The cutoff phenomenon for randomized riffle shuffles

0.34 MB·English

by  Guan-Yu Chen

THE CUTOFF PHENOMENON FOR RANDOMIZED RIFFLE
SHUFFLES
7
0
0
GUAN-YUCHENANDLAURENTSALOFF-COSTE
2

n
Abstract. We study the cutoff phenomenon for generalized riffle shuffles
a
J where,ateachstep,thedeckofcardsiscutintoarandomnumberofpacksof
multinomialsizeswhicharethenriffledtogether.
9
2

]
R
1. Introduction
P
. In this article we consider some generalizations of the standard riffle shuffle of
h
Gilbert, Shannon and Reeds (GSR-shuffle for short). The GSR-shuffle models the
t
a waytypicalcardplayersshufflecards. First,thedeckiscutintotwopacksaccording
m to an (n,1)-binomial random variable where n is the number of cards in the deck.
2
[ Next, cards are dropped one by one from one or the other pack with probability

proportional to the relative sizes of the packs. Hence, if the left pack contains
1
v a cards and the right pack b cards, the next card drops from the left pack with
7 probability a/(a+b).
2 The history of this model is described in [8, Chap. 4D] where the reader will
8
also find other equivalent definitions and a discussion of how the model relates to
1
real life card shuffling. The survey [10] gives pointers to the many developments
0
7 that arose from the study of the GSR model.
0 Early results concerning the mixing time (i.e., how many shuffles are needed
/ to mix up the deck) are described in [1, 2, 8]. In particular, using ideas of Reeds,
h
t Aldousprovedin[1]that,asymptoticallyasthenumbernofcardstendstoinfinity,
a it takes 3log n shuffles to mix up the deck if convergence is measured in total
m 2 2
variation(we use log to denote base a logarithms and log for natural, i.e., base e,
a
:
v logarithms).
i In [4], Bayer and Diaconis obtained an exact useful formula for the probability
X
distributiondescribingthe state ofthe deckafter k GSR-shuffles. Namely, suppose
r
a thatcardsarenumbered1 throughnandthatwestartwiththe deckinorder. Let
σ denote a given arrangement of the cards and let Qk(σ) be the probability that
n
the deck is in state σ after k GSR-shuffles. Then
n+2k r
(1.1) Qkn(σ)=2−kn n −
(cid:18) (cid:19)
where r is the number of rising sequences in σ. Given an arrangementof the deck,
a rising sequence is a maximal subset of cards consisting of successive face values
displayed in order. For instance, the arrangement 3,1,4,5,7,2,8,9,6 has rising
1991 Mathematics Subject Classification. 60J05.
Key words and phrases. Cutoffphenomenon, riffleshuffle.
ThefirstauthorispartiallysupportedbyNCTS,Taiwan.
ResearchpartiallysupportedbyNSFGrantsDMS0102126, 0306194and0603886.
1
2 G.-Y.CHENANDL.SALOFF-COSTE
sequences (1,2),(3,4,5,6),(7,8,9). See [2, 4] for details. By definition, the total
variation distance between two probability measures µ,ν on a set S is given by
µ ν = sup µ(A) ν(A) .
TV
k − k { − }
A S
⊂
Using the formuladisplayedin (1.1), BayerandDiaconis gavea verysharpversion
of the fact that the total variation mixing time is 3log n for the GSR-shuffle.
2 2
Theorem 1.1(BayerandDiaconis[4]). Fix c ( ,+ ). Foradeckof ncards,
∈ −∞ ∞
the total variation distance between the uniform distribution and the distribution of
a deck after k= 3log n +c GSR-shuffles is
2 2
1 2−c/4√3
e t2/2dt+O (n 1/4).
− c −
√2π Z−2−c/4√3
Thisresultillustratesbeautifullytheso-calledcutoffphenomenondiscussedin[1,
2,3,8,9,14,17]. Namely,thereisasharptransitioninconvergencetostationarity.
Indeed,the integralabovebecomes smallveryfastas ctends to + andgetsclose
∞
to 1 even faster as c tends to .
