F is a smooth´ non-negative function of compact support¯ and x0, y0, z0, A, B are real parameters. For fixed w, the first two coordinates in F are fixed and the corresponding function of the third coordinate replaces ρV in the argument above. This is essentially the only change to (10). When Craméris assumed for µab it holds also for ˚ H pw1, w2qdµabpw1qdµabpw2q as a distribution on R, and this is sufficient to apply Lemma 12 to truncate to small ξ. Open F via Fourier transform, again with α and β dual to the first two coordinates. If the Cramércondition holds it gives a rate in truncating α and β. Note that the shifts A, B introduce linear phases in the α and β integrals but this does not alter the estimation of the error, in which the integrand is bounded in absolute value. In the case where the Cramércondition is not assumed, approximate F from above and below to within ą 0 in L1 with finite sums J

ρ1,j b ρ2,j b ρ3,j j“1 ÿ 22 PERSI DIACONIS AND BOB HOUGH where each ρi,j has Fourier transform supported in an interval of length Op1q. The proof goes through as before, but does not give an eﬀective rate. The case in which one of the two abelianized variables is discrete is handled by replacing one real Fourier integral with an integration on a torus.

5. Random walk on NnpZq, proof of Theorem 3 The case n “ 2 is classical and the case n “ 3 may be deduced from Theorem 1, so we consider n ě 4. n´1 Let M : Z Ñ NnpZq be the map 1 vp1q 0 ¨ ¨ ¨ 0 vp1q p2q . p2q 0 1 v 0 . n 1 v ¨ ˛ M : Z ´ Q v “ ¨ . ˛ ÞÑ ...... 0 ˚ pn´1q ‹ ˚ pn´1q ‹ ˚ 0 0 1 v ‹ ˚ v ‹ ˚ ‹ ˚ ‹ ˚ 0 ¨ ¨ ¨ 0 1 ‹ ˝ ‚ ˚ ‹ ˝ ‚ Recall that, given m P Nn, Zpmq is the central coordinate (upper right N n´1 N corner entry). Given sequence of vectors v “ tviui“1 P pZ q the central coordinate satisﬁes the product rule

N Z M v vp1qvp2q vpn´1q. p iq “ i1 i2 ¨ ¨ ¨ in´1 ˜i“1 ¸ 1ďi ăi ă...ăi ďN ź 1 2 ÿ n´1 Write ZN ep1q epn´1q n “ i1 b ¨ ¨ ¨ b in´1 1ďi ăi ă...ăi ďN 1 2 ÿ n´1 N for the corresponding tensor. Zn,µ denotes the measure on Z obtained n´1 by pushing forward measure µ on Z via M to measureµ ˜ on NnpZq, ˚N N N then obtaining xZ, µ˜ y. Equivalently, Zn,µ is the distribution of Zn evaluated on N vectors vi drawn i.i.d. from µ. Given a probability measure ν on Z and prime p, Cauchy-Schwarz and Plancherel give

1 1 ξ 2 2 (16) νpxq ´ ď νˆ p p x mod p ˇ ˇ ˜0ıξ mod p ˇ ˆ ˙ˇ ¸ ÿ ˇ ˇ ÿ ˇ ˇ ˇ ˇ ˇ ˇ where ˇ ˇ ˇ ˇ

νˆpαq “ e´αpnqνpnq. nP ÿZ Theorem 3 thus reduces to the following estimate on the characteristic N function of Zn . RANDOM WALK ON UNIPOTENT MATRIX GROUPS 23

Proposition 15. Let n ě 4 and let µ be a measure on Zn´1 satisfying the same conditions as in Theorem 3. There exists constant C ą 0 1 such that for all N ą 0 and all 0 ă |ξ| ď 2

2 ˆN n´1 Zn,µ pξq ! exp ´CN|ξ| .

