On Hyperboundedness and Spectrum of Markov Operators

Home , Spectral gap

On hyperboundedness and spectrum of Markov operators

Laurent Miclo

Institut de Math´ematiques de Toulouse Plan of the talk

1 Høegh-Krohn and Simon’s conjecture

2 Higher order Cheeger inequalities in the ﬁnite setting

3 An approximation procedure

4 General higher order Cheeger’s inequalities

5 Quantitative links between hyperboundedness and spectrum Reversible Markov operators

On a probability space S, S,µ , a self-adjoint operator 2 2 p q M : L µ L µ is said to be Markovian if µ-almost surely, p qÑ p q 2 f L µ , f 0 M f 0 @ P p q ě ñ r sě M 1 1 r s “ So M admits a spectral decomposition: there exists a projection-valued measure El 1 1 such that p qlPr´ , s 1 M l dEl “ 1 ż´ and M is said to be ergodic if

2 f L µ , Mf f f Vect 1 @ P p q “ ñ P p q 2 namely if E1 E1 L µ Vect 1 . p ´ ´qr p qs “ p q Hyperboundedness

Stronger requirement: M has a spectral gap if there exists λ 0 2 ą such that E1 E1 λ L µ Vect 1 . Spectral gap: the p ´ ´ qr p qs “ p q supremum of such λ. The Markov operator M is hyperbounded if there exists p 2 such ą that

M L2 Lp } } pµqÑ pµq ă `8

Theorem A self-adjoint ergodic and hyperbounded Markovian operator admits a spectral gap.

This result was conjectured by Høegh-Krohn and Simon [1972], in the context of Markovian semi-groups. Semi-groups

A reversible Markovian semi-group Pt tě0: a continuous family of p 2q self-adjoint Markovian operators on L µ satisfying P0 Id and p q “

t, s 0, Pt Ps Pt s @ ě “ ` 2 2 Ergodic if for any f L µ , Pt f µ f 1 in L µ . Admits a P p q r sÑ r s p q spectral gap: this convergence is uniform over the unit ball of L2 µ . It corresponds to a spectral gap of the associated generator p q L. The semi-group Pt t 0 is said to be hyperbounded, if there exists p q ě a time T 0 such that the Markov operator PT is hyperbounded. ě It follows from the previous result, applied to M PT , that a “ reversible ergodic hyperbounded Markov semi-group admits a spectral gap. Hypercontractivity

The semi-group Pt t 0 is said to be hypercontractive, if there p q ě exist p 2 and a time T 0 such that the norm of PT from 2 ą ě L µ to Lp µ is 1. p q p q The Høegh-Krohn and Simon’s conjecture is easy to solve for a hypercontractive Markov semi-group, since not only its generator admits a spectral gap, but it also satisfies a logarithmic Sobolev inequality, property in fact equivalent to hypercontractivity. It is known that hyperboundedness is itself equivalent to a non-tight logarithmic Sobolev inequality and that the existence of a spectral gap enables to tight such an inequality. Thus the previous theorem shows that for semi-groups, hyperboundedness is equivalent of hypercontractivity. Finite Cheeger inequality Finite setting: S is a finite set whose points are given a positive probability by µ. We start with a Markovian generator L: a matrix L x, y x,y S whose off-diagonal entries are non-negative and p p qq P whose lines sum up to zero. We assume that the operator L is symmetric in L2 µ . It is well-known to be non-positive definite, let p q 0 λ1 λ2 λN “ ď ď ¨¨¨ ď be the spectrum of L. ´ Cheeger’s inequality relates λ2 to an isoperimetric quantity. The conductance associated to any subset A S with A : Ă “ H µ 1AL 1Ac j A ≔ p p qq p q µ A p q The connectivity constant of L is defined by c ι2 ≔ min max j A , j A min j A A“H,S t p q p qu “ A : 0ăµpAqď1{2 p q

What for the other eigenvalues λ3, ..., λN ? Introduce the connectivity spectrum ιn . For n N, let Dn be the set of p qnPJNK P n-tuples A1, ..., An of disjoint and non-empty subsets of S and p q

