8 Advanced L2L^{2} Techniques for Bounding Mixing Times (May 19 1999)

8.1 The comparison method for eigenvalues

xxx Revise Chapter 7, Sections 1.9 and 4, in light of this section?

The comparison method, introduced by Diaconis and Saloff-Coste [115, 116], generalizes the distinguished path method of Chapter 4, Section yyy:3 for bounding the relaxation time of a reversible Markov chain. As before, we first (xxx: delete word?) work in the setting of random walks on weighted graphs. We will proceed for given state space (vertex set) II by comparing a collection (wij)(w_{ij}) of weights of interest to another collection (w~ij)({\tilde{w}}_{ij}); the idea will be to use known results for the random walk with weights (w~ij)({\tilde{w}}_{ij}) to derive corresponding results for the walk of interest. We assume that the graph is connected under each set of weights. As in Chapter 4, Section yyy:4.3, we choose (“distinguish”) paths γxy\gamma_{xy} from xx to yy. Now, however, this need be done only for those (x,y)(x,y) with xyx\neq y and w~xy>0{\tilde{w}}_{xy}>0, but we impose the additional constraint we>0w_{e}>0 for each edge ee in the path. (Here and below, ee denotes a directed edge in the graph of interest.) In other words, roughly put, we need to construct a (wij)(w_{ij})-path to effect each given (w~xy)({\tilde{w}}_{xy})-edge. Recall from Chapter 3 yyy:(71) the definition of Dirichlet form:

(g,g)\displaystyle\mbox{${\cal E}$}(g,g) =\displaystyle= 12ijiwijw(g(j)-g(i))2,\displaystyle\frac{1}{2}\sum_{i}\sum_{j\neq i}\frac{w_{ij}}{w}(g(j)-g(i))^{2}, (8.14)
~(g,g)\displaystyle{\tilde{\cal E}}(g,g) =\displaystyle= 12ijiw~ijw~(g(j)-g(i))2.\displaystyle\frac{1}{2}\sum_{i}\sum_{j\neq i}\frac{{\tilde{w}}_{ij}}{{\tilde{% w}}}(g(j)-g(i))^{2}.
Theorem 8.1 (comparison of Dirichlet forms)

For each ordered pair (x,y)(x,y) of distinct vertices with w~xy>0{\tilde{w}}_{xy}>0, let γxy\gamma_{xy} be a path from xx to yy with we>0w_{e}>0 for every eγxye\in\gamma_{xy}. Then the Dirichlet forms (8.14) satisfy

~(g,g)A(g,g)=(g,g)ww~maxe1wexyxw~xy|γxy|1(eγxy){\tilde{\cal E}}(g,g)\leq A\mbox{${\cal E}$}(g,g)=\mbox{${\cal E}$}(g,g)\frac{% w}{{\tilde{w}}}\max_{e}\frac{1}{w_{e}}\sum_{x}\sum_{y\neq x}{\tilde{w}}_{xy}|% \gamma_{xy}|1_{(e\in\gamma_{xy})}

for every gg.

Proof. For an edge e=(i,j)e=(i,j) write Δg(e)=g(j)-g(i)\Delta g(e)=g(j)-g(i). Then

2w~~(g,g)\displaystyle 2{\tilde{w}}{\tilde{\cal E}}(g,g) =\displaystyle= xyxw~xy(g(y)-g(x))2\displaystyle\sum_{x}\sum_{y\neq x}{\tilde{w}}_{xy}(g(y)-g(x))^{2}
=\displaystyle= xyxw~xy(eγxyΔg(e))2\displaystyle\sum_{x}\sum_{y\neq x}{\tilde{w}}_{xy}\left(\sum_{e\in\gamma_{xy}% }\Delta g(e)\right)^{2}
\displaystyle\leq xyxw~xy|γxy|eγxy(Δg(e))2   by Cauchy–Schwarz\displaystyle\sum_{x}\sum_{y\neq x}{\tilde{w}}_{xy}|\gamma_{xy}|\sum_{e\in% \gamma_{xy}}(\Delta g(e))^{2}\mbox{\ \ \ by Cauchy--Schwarz}
\displaystyle\leq Aewe(Δg(e))2=A2w(g,g).   \displaystyle A\sum_{e}w_{e}(\Delta g(e))^{2}=A\cdot 2w\mbox{${\cal E}$}(g,g).% ~{}\ \ \rule{4.3pt}{4.3pt}