ˇ ˇ ´ 2 ¯ ˇ ˇ n´1 Deduction of Theoremˇ 3. Recallˇ N “ cp and let c ě 1. Apply the upper bound of Proposition 15. By (16), 2 1 ξ 2 ZN pxq ´ ď ZˆN n,µ p n,µ p ˜x mod p ˇ ˇ¸ ξPZ ˇ ˆ ˙ˇ ÿ ˇ ˇ 0ă|ÿξ|ă p ˇ ˇ ˇ ˇ 2 ˇ ˇ ˇ ˇ ˇ ˇ 2 ! exp Ćc|ξ| n´1 0ă|ξ|ă p ÿ 2 ´ ¯ ! exp pĆcq . 5.1. Proof of Proposition 15, n “ 4. We first prove Proposition 15 in the case n “ 4, which gives a good overview of the general argument. The general case introduces a further technical difficulty which is addressed the section that follows. As for the proof of Theorem 1, the proof of Proposition 15 uses word rearrangement through a group action to introduce smoothing relative to a fixed word in the generators. Let C2 be the group of two elements. For k ě 1, 2 2 2 C2 “ xτ1, τ2 : τ1 “ τ2 “ id, τ1 ` τ2 “ τ2 ` τ1y acts on blocks of indices of length 4k with the first factor determining the relative order of the first two blocks of length k, and the second factor determining the relative order of the first two blocks of length k and the second two blocks of length k. Thus if x1, x2, x3, x4 each represent a block of length k, the group action is given by

id ¨ x “ x1x2x3x4

τ1 ¨ x “ x2x1x3x4

τ2 ¨ x “ x3x4x1x2

τ1τ2 ¨ x “ x3x4x2x1.

bN As in the local limit theorem, abbreviate µ with UN . Let S “ N supp µ and WN “ S . Write

N Z “ E δ N . 4,µ UN Z4 pwq

1 ” 1 ı N 2 N 1 Given a parameter k ě 1, set N “ N pkq “ 4k . Let Gk “ pC2 q act on WN with the jth factor acting on the four contiguous blocks of X \ 24 PERSI DIACONIS AND BOB HOUGH length k with last index 4jk. For ﬁxed w set

Zkpwq “ Eτ G δ N . P k Z4 pτ¨wq N ” ı For each 1 ď k ď 4 , N Z4,µ “ EUN rZkpwqs . The key diﬀerence which distinguishes the case n “ 4 from the case of larger n is that for n “ 4, the actions of the separate factors in Gk are independent, in the sense that the characteristic function

χkpξ, wq “ EτPGk re´ξpZkpτ ¨ wqqs factors through the product structure of the Gk action. To check this, let 1 ď j ď N 1 and write p1q p1q p3q p3q wj,k,pre “ wi , wj,k,post “ wi 1ďiď4pj´1qk 4jk`1ďiďN ÿ ÿ p`q p`q @1 ď i ď 4, @1 ď ` ď 3, wj,k,i “ wt . p4jì´5qkătďp4jì´4qk ÿ p1q Thus, setting apart the jth block of size 4k, wpre is the sum of the first p3q coordinates before the block, wpost is the sum of the third coordinates p`q after the block, and wi is the sum of the `-coordinates in the ith off-set block of length k. 1 1 2 Given τ P Gk, let pτˆj, τ q denote τ with τ P C2 substituted for τj in the jth position. Suppressing the j, k subscripts, for fixed w, N N ∆j,1 “ Z4 ppτˆj, τ1q ¨ wq ´ Z4 ppτˆj, idq ¨ wq p3q p2q p2q p3q p1q “ w1 w2 ´ w1 w2 wpre

´ p2q p1q p1q¯ p2q p3q p3q p3q ` w1 w2 ´ w1 w2 w3 ` w4 ` wpost N ´ N ¯ ´ ¯ ∆j,2 “ Z4 ppτˆj, τ2q ¨ wq ´ Z4 ppτˆj, idq ¨ wq p2q p2q p3q p3q p3q p3q p2q p2q p1q “ w3 ` w4 w1 ` w2 ´ w3 ` w4 w1 ` w2 wpre