ιn ≔ min max j Ak pA1,...,AnqPDn kPJnK p q

Clearly ι1 0 λ1 and for n 2 the two deﬁnitions of ι2 coincide. “ “ “ We conjectured that there exists a mapping c : N R˚ such that Ñ ` for all S, µ, L as above, p q 2 ιn c n λn ιn (1) p q L ď ď | | The second inequality is immediate, it amounts to consider the vector space generated by the indicator functions of n disjoints subsets in the variational characterization of λn through Rayleigh quotients. Higher order Cheeger inequalities

The first inequality was recently obtained by Lee, Gharan and Trevisan [2011], in the context of weighted finite graphs S, E,ω : p q E is a set of undirected edges (which may contain loops) weighted through ω : E R . It can be transcripted for finite Markovian Ñ ` generator, taking

µ x L x, y , if e x, y with x y e E, ω e ≔ p q p q “ t u “ @ P p q L L x, x , if e x, x " | |` p q “ t u It follows Theorem (Lee, Gharan and Trevisan) There exists a universal constant η 0 such that (1) is satisﬁed ą with η n N, c n ≔ @ P p q n8 A hyperboundedness estimate

Hyperboundedness does not look very pertinent in the ﬁnite setting, let us nevertheless derive from the above theorem a quantitative bound for this property. On the ﬁnite set S, consider a Markov kernel M symmetric in L2 µ and denote its spectrum by p q

1 θ1 θ2 θN 1 “ ě ě ¨¨¨ ě ě ´

Proposition

Assume that for some n JNK, N card S , we have P “ p q θn 1 c n 4. Then we have that for any p 2, ě ´ p q{ ą p 1 2δ p 1 p n 2 ´ M 2 p p ´ q n } }L pµqÑL pµq ě 2

where δn ≔ 1 θn c n 1 2. p ´ q{ p qď { a Proof (1)

Introduce the Markovian generator L M Id, the eigenvalues of “ ´ L are λm 1 θm, m JNK. Next consider n JNK as in the ´ “ ´ P P proposition. According to Theorem [LGT], we have

ιn L λn c n λn c n δn ď | | { p q ď { p q “ a a so we can ﬁnd A1, ..., An Dn satisfying, p q P

k JnK, δnµ Ak µ 1A L 1Ac @ P p q ě r k r k ss Taking into account that

L 1Ac 1A M 1A r k s “ k ´ r k s we deduce that for any k JnK, P

1 δn µ Ak µ 1A M 1A p ´ q p q ď r k r k ss Proof (2)

For any k JnK, consider the set P

Bk ≔ x Ak : M 1A x 1 2δn t P r k sp qě ´ u We compute that

µ 1A M 1A µ 1B M 1A µ 1 M 1A r k r k ss “ r k r k ss ` r Ak zBk r k ss µ Bk 1 2δn µ Ak µ Bk ď p q ` p ´ qp p q´ p qq It follows from the two last bounds that 1 µ Ak µ Bk µ Ak 2 p q ď p q ď p q

Since the sets A1, ..., An are disjoint, there exists k JnK such that P µ Ak 1 n. Consider f 1A , it appears that p qď { “ k 2 µ f µ Ak r s “ p q Proof (3)

and since by assumption 1 2δn 0, we get by deﬁnition of Bk , ´ ě p p µ M f 1 2δn µ Bk r| r s| s ě p ´ q p p q 1 2δn p ´ q µ Ak ě 2 p q In particular, we obtain that

p p µ M f M L2pµqÑLp pµq r| r2 s|p s } } ě µ f 2 r s p 1 2δn p ´ p ´q 1 ě 2µ Ak 2 p q p 1 2δn p 1 p ´ q n 2 ´ ě 2 as announced. On the spectral Theorem

We come back to the Høegh-Krohn and Simon’s conjecture framework and approximate it by finite sets. We need a preliminary consequence of the spectral Theorem: Lemma The ergodic self-adjoint Markov operator M has no spectral gap if 2 and only if for any λ 0, E1 E1 λ L µ is of infinite ą p ´ ´ qr p qs dimension. 2 Indeed, the mapping 0, 2 λ dim E1 E1 λ L µ is r s Q ÞÑ pp ´ ´ qr p qsq non-decreasing. So if for some λ 0, 2 , we have that 2 P p s dim E1 E1 λ L µ , it appears that the Z -valued pp ´ ´ qr p qsq ă `8 ` mapping 0, λ l dim E1 E1 l has a finite number of jumps, r s Q ÞÑ p ´ ´ q which correspond to eigenvalues. A contradictory argument (1)