−∞
The aim of the present paper is to illustrate further the notion of cutoff using
somegeneralizationsoftheGSR-shuffle. Alongthiswaywewillobserveseveralphe-
nomenathathavenotbeen,tothebestofourknowledge,noticedbefore. Foradeck
ofncardsandagivenintegerm,am-riffleshuffleisdefinedasfollows. Cutthedeck
into m packs whose sizes (a ,...,a ) form a multinomial randomvector. In other
1 m
words,theprobabilityofhavingpacksofsizesa ,...,a ism n n! . Thenform
1 m − a1!...am!
anewdeckbydroppingcardsonebyonefromthesepackswithprobabilitypropor-
tional to the relative sizes of the packs. Thus, if the packs have sizes (b ,...,b )
1 m
then the next card will drop from pack i with probability b /(b + +b ). We
i 1 m
···
will refer to an m-riffle shuffle simply as an m-shuffle in what follows. Obviously
the GSR-shuffle is the same as a 2-shuffle. A 1-shuffle leaves the deck unchanged.
Theseshuffleswereconsideredin[4]wherethefollowingtwolemmasareproved.
Lemma 1.2. In distribution, an m-shuffle followed by an independent m-shuffle
′
equals an mm-shuffle.
′
Lemma 1.3. For a deck of n cards in order, the probability that after an m-shuffle
the deck is in state σ depends only of the number r =r(σ) of rising sequences of σ
and equals Q (r) where
n,m
n+m r
Qn,m(r)=m−n − .
n
(cid:18) (cid:19)
For instance, formula (1.1) for the distribution of the deck after k GSR-shuffles
follows from a direct application of these two lemmas since k consecutive indepen-
dent 2-shuffles equal a 2k-shuffle in distribution. These lemmas will play a crucial
role in this paper as well.
The model we consider is as follows. Let p=(p(1),p(2),...) be the probability
distribution of an integer valued random variable X, i.e.,
P(X =k)=p(k), k =1,2,....
A p-shuffle proceeds by picking an integer m according to p and performing an
m-shuffle. In other words, the distribution of a p-shuffle is the p-mixture of the
m-shuffle distributions. Note that casinos use multiple decks for some games and
THE CUTOFF PHENOMENON FOR RANDOMIZED RIFFLE SHUFFLES 3
thattheseareshuffledinvariousways(includingbyshufflingmachines). Themodel
above (for some appropriate p) is not entirely unrealistic in this context.
Because of Lemma 1.3, the probability that starting from a deck in order we
obtain a deck in state σ depends only on the number of rising sequences in σ and
is given by
∞
(1.2) Q (r)= p(m)Q (r)=E(Q (r)).
n,p n,m n,X
1
X
Abusing notation, if σ denotes a deck arrangement of n cards with r rising se-
quences, we write
Q (σ)=Q (r).
n,p n,p
Verygenerally,ifQisaprobabilitymeasureondeckarrangements(hencedescribes
a shuffling method), we denote by Qk the distribution of the deck after k such
shuffles, starting from a deck in order. For instance, Lemma 1.2 yields
Qk =Q .
n,m n,mk
Let U be the uniform distribution on the set of deck arrangements of n cards.
n
Althoughthiswillnotreallyplayaroleinthis work,recallthatdeckarrangements
can be viewed as elements of the symmetric group S in such a way that Qk, the
n
distribution after k successive Q-shuffles, is the k-fold convolution of Q by itself.
See,e.g.,[1,4,8,15]. Eachofthe measuresQ generatesaMarkovchainondeck
n,p
arrangements (i.e., on the symmetric group S ) whose stationary distribution is
n
U . These chains areergodicif p is notconcentratedat1. They are notreversible.
n
Note that [11] studies a similar but different model based on top m to random
shuffles. See [11, Section 2].
The goalof this paper is to study the convergenceof Qk to the uniform distri-
n,p
bution in total variation as k tends to infinity and, more precisely, the occurrence
of a total variation cutoff for families of shuffles (S ,Q ,U ) as the number
{ n n,pn n }∞1
n of cards grows to infinity and p is a fixed sequence of probability measures on
n
the integers. To illustrate this, we state the simplest of our results.
Theorem 1.4. Let p be a probability measure on the positive integers such that
∞
(1.3) µ= p(k)logk < .
∞
1
X
Fix ǫ (0,1). Then, for any k >(1+ǫ) 3 logn, we have
∈ n 2µ
nlim kQknn,p−UnkTV =0
→∞
whereas, for k <(1 ǫ) 3 logn,
n − 2µ
nlim kQknn,p−UnkTV =1.