´´ p1q ¯ ´p1q p2q ¯ p2q´ p2q ¯ ´p2q p1q ¯¯p1q p3q ` w3 ` w4 w1 ` w2 ´ w3 ` w4 w1 ` w2 wpost

´´p1q p2q p3q¯ ´ p3q ¯p1q ´ p1q p2q ¯p3 ´q ¯¯ ` w3 w4 w1 ` w2 ` w3 ` w4 w1 w2

p1q p2q ´ p3q p3q¯ ´ p1q p1q¯ p2q p3q ´ w1 w2 w3 ` w4 ´ w1 ` w2 w3 w4 N ´ N ¯ ´ ¯ ∆j,12 “ Z4 ppτˆj, τ1τ2q ¨ wq ´ Z4 ppτˆj, idq ¨ wq

“ ∆j,1 ` ∆j,2 ` δj p1q p2q p2q p1q p3q p3q δj :“ w1 w2 ´ w1 w2 w3 ` w4

´ p3q p2q p2q ¯p ´3q p1q ¯p1q ` w1 w2 ´ w1 w2 w3 ` w4 . ´ ¯ ´ ¯ RANDOM WALK ON UNIPOTENT MATRIX GROUPS 25

Since ∆j1, ∆j2, ∆j12 are each invariant under the action of the factors of Gk outside the jth position, N χkpξ, wq “ e´ξpZ4 pwqq 1 N 1 ˆ p1 ` e p∆ q ` e p∆ q ` e pp∆ ` ∆ ` δ qqq . 4 ´ξ j,1 ´ξ j,2 ´ξ j,1 j,2 j j“1 ź „  In particular, for some C ą 0, N 1 (17) χ ξ, w exp C ξδ 2 . | kp q| ď ´ } j}R{Z j“1 ź ` ˘ ´ 3 Assume without loss that |ξ| ě N 2 . Let N0 be minimal such that e1, e2, e3 may each be formed by words of length at most N0 on vectors 1 from supp µ. Let k “ max N0, 2 . Note that on average over |ξ| 3 ˆ Z ^˙ w, the tδju1ďjďN 1 are i.i.d., that δj has limiting distribution as k Ñ 8 1 at scale |ξ| , that the limiting distribution has a density on R which is positive, and thus if pµq is sufficiently small that the event Ej “ }ξδ } ě pµq occurs with probability at least p pµq ą 0 uniformly j R{Z 0 !in ξ. The estimate,) for some c ą 0, 3 ˆN 2 Z4,µ ď EUN r|χkpξ, wq|s ď exp ćN|ξ| ˇ ˇ ´ ¯ follows, see theˇ proofˇ of Lemma 12. ˇ ˇ 5.2. Proof of Proposition 15, general case. Consider n ě 5. Let n´2 n´2 C2 act on blocks of vectors of length k2 with the jth factor from n´2 j´1 C2 , j ě 1 switching the relative order of the first k2 and second j´1 k2 indices. Thus, for instance, in case n “ 5, if each of x1, ..., x8 represents a block of k consecutive indices and x “ x1x2x3x4x5x6x7x8,

τ2x “ x3x4x1x2x5x6x7x8

τ1τ3x “ τ3τ1x “ x5x6x7x8x2x1x3x4

τ1τ2τ3x “ x5x6x7x8x3x4x2x1.

1 N n´2 N 1 For k ě 1 again set N “ k2n´2 and let Gk “ pC2 q . Gk acts N on WN “ psupp µq with, for j ě 1, the jth factor of Gk acting on the contiguous subsequence of indicesX \ of length k2n´2 ending at jk2n´2. For ﬁxed k and ﬁxed w P WN , let

Z w E δ N . kp q “ τPGk Zn pτ¨wq bN Continue to abbreviate UN “ µ . For“ any k,‰ N Zn,µ “ EUN rZkpwqs .