Assume that the ergodic self-adjoint Markov operator M has no spectral gap. For any n N, let 0 ǫn 1 c n 32 be given, P ă ă ^ p p q{ q where c n is deﬁned in Theorem [LGT]. By the above lemma, we p q 2 can ﬁnd f1, ..., fn L µ , which are normalized, mutually P p q orthogonal and so that

0 ,if i j µ fi Mfj “ “ r s 1 ǫn , if i j " ě p ´ q “ To come back to the ﬁnite case, consider a non-decreasing family SN N N of ﬁnite sub-σ-algebras of S such that p q P

SN σ f1, ..., fn “ p q NPN ł A contradictory argument (2)

Fixing N N, we consider µN the restriction of µ to SN , IN the P 2 2 natural injection of L µN into L µ and EN the conditional p q p q expectation (projection operator) with respect to SN . Deﬁne furthermore MN ≔ ENMIN , which is a reversible Markov kernel on SN , SN ,µN , where SN is the ﬁnite set of atoms of SN . p q It follows from Jensen’s inequality, we have for any p 2 ą p p MN L2 Lp M L2 Lp } } pµN qÑ pµN q ď } } pµqÑ pµq Furthermore by the martingale convergence theorem, for any 2 2 f L σ f1, ..., fn ,µ , we have in L µ , P p p q q p q

lim EN f f NÑ8 r s “ A contradictory argument (3)

It can be deduced from the choice of f1, ..., fn that for N suﬃciently large, MN has n eigenvalues above 1 2ǫn. We can ´ then apply our ﬁnite hyperboundedness estimate with δn ≔ 2ǫn c n 1 4, to get { p qď { p a 1 2δ p 1 p n 2 ´ MN L2 Lp p ´ q n } } pµN qÑ pµN q ě 2 1 p 1 2´p´ n 2 ´ ě It follows that

1 p 1 p p ´p´ 2 ´ M L2 Lp MN L2 Lp 2 n } } pµqÑ pµq ě} } pµN qÑ pµN q ě and since this is true for all n N, M cannot be hyperbounded. P Extension of deﬁnitions

The previous approximation procedure also leads to the extension of Cheeger’s inequalities to the general setting of self-adjoint Markov operators in L2 µ . For any n N, define p q P Id ≔ µ f M f λn M inf max r p ´2 qr ss : f H 0 p q H : dimpHq“n µ f P zt u " r s * (in the general framework, these quantities are no longer necessarily counting the ordered eigenvalues of Id M). The ´ definition of conductance can also be extended to all non-negligible and measurable A S: P µ 1AM 1Ac j A ≔ p p qq p q µ A p q so that the connectivity spectrum ιn M n N can be defined as p p qq P before. Markovian higher order Cheeger’s inequalities

Proposition With η 0 the universal constant of Theorem [LGT], we have ą 2 η ιn M n N, p q λn M ιn M @ P n8 M ď p q ď p q | | Extension to Markovian generators? Let L be the generator of self-adjoint Markovian semi-group Pt t 0. If we deﬁne for any p q ě n N, P ≔ µ f L f λn L inf max r p´ 2qr ss : f H 0 p q H : dimpHq“n µ f P zt u " r s * then we have

1 exp tλn L n N, λn L lim ´ p´ p qq @ P p q “ tÑ0` t λn Pt lim p q “ tÑ0` t Dirichlet connectivity spectrum

A similar definition of the connectivity spectrum ιn L n N p p qq P through approximation via small times doesn’t work properly. It is convenient to introduce the Dirichlet connectivity spectrum, intermediary between the usual spectrum and the connectivity spectrum. We come back to a general self-adjoint Markov operator M. To any non-negligible A S, we associate its first Dirichlet P eigenvalue λ0 M, A given by p q µrf pId´Mqrf ss 0 ≔ λ M, A inff PDpAq µrf 2s p q 2 c # D A ≔ f L µ : f 0 µ-a.s. on A p q P p q “ ( The Dirichlet connectivity spectrum Λn M n N of M is defined by p p qq P

n N, Λn M ≔ min max λ0 M, Ak @ P p q pA1,...,AnqPDn kPJnK p q Usual and Dirichlet connectivity spectra