→∞
In words, this theorem establishes a total variation cutoff at time 3 logn (see
2µ
the definition of cutoff in Section 2 below). If p is concentrated at 2, i.e., Q
n,p
representsaGSR-shuffle,thenµ=log2and 3 logn= 3log n inaccordancewith
2µ 2 2
the results of Aldous [1] and Bayer-Diaconis [4] (e.g., Theorem 1.1).
The results we obtain are more general and more precise than Theorem 1.4 in
severaldirections. First,wewillconsiderthecasewheretheprobabilitydistribution
p = p depends on the size n of the deck. This is significant because we will not
n
4 G.-Y.CHENANDL.SALOFF-COSTE
imposethatthe sequencep convergesasn tends toinfinity. Second,andthis may
n
be a little surprising at first, (1.3) is not necessary for the existence of a cutoff
and we will give sufficient conditions that are weaker than (1.3). Third, under
stronger moment assumptions, we will describe the optimal window size of the
cutoff. For instance, Theorem 1.1 says that, for the GSR-shuffle, the window size
is of order 1 with a normal shape. This result generalizes easily to any m-shuffle
where m is a fixed integer greater or equal to 2. See Remark 3.1 and Theorem 5.4
below. Suppose now that instead of the GSR-shuffle we consider the p-shuffle with
p(2) = p(3) = 1/2. In this case, µ = log√6. Theorem 1.4 gives a total variation
cutoff at time 3log n. We will show that this cutoff has optimal window size of
2 √6
order√logn. Thuspickingatrandombetween2and3shuffleschangesthewindow
size significantly when compared to either pure 2-shuffles or pure 3-shuffles.
We close this introduction with a remark concerning the spectrum of these gen-
eralizedriffleshufflesandhowitrelatestothe windowofthecutoff. AsLemma1.2
makes clear, all riffle shuffles commute. Although riffle shuffles are not reversible,
they are all diagonalizable with real positive eigenvalues and their spectra can be
computedexplicitly(thisisanotheralgebraic“miracle”attachedtotheseshuffles!).
See[4,5,6]. Inparticular,thesecondlargesteigenvalueofanm-shuffleis1/mwith
thesameeigenspaceforallm 2. See[13]forastrongerresultimplyingthisstate-
≥
ment. Thus, the second largest eigenvalue of a p-shuffle is β = k 1p(k). By
−
definition,the relaxationtime ofafinite Markovchainistheinverseofthe spectral
P
gap (1 β) 1 and one might expect that, quite generally, for families of Markov
−
−
chains presenting a cutoff, this quantity would give a good control of the window
of the cutoff. The generalized riffle shuffles studied here provided interesting (al-
beit non-reversible)counterexamples: Take, for instance, the case discussed earlier
where p(2) = p(3) = 1/2. Then β = 5 and (1 β) 1 = 12, independently of the
12 − − 7
number n of cards. However, as mentioned above, the optimal window size of the
cutoff for this family is √logn. For generalized riffle shuffles, the window size of
the cutoff and the relaxation time appear to be disconnected.
2. The cutoff phenomenon
The following definition introduces the notion of cutoff for a family of ergodic
Markov chains.
Definition 2.1. Let (S ,K ,π ) be a family of ergodic Markov chains where
{ n n n }∞1
S denotes the state space, K the Markovkernel, and π the stationary distribu-
n n n
tion. This family satisfies a total variation cutoff with critical time t > 0 if, for
n
any fixed ǫ (0,1),
∈
0 if k >(1+ǫ)t
nl→im∞xs∈uSpnkKnkn(x,·)−πnkTV =(cid:26) 1 if knn <(1−ǫ)tnn.
This definitionwasintroducedin[2]. Amorethoroughdiscussionisin[9]where
many examples are described. Note that this definition does not require that the
critical time t tends to infinity (in [9], the corresponding definition requires that
n
t tends to infinity). The positive times t can be arbitrary and thus can have
n n
severallimit points in [0, ]. Examples of families having a cutoff with a bounded
∞
critical time sequence will be given below. Theorem 1.4 above states that, under
assumption (1.3), a p-shuffle has a total variation cutoff with critical time t =
n
3 logn.