The action of Gk on w has a dual action on a linear space of dual n- tensors. Let IN be the collection of a multi-indices i “ pi1, i2, ..., in´1q 26 PERSI DIACONIS AND BOB HOUGH satisfying 1 ď i1 ă i2 ă ... ă in´1 ď N. Given i P IN and k ě 1, let Si,k Ă Sn´1 be the subset of permutations Si,k “ tστ,i : τ P Gku where

@1 ď j ď n ´ 1, στ,ipjq “ #t1 ď k ď n ´ 1 : τpikq ď τpijqu.

That is, στ,ipjq is the position of τ ¨ ij when τ ¨ i is sorted to be in increasing order. Put another way, calculate (note τ 2 “ e) ep1q epnq τ w ep1q epnq w i1 b ¨ ¨ ¨ b in p ¨ q “ τ¨i1 b ¨ ¨ ¨ b τ¨in p q σ 1 σ n “ ep τ,ip qq b ¨ ¨ ¨ b ep τ,ip qqpwq π1pi,τq πnpi,τq where π1pi, τq, ..., πnpi, τq represent τ ¨ i1, ..., τ ¨ in in increasing order. Let X epσp1qq epσpn´1qq : i , σ S . N,k “ i1 b ¨ ¨ ¨ b in´1 P IN P i,k

The action of τ P!Gk is defined on a representative set within) XN,k by, for each i P IN , (18) τ ep1q epn´1q epστ,ip1qq epστ,ipn´1qq. ¨ i1 b ¨ ¨ ¨ b in´1 “ i1 b ¨ ¨ ¨ b in´1 The following´ lemma justifies that¯ this definition extends to a unique group action of Gk on all of XN,k. 1 Lemma 16. Let τ, τ P Gk and i P IN satisfy στ,i “ στ 1,i. Then for 2 any τ P Gk, στ`τ 2,i “ στ 1`τ 2,i. In particular, (18) extends to a unique group action on XN,k. Proof. This follows, since, for any 1 ď i ă j ď N there is at most one n´2 factor of G “ C2 in Gk, and one index `, 1 ď ` ď n ´ 2 of G which exchanges the order of i and j. To define the group action in general, given τ P Gk, i P IN and σ P Si,k choose any τ 0 such that σ “ στ 0,i. 1 Let σ “ στ 0`τ,i. Then 1 1 τ epσp1qq epσpn´1qq epσ p1qq epσ pn´1qq. ¨ i1 b ¨ ¨ ¨ b in´1 “ i1 b ¨ ¨ ¨ b in´1 The definition is clearly unique, since (18) surjects on XN,k.

The actions of SN on WN and on XN , although not adjoint, are N compatible on Zn , in the sense that for any τ, N N pτ ¨ Zn qpwq “ Zn pτ ¨ wq so that Z w E δ N . kp q “ τPGk pτ¨Zn qpwq Note that when n ě 5, although Gk is a product group, the separate “ N‰ factors τi do not act independently in τ ¨ Zn in the sense that the characteristic function N χkpξ, wq “ EτPGk e´ξ Zn pτ ¨ wq need not factor as a product. A pleasant“ ` feature˘‰ of the general case is that this diﬃculty is rectiﬁed by estimating instead of χkpξ, wq, a RANDOM WALK ON UNIPOTENT MATRIX GROUPS 27 function Fkpξ, wq which is the result of applying the Gowers-Cauchy- n´2 N 1 Schwarz inequality to χkpξ, wq. To describe this, write Gk “ pC2 q “ N 1 n´2 pC2 q , and thus