Remarks. (a) Suppose the comparison weights (w~ij)({\tilde{w}}_{ij}) are given by

w~ij=wiwj/w   for i,jIi,j\in I.{\tilde{w}}_{ij}=w_{i}w_{j}/w\mbox{\ \ \ for $i,j\in I$.}

The corresponding discrete-time random walk is then the “trivial”walk with w~=w{\tilde{w}}=w and

w~i=wi, p~ij=πj, π~j=πj{\tilde{w}}_{i}=w_{i},\ \ {\tilde{p}}_{ij}=\pi_{j},\ \ {\tilde{\pi}}_{j}=\pi_{j}

for all i,ji,j, and

~(g,g)\displaystyle{\tilde{\cal E}}(g,g) =\displaystyle= 12ijiπiπj(g(j)-g(i))2=varπg\displaystyle\frac{1}{2}\sum_{i}\sum_{j\neq i}\pi_{i}\pi_{j}(g(j)-g(i))^{2}={{% \rm var}}_{\pi}\,g
=\displaystyle= g22   provided iπig(i)=0\sum_{i}\pi_{i}g(i)=0.\displaystyle\|g\|^{2}_{2}\mbox{\ \ \ provided $\sum_{i}\pi_{i}g(i)=0$.}

In this case the conclusion of Theorem 1 reduces to

g22(g,g)wmaxe1wexyxπxπy|γxy|1(eγxy).\|g\|^{2}_{2}\leq\mbox{${\cal E}$}(g,g)w\max_{e}{\textstyle\frac{1}{w_{e}}}% \sum_{x}\sum_{y\neq x}\pi_{x}\pi_{y}|\gamma_{xy}|1_{(e\in\gamma_{xy})}.

This inequality was established in the proof of the distinguished path theorem (Chapter 4 Theorem yyy:32), and that theorem was an immediate consequence of the inequality. Hence the comparison Theorem 1 may be regarded as a generalization of the distinguished path theorem.

[xxx For NOTES: We’ve used simple Sinclair weighting. What about other weighting in use of Cauchy–Schwarz? Hasn’t been considered, as far as I know.]

(b) When specialized to the setting of reversible random flights on Cayley graphs described in Chapter 7 Section yyy:1.9, Theorem 1 yields Theorem yyy:14 of Chapter 7. To see this, adopt the setup in Chapter 7 Section yyy:1.9, and observe that the word

x=g1g2gd   (with each gi𝒢g_{i}\in\mbox{${\cal G}$})x=g_{1}g_{2}\cdots g_{d}\mbox{\ \ \ (with each $g_{i}\in\mbox{${\cal G}$}$)} (8.15)

corresponds uniquely to a path

γid,x=(id,g1,g1g2,,g1g2gd=x)\gamma_{{\rm id},x}=({\rm id},g_{1},g_{1}g_{2},\ldots,g_{1}g_{2}\cdots g_{d}=x) (8.16)

in the Cayley graph corresponding to the generating set 𝒢{\cal G} of interest. Having built paths γid,x\gamma_{{\rm id},x} for each xIx\in I, we then can build paths γyz\gamma_{yz} for y,zIy,z\in I by exploiting vertex-transitivity, to wit, by setting

γyz=(y,yg1,yg1g2,,yg1g2gd=z)\gamma_{yz}=(y,yg_{1},yg_{1}g_{2},\ldots,yg_{1}g_{2}\cdots g_{d}=z)

where y-1z=xy^{-1}z=x and the path γid,x\gamma_{{\rm id},x} is given by (8.16). In Theorem 1 we then have both stationary distributions π\pi and π~{\tilde{\pi}} uniform,

w~xy=μ~(x-1y)/n,   w~=1,   |γxy|=|γid,x-1y|=d(id,x-1y),{\tilde{w}}_{xy}={\tilde{\mu}}(x^{-1}y)/n,\ \ \ {\tilde{w}}=1,\ \ \ |\gamma_{% xy}|=|\gamma_{{\rm id},x^{-1}y}|=d({\rm id},x^{-1}y),

and, if e=(v,vg)e=(v,vg) with vIv\in I and g𝒢g\in\mbox{${\cal G}$},

we=μ(g)/n,   w=1,   1(eγxy)=1((x-1v,x-1vg)γid,x-1y).w_{e}=\mu(g)/n,\ \ \ w=1,\ \ \ 1_{(e\in\gamma_{xy})}=1_{((x^{-1}v,x^{-1}vg)\in% \gamma_{{\rm id},x^{-1}y})}.