In the finite setting, Lee, Gharan and Trevisan also proved: Theorem (Lee, Gharan and Trevisan) There exists a universal constant η 0 such that for any finite ą self-adjoint Markov operator M, we have p η n N, Λn M λn M Λn M @ P n6 p q ď p q ď p q p There is no difficulty in applying the spatial approximation procedure to the Dirichlet connectivity spectrum, so that the finiteness assumption can be indeed removed. But due to its spectral features, the Dirichlet connectivity spectrum also well-behaves under small times approximations. Dirichlet conditions

For any non-negligible A S, resorting to the Dirichlet semi-group P 2 t 0, PA,t : L µ f 1APt 1Af @ ě p q Q ÞÑ r s it can be proven that

Λn Pt lim p q Λn L tÑ0` t “ p q ≔ min max λ0 L, Ak pA1,...,AnqPDn kPJnK p q

where

µrf p´Lqrf ss 0 ≔ λ L, A inff PDpL,Aq µrf 2s p q c # D L, A ≔ f D L : f 0 µ-a.s. on A p q t P p q “ u With these definitions, Theorem [LGT2] extends to the generator L. Riemannian setting Let us return to the Riemannian setting of Cheeger [1970]. The state space S is a compact Riemannian manifold and L is the associated Laplacian △ operator. Due to the regularizing properties of the corresponding heat semi-group, Dn can be restricted to contain only subsets with smooth boundaries. If A is such a subset, a minimizer function exists for λ0 L, A , so going p q through the proof of original Cheeger’s inequality, via co-area formula, it appears that we can find a subset B A with smooth Ă boundary B such that B 2 σ B pB q 4λ0 △, A µ B ď p q ˆ p q ˙ σ is the dim S 1 -dimensional measure associated to µ, the p p q´ q Riemannian probability. This observation leads to define the connectivity spectrum ιn △ n N of △ through p p qq P σ Ak n JNK, ιn △ ≔ min max pB q @ P p q 1 D kPJnK µ Ak pA ,...,AnqP n p q p Cheeger and Buser inequalities The Riemannian higher order Cheeger inequalities follow: Theorem There exists a universal constant η 0 such that for any compact ą Riemannian manifold S, we have p η 2 n N, λn △ ι △ @ P p q ě n6 np q p Adapting the proof of Ledoux [1994] to the context of Dirichlet semi-groups, we can also extend the inequalities of Buser [1982]: Theorem there exists a constant C 0 depending only on the dimension of ą S, such that for all n N, P 2 λn △ C ?Kιn △ ι △ p q ď p p q` np qq where K 0 is a lower bound on the Ricci curvature of S. ´ ď Witten Laplacian

But it seems that the inequalities of Theorem [LGT2] are more interesting that those of Cheeger. For instance they can be used to deduce the exponential behavior of the small eigenvalues of Witten Laplacians at small temperature. Still on a compact Riemannian manifold, Witten Laplacians have the form Lβ △ β ∇U, ∇ , “ ¨´ x 1 ¨y where β 0 is the inverse temperature and U : S R is a C ě Ñ potential. An open and connected set B S is said to be a well, if U is Ă constant on B and if for any x B, U x U B . The height B P p qă pB q of a well B is given by h B U B minB U. For n N, let ln p q“ pB q´ P the highest l 0 such that n disjoint wells of height l can be ě found in S. Then we have

´1 n N, ln ≔ lim β ln λn Lβ @ P ´ βÑ`8 p p qq On a result of Wang Consider the traditional L2 to L4 hypercontractivity. Wang [2004] 4 has shown that if M L2 L4 2, then the conjecture of } } pµqÑ pµq ă Høegh-Krohn and Simon is true and it is possible to deduce a quantitative lower bound on the spectral gap. Next result shows that this last part cannot be extended: Proposition

