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 With L ≔ maxxPS L x, x , the Cheeger’s inequality states that | | | p 2 q| ι2 λ2 ι2 8 L ď ď | | Connectivity spectrum 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].