1 1 EτPGk rfpτqs “ E N ¨ ¨ ¨ E N fpτ 1, ..., τ n 2q . τ 1PC2 τ n´2PC2 ´ Then, setting apart one expectation at a time“ and applying‰ Cauchy- Schwarz, 2n´2 |χkpξ, wq|

n´2´|S| ď Eτ ,τ 1 CN1 ¨ ¨ ¨ Eτ ,τ 1 CN1 e´ξ p´1q τ S ¨ w 1 1P 2 n´2 n´2P 2 » ¨ ˛fi SĂrn´2s ÿ – ˝ ‚fl n´2´|S| “ Eτ,τ 1 G e ξ p´1q τ ¨ w P k » ´ ¨ S ˛fi SĂrn´2s ÿ “: Fkpξ, wq–, ˝ ‚fl where τ i P S τ “ pτ , ..., τ q, τ “ i S S,1 S,n´2 S,i τ 1 i R S " i Lemma 17. Fkpξ, wq factors as the product 1 N 1 F pξ, wq F pξ, wq “ 1 ´ ` k,j k 2n´2 2n´2 j“1 ź ˆ ˙ where Fk,jpξ, wq is a function of wj,k “ pω1, ..., ω2n´2 q with the

ωi “ w` p2n´2pj´1qì´1qkă`ďp2n´2pj´1qìqk ÿ n´2 the sum of consecutive blocks of length k in w. Identify C2 with n´2 n´2 t0, 1u and write |τ| “ i“1 1pτi ‰ 0q. Then ř |τ 1| 2n´2 1 Fk,jpξ, wq “ Eτ Cn´2 e´ξ p´1q Zn ppτ ` τ q ¨ wj,kq P 2 » ¨ ˛fi τ 1 Cn´2 Pÿ2 n´2– ˝ ‚fl with the action of C2 on blocks of size 1 in wj,k. 1 Proof. Consider for fixed τ, τ P Gk the sum N 1 n´2´|S| N Zn pτ, τ qpwq “ p´1q τ S ¨ Zn pwq. SĂrn´2s ÿ After replacing w with τ 1w and τ with τ ` τ 1 it suffices to consider τ 1 “ id. Consider the action of n´2´|S| τˆ “ p´1q τS SĂrn´2s ÿ 28 PERSI DIACONIS AND BOB HOUGH on a tensor e ep1q ep2q epn´1q, 1 i i i N “ i1 b i2 b ¨ ¨ ¨ b in´1 ď 1 ă 2 ă ¨ ¨ ¨ ă n´1 ď N n´2 appearing in Zn . Let G “ C2 identified with subsets S of rn ´ 2s, 0 n´2 let G “ stabpiq ď C2 be the subgroup consisting of S for which 1 n´2 0 τ S ë “ e, and let G “ C2 {G . By the group action property, for all 0 1 0 x P G , for all y P G , τ x`ye “ τ ye so that when G ‰ t1u,τ ˆ ¨ e “ 0. A necessary and sufficient condition for G0 “ t1u is that, for some 1 ď j ď N 1, n´2 n´2 2 pj ´ 1qk ă i1 ď 2 pj ´ 1qk ` k n´2 `´2 n´2 `´1 @1 ă ` ď n ´ 1 2 pj ´ 1qk ` 2 k ă i` ď 2 pj ´ 1qk ` 2 k,

n´2 and τj “ 1n´2 P C2 . In words, the indices must all belong to a n´2 common block of length 2 k acted on by a single factor from Gk, `´1 within this block, the ﬁrst 2 k elements must contain i` and the `´1 second 2 k must contain i``1 for ` “ 1, 2, ..., n ´ 2, and the factor τj n´2 acting on the block must be the element 1n´2 of the hypercube C2 . The product formula given summarizes this condition. The remainder of the proof of the general case of Proposition 15 now follows essentially as in the case n “ 4.