2µ
THE CUTOFF PHENOMENON FOR RANDOMIZED RIFFLE SHUFFLES 5
Informally, a family has a cutoff if convergence to stationarity occurs in a time
interval of size o(t ) around the critical time t . The size of this time interval can
n n
be thought of as the “window” of the cutoff. The next definition carefully defines
the notion of the window size of a cutoff.
Definition 2.2. Let (S ,K ,π ) be a family of ergodic Markov chains as in
{ n n n }∞1
Definition 2.1. We say that this family presents a (t ,b ) total variation cutoff if
n n
the following conditions are satisfied:
(1) For all n=1,2,..., we have t >0 and lim b /t =0.
n n n
n
(2) For c R 0 and n 1, set →∞
∈ −{ } ≥
t +cb if c>0
n n
k =k(n,c)= ⌈ ⌉ .
( tn+cbn if c<0
⌊ ⌋
The functions f,f defined by
f(c)=limsup sup Kk(x, ) π for c=0
k n · − nkTV 6
n→∞ x∈Sn
and
f(c)=liminf sup Kk(x, ) π for c=0
n→∞ x∈Snk n · − nkTV 6
satisfy
lim f(c)=0, lim f(c)=1.
c c
→∞ →−∞
Definition 2.3. Referring to Definition 2.2, a (t ,b ) total variationcutoff is said
n n
to be optimal if the functions f,f satisfy f(c)>0 and f( c)<1 for all c>0.
−
Note that any family having a (t ,b ) cutoff (Definition 2.2) has a cutoff with
n n
criticaltime t (Definition 2.1). The sequence (b ) in Definition 2.2 describes an
n n ∞1
upperboundontheoptimalwindowsizeofthecutoff. Forinstancethemainresult
of Bayer and Diaconis [4], i.e., Theorem 1.1 above, shows that the GSR-shuffle
family presents a (t ,b ) total variation cutoff with t = 3log n and b = 1.
n n n 2 2 n
Theorem 1.1 actually determines exactly “the shape” of the cutoff, that is, the
two functions f,f of Definition 2.2. Namely, for the GSR-shuffle family and t =
n
3log n, b =1, we have
2 2 n
1 2−c/4√3
f(c)=f(c)= e−t2/2dt.
√2π Z−2−c/4√3
This shows that this cut-off is optimal (Definition 2.3).
The optimality introduced in Definition 2.3 is very strong. If a family presents
anoptimal (t ,b ) total variationcut-off and also a (s ,c ) totalvariationcut-off,
n n n n
then t s and b =O(c ). In words, if (t ,b ) is an optimal cut-off then there
n n n n n n
∼
are no cut-offs with a window significantly smaller than b . For a more detailed
n
discussion of the cutoff phenomena and their optimality, see [7].
3. Cutoffs for generalized riffle shuffles
In this section we state our main results and illustrate them with simple exam-
ples. They describe total variation cutoffs for generalized riffle shuffles, that is, for
6 G.-Y.CHENANDL.SALOFF-COSTE
the p-shufflesdefined in the introduction. Moreprecisely,for eachn (n is the num-
ber of cards), fix a probability distribution p = (p (1),p (2),...) on the integers
n n n
and consider the family of Markov chains (i.e., shuffles)
{(Sn,Qn,pn,Un)}∞1 .
Here S is the set of all deck arrangements (i.e., the symmetric group) and U is
n n
the uniform measure on S . For any x [0, ], set
n
∈ ∞
1 x/4√3
(3.1) Ψ(x)= e−t2/2dt.
√2π Z−x/4√3
We start with the simple case where the probability distributions p is concen-
n
trated on exactly one integer m and use the notation Q for an m -shuffle.
n n,mn n
Theorem 3.1. Let (m ) be any sequence of integers all greater than 1 and set
n ∞1
3logn
µ =logm , t = .
n n n
2µ
n
Then the family (S ,Q ,U ) presents a (t ,µ 1) total variation cutoff.
{ n n,mn n }∞1 n −n
Remark 3.1. When m = m is constant Theorem 3.1 gives a (3log n,1) total
n 2 m
variation cutoff. In this case, for k = 3log n +c, one has the more precise result
2 m
that Qk U = Ψ(m c)+O (n 1/4). In particular, for m = 2, this is the
k n,m− nkTV − c −
Theorem of Bayer and Diaconis stated as Theorem 1.1 in the introduction.