For any K 2 and for any ǫ 0, 1 , we can ﬁnd a self-adjoint ě P p q ergodic Markov operator M whose spectral gap is ǫ and which is 4 such that M L2 L4 K. } } pµqÑ pµq “ Indeed, the example is very simple and is constructed on the two points state space S ≔ 0, 1 , endowed with the probability t u measure µ ≔ η, 1 η , η 0, 1 2 , and with the self-adjoint p ´ 2q P p { s Markov operator in L µ given by p q M ≔ 1 ǫ Id ǫµ p ´ q ` where ǫ 0, 1 , whose spectral gap is ǫ. P p s Changing the indices

To get quantitative bounds, we need to look at other eigenvalues than the spectral gap. Here is another result of Lee, Gharan and Trevisan: Theorem (Lee, Gharan and Trevisan) There exists a universal constant η 0 such that if ą c n ≔ η ln 1 n , then for any ﬁnite self-adjoint Markov p q { p ` q operator,

r 2 n N, λ2n c n ι @ P ě p q n Again there is no diﬃculty to extend this result to general r self-adjoint Markov operators. Lee, Gharan and Trevisan checked that the logarithmic order of c n is optimal, by applying hypercontractivity results due to p q Bonami [1970] and Beckner [1975] to noisy hypercubes. We will rrecover this optimality via the hypercontractivity of the Ornstein-Uhlenbeck process. A quantitative version Consider a Young function Φ satisfying Φ r lim p2 q rÑ`8 r “ `8 meaning that the Orlicz’s norm of LΦ is stronger than the L2 norm. Deﬁne ?n n N, k n @ P p q “ Φ´1 n p q A careful examination of the previous proofs leads to: Theorem Let M be a self-adjoint Markov operator. If n N is such that P k n 2?2 M L2 LΦ , then we are assured of p qě } } Ñ c n λ2n M p q p q ě 16 In particular the top of the spectrum ofr M consists of 2n eigenvalues 1, 1 λ2 M , ..., 1 λ2n M (with multiplicities). ´ p q ´ p q Ornstein-Uhlenbeck generator

Let Pt t 0 be the self-adjoint Markov semi-group associated to p q ě the Ornstein-Uhlenbeck generator deﬁned by

2 f C R , x R, L f x ≔ f 2 x xf 1 x @ P b p q @ P r sp q p q´ p q which is essentially self-adjoint on L2 γ , where γ is the normal p q centered Gaussian distribution. It is well-known (see Nelson [1973] and Gross [1975]) that for any p 2, ą , if t 1 ln p 1 `8 ă 2 p ´ q Pt L2pγqÑLp pγq 1 } } “ # 1 ,if t 2 ln p 1 ě p ´ q (for a less radical transition, consider the Young function given by 2 2 2 Φ r ≔ r 1 ln r 1 r , then Pt L2 LΦ is of order p q p ` q p ` q´ } } pγqÑ pγq 1 ?t for t 0 small). { ą A disappointing application?

Let us apply the previous theorem to M Pt for t 0 and “ ą relatively to the usual Lebesgue space Lp γ with p 2. More p q ą precisely, for n N, consider P 2ln n pn ≔ p q ln n 2ln 2?2 p q´ p q 1 tn ≔ ln pn 1 2ln 2?2 ln n 2 p ´ q „ p q{ p q

and let n0 N be the smallest integer such that pn 2. We obtain P ą c n n n0 λn L p q (2) ě ùñ p qě 16tn The r.h.s. is of order 1, which is quite disappointing, since it is well-known that λn L n 1 for all n N! p q“ ´ P Could our quantitative estimates (or Theorem [LGT3]) be improved to get a lower bound of λn L going to inﬁnity with n? p q Tensorization

This is not possible, because the hypercontractivity property is stable by tensorization. More precisely, for N N, consider the bN 2 bN P pNq semi-group Pt tě0 acting on L γ . The generator L of bN p q p q P t 0 corresponds to the sum of N copies of L, each acting on p t q ě diﬀerent coordinates of RN. In particular, we get

pNq n J2, N 1K, λn L λ2 L @ P ` p q “ p q This forbids a lower bound going to inﬁnity with n in (2) and shows the optimality of the previous estimates.