Appendix A. The characteristic function of a Gaussian measure on the Heisenberg group This section gives the proof of Theorem 9, which gives a rate of convergence to Gaussian measure on the Heisenberg group. Recall that ˚ x H pxq bN Ipα, ξ; Nq “ e´α ? e´ξ dν2 pxq 2 N N N żpR q ˆ ˙ ˆ ˙ 1 }x}2 where ν2pxq “ 2π exp ´ 2 . First consider the case´ α “¯ 0. Integrate away xp1q to obtain,

1 1 t 2 2 Ip0, ξ; Nq “ N exp ´ y 1 ´ ξ0 IN ` ξ0 H y dy 2π 2 N 2 p q żR ˆ ˙ `` ˘ ˘ where πξ ξ “ ,H “ N ´ 2|i ´ j|; 0 N i,j this follows from N 2 t δpjăiq y pH ´ IN qy “ p´1q yj . i“1 ˜j‰i ¸ ÿ ÿ RANDOM WALK ON UNIPOTENT MATRIX GROUPS 29

Thus, 1 Ip0, ξ; Nq “ . 2 2 det pp1 ´ ξ0 q IN ` ξ0 Hq Let a N´1

U´ “ IN ´ ei b ei`1. i“1 ÿ Then

t 2 2 U´ 1 ´ ξ0 IN ` ξ0 H U´ N 1 `` ˘ 1 ´˘ξ2 ´ “ p1 ` ξ2q 2I ´ 0 pe b e ` e b e q 0 N 1 ` ξ2 i i`1 i`1 i « 0 i“1 ÿ 2ξ2 N pN ` 1qξ2 ´ 1 ´ 0 pe b e ` e b e q ` 0 e b e . 1 ` ξ2 1 i i 1 1 ` ξ2 1 1 0 i“1 0 ﬀ ÿ 2 1´ξ0 Set η “ 2 and deﬁne sequences 1`ξ0 η2 ε1 “ 2, @i ě 1, εi`1 “ 2 ´ εi i

π0 “ 1, @i ě 1, πi “ εi j“1 ź ηδi δ1 “ 1, @i ě 1, δi`1 “ 1 ` εi These parameters have the following behavior with proof postponed until the end of this section.

1 Lemma 18. For ξ P 0,N 2 the following asymptotics hold ´ ı sinhp2πξq 1 ` ξ2 π “ N 1 ` O N 2πξ N ˆ ˆ ˙˙ 2πξ 1 ` ξ2 ε “ 1 ` cothp2πξq 1 ` O N N N ˆ ˆ ˙˙ N tanh πξ 1 ` ξ2 δ “ 1 ` O N 2πξ N ˆ ˆ ˙˙ N´1 δ2 N 3 1 ` ξ2 j “ r2πξ ´ 2 tanh πξs 1 ` O . ε 8π3ξ3 N j“1 j ÿ ˆ ˆ ˙˙ Set N´1 e b e 1 N e b e L “ I ` η i`1 i ,D “ i i . ε N ε ε p1 ` ξ2q ε i“1 N´i`1 0 i“1 N`1´i ÿ ÿ 30 PERSI DIACONIS AND BOB HOUGH

Then 1 1 2 t t 2 2 2 Dε LεU´ 1 ´ ξ0 IN ` ξ0 H U´LεDε “ IN ` P where P is the rank`` two symmetric˘ matrix˘ 2 N 2 ´2ξ0 δN`1í pN ` 1qξ0 ´ 1 P “ ? pe1 b ei ` ei b e1q ` e1 b e1. 1 ` ξ2 ε ε ε p1 ` ξ2q 0 i“1 N Ní`1 N 0 ÿ Then, for some orthogonal matrix O, and λ` ě λ´, t O pIN ` P qO “ pλè1 b e1 ` λé2 b e2q ‘ IN´2. By direct calculation (expand by the top row), 1 pN ` 1qξ2 (19) det pI ` P q “ λ λ “ 1 ´ ` 0 N ` ´ ε p1 ` ξ2q ε p1 ` ξ2q ˆ N 0 ˙ N 0 4ξ2 δ 4ξ4 N´1 δ2 ´ 0 N ´ 0 i 1 ` ξ2 ε p1 ` ξ2q2ε ε 0 N 0 N j“1 i ÿ πξ coth πξ 1 ` ξ2 “ 1 ` O . N N ˆ ˆ ˙˙ Since 1 ` ξ2 N sinh 2πξ (20) det pD q´1 “ p1 ` ξ2qN π “ 1 ` O , ε 0 N N 2πξ ˆ ˆ ˙˙ 1 ` ξ2 det 1 ´ ξ2 I ` ξ2H “ pcosh πξq2 1 ` O . 0 N 0 N ˆ ˆ ˙˙ Now consider`` the˘ general case˘ in which α ‰ 0. Treat x as N vec- 2 2 N tors in R . When SO2pRq acts diagonally on pR q rotating each xi simultaneously, H˚ and the Gaussian density are preserved. Thus, Ipα, ξ; Nq “ Ipp0, }α}qt, ξ; Nq. Calculate