Next we give a more explicit version of Theorem 3.1 which requires some addi-
tional notation. For any real t>0, set
1/2 if 0<t<1/2
t =
{ } k if k 1/2 t<k+1/2 for some k =1,2,...,
(cid:26) − ≤
(this is a sort of “integer part” of t) and
1/2 if 0<t<1/2
d(t)=
t t if 1/2 t< .
(cid:26) −{ } ≤ ∞
Theorem 3.2. Let (m ) be any sequenceof integers all greater than 1. Consider
n ∞1
the family of shuffles (S ,Q ,U ) and let µ , t be as in Theorem 3.1.
{ n n,mn n }∞1 n n
(A) Assume that lim m = , that is, lim µ = . Then, we have:
n n
n ∞ n ∞
(1) The famil→y∞(S ,Q ,U ) a→lw∞ays has a ( t ,b ) cutoff for any
{ n n,mn n }∞1 { n} n
positive b =o(1), that is,
n
lim inf Qk U =1, lim sup Qk U =0.
n→∞k<{tn}k n,mn − nkTV n→∞k>{tn}k n,mn − nkTV
(2) If lim d(t )µ = then there is a (t ,0) cutoff, that is,
n n n
n | | ∞
→∞
lim inf Qk U =1, lim sup Qk U =0.
n→∞k≤tnk n,mn − nkTV n→∞k≥tnk n,mn − nkTV
(3) If liminf d(t )µ < then there exists a sequence (n ) tending to
n | n | n ∞ i ∞1
infin→ity∞such that
0<liim→∞infkQ{ntin,mi}ni −UnikTV ≤limi→s∞upkQ{ntin,mi}ni −UnikTV <1.
In particular, there is no (t ,0) total variation cutoff.
n
THE CUTOFF PHENOMENON FOR RANDOMIZED RIFFLE SHUFFLES 7
(4) If lim d(t )µ =L [ , ] exists then
n n
n ∈ −∞ ∞
→∞
(3.2) nlim kQn⌊{,mtnn}⌋−UnkTV =Ψ(eL).
→∞
(B) Assume that (m ) is bounded. Then t tends to infinity, there is a (t ,1)
n ∞1 n n
total variation cutoff and, for any fixed k Z, we have
∈
0<linm→i∞nfkQn{t,nm}n+k−UnkTV ≤linm→s∞upkQ{nt,nm}n+k−UnkTV <1.
In particular, the (t ,1) cutoff is optimal.
n
Example 3.1. To illustratethis result,considerthe casewherem = nα forsome
n
⌊ ⌋
fixed α>0. In this case, we have
3logn 3
µ αlogn, t = as n tends to infinity.
n n
∼ 2µ ∼ 2α
n
(a) Assume that 3 (k,k+1) for some k=0,1,2,.... Then d(t )µ
2α ∈ | n | n →∞
and
lim Qk U =1, lim Qk+1 U =0.
n k n,mn − nkTV n k n,mn − nkTV
→∞ →∞
(b) Assume that 3 =k forsomeintegerk =1,2,.... Then d(t ) =O(n α).
2α | n | −
Hence d(t )µ 0 as n tends to infinity. Theorem 3.2(1) shows that we
n n
| | →
have a (k,b ) cutoff where b is an arbitrary sequence of positive numbers
n n
tending to 0. That means that
lim Qk 1 U =1, lim Qk+1 U =0.
n k n−,mn − nkTV n k n,mn − nkTV
→∞ →∞
Moreover Theorem 3.2(4) gives lim Qk U =Ψ(1).
n→∞k n,mn − nkTV
Example 3.2. Consider the case where m = (logn)α , α>0. Then
n
⌊ ⌋
3logn
µ αloglogn, t as n tends to infinity.
n n
∼ ∼ 2αloglogn
Note that t tends to infinity and the window size µ 1 goes to zero.
n −n
We now state results concerning general p-shuffles. We will need the following
notation. For each n, let p be a probability distribution on the integers. Let X
n n
be a random variable with distribution p . Assume that p is not supported on a
n n
single integer and set
logX µ
µ =E(logX ), σ2 =Var(logX ), ξ = n − n.
n n n n n σ
n
Consider the following conditions which may or may not be satisfied by p :
n
logn
(3.3) lim = .
n→∞ µn ∞
(3.4) ∀ǫ>0, nl→im∞E(cid:16)ξn21{ξn2>ǫµ−n1logn}(cid:17)=0.