Thus AA of Theorem 1 equals

maxvI,  g𝒢1μ(g)xyxμ~(x-1y)d(id,x-1y)1(x-1v,x-1vgγid,x-1y),\max_{v\in I,\ g\in\mbox{${\cal G}$}}\frac{1}{\mu(g)}\sum_{x}\sum_{y\neq x}{% \tilde{\mu}}(x^{-1}y)d({\rm id},x^{-1}y)1_{(x^{-1}v,x^{-1}vg\in\gamma_{{\rm id% },x^{-1}y})},

which reduces easily to KK of Theorem yyy:14 of Chapter 7. Since π\pi and π~{\tilde{\pi}} are both uniform, the extremal characterization

τ2=sup{g22/(g,g):  iπig(i)=0}\tau_{2}=\sup\{\|g\|^{2}_{2}/\mbox{${\cal E}$}(g,g):\ \sum_{i}\pi_{i}g(i)=0\} (8.17)

gives Theorem yyy:14 of Chapter 7.

Theorem 8.1 compares Dirichlet forms. To compare relaxation times using the extremal characterization (8.17), we compare L2L^{2}-norms using the same “direct” technique as for Chapter 3 Lemma yyy:26. For any gg,

g22g22maxi(πi/π~i)\|g\|^{2}_{2}\leq\|g\|^{\sim 2}_{2}\max_{i}(\pi_{i}/{\tilde{\pi}}_{i}) (8.18)

where, as usual, πi:=wi/w\pi_{i}:=w_{i}/w and π~i=w~i/w~{\tilde{\pi}}_{i}={\tilde{w}}_{i}/{\tilde{w}}. So if gg has π\pi-mean 00 and π~{\tilde{\pi}}-mean bb, then

g22(g,g)g-b22(g-b,g-b)Ag-b22~(g-b,g-b)maxi(πi/π~i).\frac{\|g\|^{2}_{2}}{\mbox{${\cal E}$}(g,g)}\leq\frac{\|g-b\|^{2}_{2}}{\mbox{$% {\cal E}$}(g-b,g-b)}\leq\frac{A\|g-b\|^{\sim 2}_{2}}{{\tilde{\cal E}}(g-b,g-b)% }\max_{i}(\pi_{i}/{\tilde{\pi}}_{i}). (8.19)


Corollary 8.2 (comparison of relaxation times)

In Theorem 1,



A\displaystyle A :=\displaystyle:= ww~maxe1wexyxw~xy|γxy|1(eγxy),\displaystyle\frac{w}{{\tilde{w}}}\max_{e}\frac{1}{w_{e}}\sum_{x}\sum_{y\neq x% }{\tilde{w}}_{xy}|\gamma_{xy}|1_{(e\in\gamma_{xy})},
a\displaystyle a :=\displaystyle:= mini(π~i/πi).\displaystyle\min_{i}({\tilde{\pi}}_{i}/\pi_{i}).

xxx Perhaps restate as



B:=(maxe1we)(maxiwiw~i)B:=\left(\max_{e}\frac{1}{w_{e}}\sum\sum\cdots\right)\left(\max_{i}\frac{w_{i}% }{{\tilde{w}}_{i}}\right)

(and similarly for Corollaries 4 and 7)?

xxx Remark about best if π=π~\pi={\tilde{\pi}}?

Here is a simple example, taken from [115], showing the improvement in Corollary 8.2 over Theorem yyy:32 of Chapter 4 provided by the freedom in choice of benchmark chain.

xxx NOTE: After the fact, I realized that the following example was already Example yyy:20 of Chapter 7; must reconcile.

Example 8.3

Consider a card shuffle which transposes the top two cards in the deck, moves the top card to the bottom, or moves the bottom card to the top, each with probability 1/31/3. This example fits the specialized group framework of Chapter 7 Section yyy:1.9 (see also Remark (b) following Theorem 8.1 above) with II taken to be the symmetric group on mm letters and

𝒢:={(1 2),(mm-1m-2 1),(1 2m)}\mbox{${\cal G}$}:=\{(1\ 2),(m\ m-1\ m-2\ \cdots\ 1),(1\ 2\ \cdots\ m)\}

in cycle notation. [If the order of the deck is represented by a permutation σ\sigma in such a way that σ(i)\sigma(i) is the position of the card with label ii, and if permutations are composed left to right, then σ(mm-1m-2 1)\sigma\cdot(m\ m-1\ m-2\ \cdots\ 1) is the order resulting from σ\sigma by moving the top card to the bottom.]