1 2 e1 r1, 1, ¨ ¨ ¨ , 1sU L Dε “ . ´ ε 2 N p1 ` ξ0 q

1 a 1 2 It follows that after making the change of coordinates y “: U1LεDε y ? 2π}α} the phase has magnitude 2 and is now in the e1 direction. Let NN p1`ξ0 q v`, v´ be unit vectors generating the eigenspaces λ`, λ´ respectively. Since e1 lies in the span of v`, v´ it follows ´2π2}α}2 xv , e y2 xv , e y2 Ipα, ξ; Nq “ exp ` 1 ` ´ 1 Ip0, ξq. N p1 ` ξ2q λ λ ˆ N 0 ˆ ` ´ ˙˙ Calculate ξ2 (21) T “ λ ` λ “ 2 ` et P e “ 1 ` O ` ´ 1 1 N ˆ ˙ RANDOM WALK ON UNIPOTENT MATRIX GROUPS 31 so that 1 ` ξ2 πξ coth πξ pλ , λ q “ 1 ` O 1, . ` ´ N N ˆ ˆ ˙˙ ˆ ˙ Also, 2 2 xv`, e1y ` xv´, e1y “ 1 1 ` ξ2 λ xv , e y2 ` λ xv , e y2 “ 1 ` et P e “ O ` ` 1 ´ ´ 1 1 1 N ˆ ˙ so that 1 ` ξ2 1 ` ξ2 xv , e y2 “ O , xv , e y2 “ 1 ` O . ` 1 N ´ 1 N ˆ ˙ ˆ ˙ It follows that xv , e y2 xv , e y2 N 1 ` ξ2 ` 1 ` ´ 1 “ 1 ` O . λ λ πξ coth πξ N ` ´ ˆ ˆ ˙˙ In particular