Condition(3.4)shouldbeunderstoodasaLindebergtypecondition. Wewillprove
inLemma7.1that(3.4)implies(3.3). Example3.5showsthattheconverseisfalse.
8 G.-Y.CHENANDL.SALOFF-COSTE
Theorem 3.3. Referring to the notation introduced above, assume that
0<µ ,σ <
n n
∞
and set
3logn 1 σ2logn
t = , b = max 1, n .
n n
2µn µn ( s µn )
Assume that the sequence (p ) satisfies (3.4). Then the family (S ,Q ,U )
n { n n,pn n }∞1
presentsa(t ,b )totalvariation cutoff. Moreover, ifthe window sizeb is bounded
n n n
from below by a positive real number, then the (t ,b ) total variation cut-off is
n n
optimal.
Example 3.3. Assume p =p is independent of n and
n
∞ ∞
µ= p(k)logk < , σ2 = µ logk 2p(k)< .
∞ | − | ∞
1 1
X X
Then condition (3.4) holds and
3
t = logn, b logn
n n
2µ ≈
p
where b √logn means that the ratio b /logn is bounded above and below by
n n
≈
positive constants. Thus Theorem 3.3 yields an optimal ( 3 logn,√logn) total
2µ
variation cutoff.
Example 3.4. Assume that p is concentrated equally on two integers m < m
n n ′n
and write m =m k2. Thus p (m )=p (m k2)=1/2 and
′n n n n n n n n
µ =logm k , σ =logk .
n n n n n
In this case, Condition (3.4) is equivalent to (3.3), that is
µ =log(m k )=o(logn).
n n n
Assuming that (3.3) holds true, Theorem3.3 yields a total variationcutoff at time
3logn
t =
n
2logm k
n n
with window size
1 (logk )2logn
n
b = max 1, .
n
logmnkn ( s logmnkn )
For instance, assume that m = m +1 with m tending to infinity. Then (3.3)
′n n n
becomes logm =o(logn) and we have
n
1 (logn)1/2
b = max 1, .
n logm m (logm )1/2
n (cid:26) n n (cid:27)
Specializing further to m (logn)α with α (0, ) yields
n
≈ ∈ ∞
3logn
t
n
∼ 2αloglogn
and
(loglogn) 1 if α [1/2, )
bn ≈ (logn)1/2−α(logl−ogn)−3/2 if α∈(0,1/∞2).
(cid:26) ∈
THE CUTOFF PHENOMENON FOR RANDOMIZED RIFFLE SHUFFLES 9
In particular, b = o(1) when α 1/2 but tends to infinity when α (0,1/2).
n
≥ ∈
Compare with Example 3.2 above.
Regarding Theorem 3.3, one might want to remove the hypothesis of existence
ofasecondmomentconcerningthe randomvariableslogX . Itturns outthatitis
n
indeedpossible butatthe priceoflosingcontrolofthe windowofthe cutoff. What
may be more surprising is that one can also obtain results without assuming that
the first moment µ is finite. In some cases, it might be possible to control the
n
window size by using convergence to symmetric stable law of exponent α (1,2)
∈
but we did not pursue this here.
Theorem 3.4. Referring to the notation introduced above, assume that µ > 0
n
(includingpossiblyµ = ). Assumefurtherthatthereexistsasequencea tending
n n
∞
to infinity and satisfying
(logn)EZ2 logn
(3.5) a =O(logn), lim n =0, lim = ,
n n→∞ a2nEYn n→∞ EYn ∞
where Y = Z = logX if logX a , and Y = 0, Z = a if logX > a .
n n n n n n n n n n
≤
Then the family (S ,Q ,U ) presents a total variation cutoff with critical
{ n n,pn n }∞1
time
3logn
t = .
n
2EY
n
Remark 3.2. In Theorem 3.4, if (3.5) holds for some sequence (a ) then it also
n
holds for any sequence (da ) with d>0. Moreover,for all d>0,
n
E (logX )1 EY .
n {logXn≤dan} ∼ n
This is proved in Lemma 8(cid:0).2 below. (cid:1)
Example 3.5. Assume p ( ei ) = c 1i 2 for all 1 i logn , where c =
n ⌊ ⌋ −n − ≤ ≤ ⌊ ⌋ n
1+2 2+3 2+ +( logn ) 2. Note thatc c=π2/6asn . Inthiscase,
− − − n
··· ⌊ ⌋ → →∞
µ c 1loglogn, σ2 c 1logn and for ǫ>0
n ∼ − n ∼ −
ǫ
E ξn21{ξn2<ǫµ−n1logn} ∼ loglogn.