We obtain a representation (8.15) for any given permutation xx by writing

x=hmhm-1h2x=h_{m}h_{m-1}\cdots h_{2}

in such a way that

(hmhi)-1(j)=x-1(j)   for ijmi\leq j\leq m(h_{m}\cdots h_{i})^{-1}(j)=x^{-1}(j)\mbox{\ \ \ for $i\leq j\leq m$} (8.20)

(i.e., hmhih_{m}\cdots h_{i} and xx agree in positions ii through mm) and each hih_{i} is explicitly represented as a product of generators. To accomplish this, we proceed inductively. Suppose that (8.20) holds for given i{3,,m+1}i\in\{3,\ldots,m+1\}, and that (hmhi)(x-1(i-1))=li=l(h_{m}\cdots h_{i})(x^{-1}(i-1))=l_{i}=l, with 1li-11\leq l\leq i-1. Then let

hi-1\displaystyle h_{i-1} :=\displaystyle:= (mm-1m-2 1)l-1[(1 2)(mm-1m-2 1)]i-l-1\displaystyle(m\ m-1\ m-2\ \cdots\ 1)^{l-1}[(1\ 2)(m\ m-1\ m-2\ \cdots\ 1)]^{i% -l-1}
 (mm-1m-2 1)m-i+2.\displaystyle\ \ \cdot(m\ m-1\ m-2\ \cdots\ 1)^{m-i+2}.

In words, beginning with hmhih_{m}\cdots h_{i}, we repeatedly move the top card to the bottom until card x-1(i-1)x^{-1}(i-1) has risen to the top; then we repeatedly transpose and shift until the top m-i+2m-i+2 cards, in order, are x-1(i-1),,x-1(m)x^{-1}(i-1),\ldots,x^{-1}(m); and finally we cut these m-i+2m-i+2 cards to the bottom.

xxx Either revise Section 1.9 of Chapter 7 to delete requirement of geodesic paths, or explain one can erase cycles.

It follows that the diameter Δ\Delta of the Cayley graph associated with 𝒢{\cal G} satisfies

Δi=2m+1[(li-1)+2(i-li-1)+(m-i+2)]3(m2)\Delta\leq\sum_{i=2}^{m+1}[(l_{i}-1)+2(i-l_{i}-1)+(m-i+2)]\leq 3{m\choose 2}

and so by Chapter 7 Corollary yyy:15 that τ227(m2)2<274m4.\tau_{2}\leq 27{m\choose 2}^{2}<\frac{27}{4}m^{4}.

To improve this bound on the relaxation time we compare the chain of interest to the random transposition chain of Chapter 7 Example yyy:18 and employ Corollary 8.2, or rather its specialization, (yyy:Theorem 14) of Chapter 7.

xxx Continue as in Chapter 7 Example yyy:20 to get

τ2τ~23β2,   τ~2=m2,   τ2272m3.\frac{\tau_{2}}{{\tilde{\tau}}_{2}}\leq 3\beta^{2},\ \ \ {\tilde{\tau}}_{2}=% \frac{m}{2},\ \ \ \tau_{2}\leq\frac{27}{2}m^{3}.

xxx Test function on page 2139 of [115] shows this is right order.

Corollary 8.2 can be combined with the inequality

τ^τ2(12log1π*+1){\hat{\tau}}\leq\tau_{2}\left(\frac{1}{2}\log\frac{1}{\pi_{*}}+1\right) (8.21)

from (8.7) to bound the L2L^{2} threshold parameter τ^{\hat{\tau}} for the chain of interest, but Theorem 8.1 sometimes affords a sharper result. From the Courant–Fischer “min–max” theorem ([183], Theorem 4.2.11) it follows along the same lines as in Chapter 3 Section yyy:6.3 that

λ-1=infρ(h1,h2,,hm-1),   m=2,,n,\lambda^{-1}=\inf\rho(h_{1},h_{2},\ldots,h_{m-1}),\ \ \ m=2,\ldots,n, (8.22)

where h11h_{1}\equiv 1 and xxx Say the conditions better!