´2π}α}2 exp 2 2 ξ coth πξ p1 ` }α} qp1 ` ξ q Ipα, ξ; Nq “ 1 ` O cosh´ πξ ¯ N ˆ ˆ ˙˙ 2 1´ξ0 Proof of Lemma 18. Recall η “ 2 . πn satisfies the recurrence 1`ξ0 2 πn “ 2πn´1 ´ η πn´2, π0 “ 1, π1 “ 2. The following closed forms hold, 2n`2 2n`2 p1 ` ξ0q ´ p1 ´ ξ0q πn “ 2 n 4ξ0p1 ` ξ0 q 2n 2n 2ξ0 p1 ` ξ0q ` p1 ´ ξ0q εn “ 1 ` 2 2n 2n 1 ` ξ0 p1 ` ξ0q ´ p1 ´ ξ0q n n 1`ξ0 ` 1´ξ0 ´ 2 1 1´ξ0 1`ξ0 1 δn “ n n ` ´ 1`¯ξ ´ 1´¯ξ 2ξ0 0 ´ 0 2 1´ξ0 1`ξ0 ´ ¯ ´ ¯ The formula for πn is immediate from the recurrence relation, since 2 2 p1 ` ξ0q p1 ´ ξ0q 2 , 2 1 ` ξ0 1 ` ξ0 2 2 are the two roots of x ´ 2x ` η “ 0. The formula for εn follows from πn εn “ . The formula for δn is obtained on summing the geometric πn´1 series ηn´1 n´1 π δ “ j , n π ηj n´1 j“0 ÿ 32 PERSI DIACONIS AND BOB HOUGH and use 2n`2 2n`2 πn p1 ` ξ0q ´ p1 ´ ξ0q n “ 2 n η 4ξ0p1 ´ ξ0 q 1 ´ ξ2 1 ` ξ n`1 1 ´ ξ n`1 “ 0 0 ´ 0 . 4ξ 1 ´ ξ 1 ` ξ 0 «ˆ 0 ˙ ˆ 0 ˙ ff The claimed asymptotics for π, ε, δ are straightforward. Using the estimates for ε and δ yields 2 2 2 2 jξ δj 1 ξ N cosh N ´ 1 “ 1 ` O ` 2 jξ εj j N ξ sinh ˆ ˆ ˙˙ ˜ ` ˘N ¸ 2 1 ξ2 N 2 jξ` ˘ “ 1 ` O ` tanh . j N ξ2 2N ˆ ˆ ˙˙ ˆ ˙ Approximating with a Riemann sum, N´1 δ2 ξ2 N 3 1 tξ 2 j “ 1 ` O tanh dt ε N ξ2 2 j“1 j 0 ÿ ˆ ˆ ˙˙ ż ˆ ˙ which gives the claimed estimate. References [1] G. K. Alexopoulos “Random walks on discrete groups of polynomial volume growth.” Ann. Probab. 30 (2002), no. 2, 723–801. [2] G. K. Alexopoulos “Random walks on discrete groups of polynomial volume growth.” Ann. Probab. 30 (2002), no. 2, 723–801. [3] P. Baldi and L. Caramellino. “Large and moderate deviations for random walks on nilpotent groups.” J. Theoret. Probab. 12 (1999), no. 3, 779–809. MR1702883 (2001b:60013) [4] Balog, A. “On the distribution of integers having no large prime factors.” JournéesArithmétiques, Besan¸con,Astérisque147/148 (1985): 27–31. [5] E. F. Breuillard, Equidistribution of random walks on nilpotent Lie groups and homogeneous spaces, ProQuest LLC, Ann Arbor, MI, 2004. [6] Breuillard, Emmanuel. “Local limit theorems and equidistribution of random walks on the Heisenberg group.” Geometric and functional analysis GAFA 15.1 (2005):35–82. [7] Breuillard, Emmanuel. “Equidistribution of dense subgroups on nilpotent Lie groups.” Ergodic Theory Dynam. Systems 30 (2010), no. 1, 131150. [8] Daniel Bump, Persi Diaconis, Angela Hicks, Laurent Miclo, and Harold Widom. An exercise (?) in Fourier analysis on the Heisenberg group. arXiv:1502.04160, 2015. [9] P. Crépel and A. Raugi. “Théorèmecentral limite sur les groupes nilpotents.” Ann. Inst. H. Poincaré sec B, Prob. and Stat., vol XIV, 2. (1978): 145–164. [10] Coulhon, Saloff-Coste, and Varopoulos. Analysis and geometry on groups. Cambridge tracts on mathematics, Cambridge University Press, 1992. [11] Chung, Fan and Linyuan Lu. “Concentration inequalities and martingale inequalities: a survey.” Internet mathematics 3.1 (2006): 79–127. [12] P. Diaconis. “Threads through group theory.” In Character theory of finite groups, 33–47, Contemp. Math., 524, Amer. Math. Soc., Providence, RI (2010). RANDOM WALK ON UNIPOTENT MATRIX GROUPS 33

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(Persi Diaconis) Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, CA, 94305, USA E-mail address: [email protected]

(Bob Hough) Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, CA, 94305, USA Current address: School of Mathematics, Institute of Advanced Study, 1 Einstein Drive, Princeton, NJ, 08540 E-mail address: [email protected]