h i r
Hence the Lindeberg type condition (3.4) does not hold and Theorem 3.3 does not
apply. However, if we consider a = logn and try to apply Theorem 3.4, we have
n
EY = µ c 1loglogn and EZ2 c 1logn. This implies that (3.5) holds and
n n ∼ − n ∼ −
yields a total variation cutoff with critical time π2logn .
4loglogn
The untruncated version of this example is p ( ei ) = p( ei ) = c 1i 2, i =
n − −
⌊ ⌋ ⌊ ⌋
1,2,... and c = π2/6. In this case, µ = µ = . Theorem 3.4 applies with
n
∞
a =logn and yields a total variation cutoff with critical time π2logn .
n 4loglogn
We endthis sectionwitharesultwhichis asimple corollaryofTheorem3.4and
readily implies Theorem 1.4.
Theorem 3.5. Let X ,p ,µ be as above. Assume that
n n n
(3.6) µ =E(logX )=o(logn)
n n
and that, for any fixed η >0,
(3.7) E[(logX )1 ]=o (µ ).
n {logXn>ηlogn} η n
Then the family (S ,Q ,U ) has a total variation cutoff at time t = 3logn.
{ n n,pn n }∞0 n 2µn
10 G.-Y.CHENANDL.SALOFF-COSTE
Example 3.6. Suppose p = p and 0 < µ = µ < as in Theorem 1.4. Then
n n
∞
condition(3.6)-(3.7)areobviouslysatisfied. ThusTheorem1.4followsimmediately
from Theorem 3.5 as mentioned above.
Remark 3.3. Condition (3.7) holds true if X satisfies the (logarithmic) moment
n
condition that there exists ǫ>0 such that
E([logX ]1+ǫ)
n
=o(µ ).
(logn)ǫ n
4. An application: Continuous-time card shuffling
In this section, we consider the continuous-time version of the previous card
shuffling models where the waiting times between two successive shuffles are inde-
pendent exponential(1) random variables. Thus, the distribution of card arrange-
ments attime t startingfromthe deck inorderis givenby the probability measure
Hn,t =e−t(I−Qn,pn) defined by
∞ tk
(4.1) H (σ)=H (r)=e t Qk (r) for σ S ,
n,t n,t − k! n,pn ∈ n
k=0
X
where r is the number of rising sequences of σ.
The definition of total variation cutoff and its optimality for continuous time
families is the same as in Definitions 2.1, 2.2 and 2.3 except that all times are
now taken to be non-negative reals. To state our results concerning the family
(S ,H ,U ) of continuous time Markov chains associated with p -shuffles,
{ n n,t n }∞1 n
n=1,2,..., we keep the notation introduced in Section 3. In particular, we set
3logn
µ =E(logX ), σ2 =Var(logX ), t = ,
n n n n n 2µ
n
where X denotes a random variable with distribution p , and, if µ ,σ (0, ),
n n n n
∈ ∞
logX µ
n n
ξ = − .
n
σ
n
We will obtain the following theorems as corollaries of the discrete time results of
Section 3. Our first result concerns the case where each p is concentrated on one
n
integer as in Theorem 3.1.
Theorem 4.1. Assumethat for each n there is an integer m such that p(m )=1
n n
Then µ =logm , t = 3logn and the family = (S ,H ,U ) presents a
n n n 2logmn F { n n,t n }∞1
total variation cutoff if and only if
logn
lim = .
n→∞logmn ∞
Moreover, if this condition is satisfied then has an optimal t ,√t total vari-
n n
F
ation cutoff.
(cid:0) (cid:1)
Compare with the discrete time resultstated in Theorem3.1and with Example
3.1 which we now revisit.
Example 4.1. Assume that P(X = nα )=1 for a fixed α>0 as in Example 3.1.
n
⌊ ⌋
According to Theorem 4.1, the continuous time family does not present a total
variation cutoff in this case since lim logn = 1/α <F . Recall from Example
n→∞ µn ∞
3.1 that the corresponding discrete time family has a cutoff.