:=\displaystyle:= sup{g22/(g,g):  iπihj(i)g(i)=0   for j=1,,m-1j=1,\ldots,m-1}\displaystyle\sup\{\|g\|^{2}_{2}/\mbox{${\cal E}$}(g,g):\ \sum_{i}\pi_{i}h_{j}% (i)g(i)=0\mbox{\ \ \ for $j=1,\ldots,m-1$}\}

and the inf\inf in (8.22) is taken over all vectors h1,,hm-1h_{1},\ldots,h_{m-1} that are orthogonal in L2(π)L^{2}(\pi) (or, equivalently, that are linearly independent). Using (8.19), Corollary 8.2 now generalizes to

Corollary 8.4 (comparison of eigenvalues)

In Theorem 8.1, the eigenvalues λm\lambda_{m} and λ~m{\tilde{\lambda}}_{m} in the respective spectral representations satisfy


with AA and aa as defined in Corollary 8.2.

Here is a simple example [116] not possessing vertex-transitivity:

xxx NOTE: This is a DIRECT comparison!: see Chapter 3 Section 6.4.

Example 8.5

Random walk on a dd-dimensonal grid.

To keep the notation simple, we let d=2d=2 and consider the grid I:={0,,m1-1}×{0,,m2-1}I:=\{0,\ldots,m_{1}-1\}\times\{0,\ldots,m_{2}-1\} as an (unweighted) subgraph of 𝐙2{\bf Z}^{2}. The eigenvalues λl\lambda_{l} are not known in closed form. However, if we add self-loops to make a benchmark graph where II is regular with degree 44, then the eigenvalues λ~l{\tilde{\lambda}}_{l} for the continuous-time walk are

1-12(cosπrm1+cosπsm2),   0rm1-1,   0sm2-1.1-\frac{1}{2}\left(\cos\frac{\pi r}{m_{1}}+\cos\frac{\pi s}{m_{2}}\right),\ \ % \ 0\leq r\leq m_{1}-1,\ \ \ 0\leq s\leq m_{2}-1.

xxx Product chain. Add discussion of all eigenvalues to Section yyy:6.2 of Chapter 4?

xxx P.S. See Chapter 5, (66).

In particular, assuming m1m2m_{1}\geq m_{2} we have

τ~2=2(1-cosπm1)-1.{\tilde{\tau}}_{2}=2\left(1-\cos\frac{\pi}{m_{1}}\right)^{-1}. (8.23)

Now we apply Corollary 8.4 to bound the eigenvalues λl\lambda_{l}. In Theorem 8.1, the two graphs agree except for self-loops, so



a=miniπ~iπi=ww~miniw~iwi,a=\min_{i}\frac{{\tilde{\pi}}_{i}}{\pi_{i}}=\frac{w}{{\tilde{w}}}\min_{i}{{% \tilde{w}}_{i}}{w_{i}},


Aa=maxiwiw~i1.\frac{A}{a}=\max_{i}\frac{w_{i}}{{\tilde{w}}_{i}}\leq 1.

Thus λl-1λ~l-1\lambda^{-1}_{l}\leq{\tilde{\lambda}}^{-1}_{l} for 1ln:=m1m21\leq l\leq n:=m_{1}m_{2}; in particular,

τ2τ~2.\tau_{2}\leq{\tilde{\tau}}_{2}. (8.24)

Comparing the other way around gives

λ~l-1(maxiw~iwi)λl-1=2λl-1,   1ln{\tilde{\lambda}}^{-1}_{l}\leq\left(\max_{i}\frac{{\tilde{w}}_{i}}{w_{i}}% \right)\lambda^{-1}_{l}=2\lambda^{-1}_{l},\ \ \ 1\leq l\leq n

and in particular


The result 12λ~l-1λl-1λ~l-1\frac{1}{2}{\tilde{\lambda}}^{-1}_{l}\leq\lambda^{-1}_{l}\leq{\tilde{\lambda}}% ^{-1}_{l} extends to general dd, for which (for example)


where I={0,,m1-1}××{0,,md-1}I=\{0,\ldots,m_{1}-1\}\times\cdots\times\{0,\ldots,m_{d}-1\} and m:=maximim:=\max_{i}m_{i}.

Example 8.6

Random walk on a thinned grid.

As a somewhat more interesting example, suppose we modify the grid in 𝐙2{\bf Z}^{2} in Example 8.5 by deleting at most one edge from each unit square.

xxx Copy picture on page 700 in [116] as example?

Again we can apply Corollary 8.4, using the same benchmark graph as in Example 8.5. In Theorem 8.1, w~xy>0{\tilde{w}}_{xy}>0 for xyx\neq y if and only if xx and yy are neighboring vertices in the (unthinned) grid {0,,m1-1}×{0,,m2-1}\{0,\ldots,m_{1}-1\}\times\{0,\ldots,m_{2}-1\}. We can choose γxy\gamma_{xy} to have length 11 (if the edge joining xx and yy has not been deleted) or 33 (if it has). For any directed edge ee in the grid, there are at most two paths of length 33 and at most one path of length 11 passing through ee. Thus A7w/w~A\leq 7w/{\tilde{w}}, and so A/a7maxi(wi/w~i)7A/a\leq 7\max_{i}(w_{i}/{\tilde{w}}_{i})\leq 7; comparing the other way around is even easier (all paths have length 11), and we find

14λ~l-1λl-17λ~l-1,   2ln.{\textstyle\frac{1}{4}}{\tilde{\lambda}}^{-1}_{l}\leq\lambda^{-1}_{l}\leq 7{% \tilde{\lambda}}^{-1}_{l},\ \ \ 2\leq l\leq n.

xxx REMINDER: NOTES OR ELSEWHERE?: Mention exclusion process [149, 116].

Example 8.7

The nn-path with end self-loops.

The comparison technique does not always provide results as sharp as those in the preceding two examples, even when the two chains are “close.” For example, let the chain of interest be the nn-path, with self-loops added at each end added to make the graph regular with degree 22, and let the benchmark graph be the nn-cycle (Chapter 5, Example yyy:7). Use of Corollary 8.2 gives only τ2nτ~2\tau_{2}\leq n{\tilde{\tau}}_{2}, whereas in fact τ2=(1-cosπn)-12π2n2\tau_{2}=\left(1-\cos\frac{\pi}{n}\right)^{-1}\sim\frac{2}{\pi^{2}}n^{2} and τ~2=(1-cos2πn)-112π2n2.{\tilde{\tau}}_{2}=\left(1-\cos\frac{2\pi}{n}\right)^{-1}\sim\frac{1}{2\pi^{2}% }n^{2}.

It is difficult in general to use Corollary 8.4 to improve upon (8.21). However, when both the chain of interest and the benchmark chain are symmetric reversible chains (as defined in Chapter 7 Section yyy:1.1), it follows from Chapter 4 yyy:(14) by averaging over ii that

d^(t)d^~(aAt),   t0,{\hat{d}}(t)\leq{\tilde{{\hat{d}}}}\left(\frac{a}{A}t\right),\ \ \ t\geq 0,

and hence from (8.1) we obtain

Corollary 8.8 (comparison of L2L^{2} mixing times)

In Theorem 8.1, if both the graph of interest and the benchmark graph are vertex-transitive, then the L2L^{2} mixing time parameters τ^{\hat{\tau}} and τ^~{\tilde{{\hat{\tau}}}} satisfy

Example 8.9

Returning to the slow card-shuffling scheme of Example 8.3 with random transpositions benchmark, it is known from group representation methods [123, 120] which make essential use of all the eigenvalues λ~r{\tilde{\lambda}}_{r}, not just λ~2{\tilde{\lambda}}_{2}, that

τ^~12mlogm   as mm\to\infty.{\tilde{{\hat{\tau}}}}\sim{\textstyle\frac{1}{2}}m\log m\ \ \ \mbox{as $m\to% \infty$.}

Since a=1a=1 and AA (=K=K of Chapter 7, Theorem yyy:14) 27m2\leq 27m^{2}, it follows that

τ^(1+o(1))272m3logm.{\hat{\tau}}\leq(1+o(1)){\textstyle\frac{27}{2}}m^{3}\log m. (8.25)

This improves upon Example 8.3, which combines with (8.21) to give only

τ^(1+o(1))274m4logm.{\hat{\tau}}\leq(1+o(1)){\textstyle\frac{27}{4}}m^{4}\log m.

xxx Show truth is τ^=Θ(m3logm){\hat{\tau}}=\Theta(m^{3}\log m)?