The L p convergence of finite Markov chains Guan-Yu Chen Department of Applied Mathematics, NCTU August 10, 2010 G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 1 / 25
Outline Ergodic Markovian semigroups 1 L p mixing 2 The L p cutoff 3 Recent works & references 4 G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 2 / 25
Ergodic Markovian semigroups Basic setting S : a set equipped with a σ -field B . T : the time, either { 0 , 1 , 2 , ... } or [0 , ∞ ). p ( t , x , · ): a family of probabilities (or transition functions) for all x ∈ S and t ∈ T . P t : an operator defined by � P t f ( x ) = f ( y ) p ( t , x , dy ) S for any bounded B -measurable function f . µ P t : a probability on S given by � µ P t ( A ) = p ( t , x , A ) µ ( dx ) . S G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 3 / 25
Ergodic Markovian semigroups Markov transition functions & Invariant measures p ( t , x , · ) (or P t ) is called Markovian if for any A ∈ B and x ∈ S , � p ( s + t , x , A ) = p ( t , y , A ) p ( s , x , dy ) , p (0 , x , { x } ) = 1 . S A Markov process ( X t ) t ∈ T with filtration F t = σ ( X s : s ≤ t ) ⊂ B has p ( t , x , · ) as transition function provided � E ( f ◦ X t |F s ) = f ( y ) p ( t − s , X s , dy ) = P t − s f ( X s ) S for all 0 < s < t and bounded measurable function f . G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 4 / 25
Ergodic Markovian semigroups A probability π is called invariant if π P t = π for all t ∈ T . If µ ≪ π , then µ P t ≪ π for all t > 0 and d ( µ P t ) / d π = P ∗ t ( d µ/ d π ), where P ∗ t is the adjoint of P t in L 2 ( π ). Suppose |S| < ∞ . In either of the following cases, (i) T = [0 , ∞ ), P t = e − t ( I − K ) where K is an irreducible stochastic matrix; (ii) T = { 0 , 1 , 2 , ... } , p (1 , · , · ) is irreducible and aperiodic ; there is a unique invariant probability, say π , and t →∞ p ( t , x , y ) = π ( y ) lim ∀ x , y . G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 5 / 25
L p mixing L p distance & mixing time Let p ( t , x , · ) be a Markov transition function with invariant probability π . For p ∈ [1 , ∞ ], the L p distance is defined by D p ( µ, t ) = � d ( µ P t ) / d π − 1 � p and the sup- L p distance is defined by D p ( t ) := sup µ D p ( µ, t ) = sup x D p ( δ x , t ) . G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 6 / 25
L p mixing For t , s ∈ T such that t + s ∈ T , D p ( µ, t + s ) ≤ D p ( µ, s )[ D p ( t ) ∧ 1] . This implies (a) t �→ D p ( µ, t ) is non-increasing. (b) t �→ D p ( t ) is non-increasing and submultiplicative. The L p mixing time is defined by T p ( µ, ǫ ) = inf { t ∈ T | D p ( µ, t ) ≤ ǫ } and T p ( ǫ ) = inf { t ∈ T | D p ( t ) ≤ ǫ } . G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 7 / 25
L p mixing Transitive group action Assume that S is a compact topological space and G is a compact group acting continuously and transitively on S . Then, S = G / G o where G o is the stabilizer of any fixed point o ∈ S . If p ( t , x , · ) is invariant under the action of G , i.e., p ( t , gx , gA ) = p ( t , x , A ) ∀ g ∈ G , x ∈ S , A ⊂ S , then D p ( δ x , t ) = D p ( δ y , t ) for all x , y ∈ G . π ( A ) = u ( φ − 1 ( A )), where φ : G → G / G o is the canonical projection map and u is the normalized Haar measure on G . G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 8 / 25
L p mixing Random walks on finite groups Let Q be a probability on a finite group G . A random walk on G driven by Q is a Markov chain whose transition function p ( t , x , y ) satisfies: (i) If T = { 0 , 1 , ... } , then p ( t , x , y ) = Q ∗ t ( x − 1 y ) i =0 e − t t i i ! Q ∗ i ( x − 1 y ) (ii) If T = [0 , ∞ ), then p ( t , x , y ) = � ∞ where Q ∗ i is the convolution of Q for i times. The uniform distribution on G is invariant. D p ( t ) = D p ( δ x , t ) for all x ∈ G . Remark: D p ( δ x , t ) = D p ( δ y , t ) for all x , y does not necessarily imply D p ( µ, t ) = D p ( ν, t ) for any probabilities µ, ν in general. G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 9 / 25
L p mixing Spectral gap The spectral gap (of a Markov transition function) λ is defined to be the largest constant c such that � P t f − π ( f ) � 2 ≤ e − ct � f � 2 ∀ f . If T = [0 , ∞ ) and P t is strongly continuous on L 2 ( π ), then λ = inf {�− Af , f �| f ∈ Dom( A ) , real-valued , π ( f ) = 0 , � f � 2 = 1 } where A is the infinitesimal generator of P t . G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 10 / 25
L p mixing If T = { 0 , 1 , 2 , ... } , then e − λ is the second largest singular value of P 1 on L 2 ( π ). That is, λ = − log( � P 1 − E π � L 2 ( π ) → L 2 ( π ) ) . If S is finite, then for p ∈ [1 , ∞ ], ω ≤ T p (1 / e ) ≤ 1 1 2 λ (1 + log π − 1 ∗ ) where e − ω t is the spectral radius of P t and π ∗ = min x π ( x ). G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 11 / 25
L p mixing Logarithmic Sobolev inequality & Poincar´ e inequality Gross (1975) introduced the concept of logarithmic Sobolev inequality on continuous spaces. Diaconis & Saloff-Coste (1996) introduced a discrete version as follows. Write P t = e − t ( I − K ) and let π be the invariant probability. Define the Dirichlet form associated with K by E ( f , g ) = ℜ� f , g � and set L ( f ) = π ( | f | 2 log | f | 2 ) − π ( | f | 2 ) log π ( | f | 2 ). The logarithmic Sobolev inequality is what follows. c L ( f ) ≤ E ( f , f ) ∀ f . G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 12 / 25
L p mixing The logarithmic Sobolev constant denoted by α is the largest constant such that the above inequality holds. That is, � E ( f , f ) � � � α = inf � L ( f ) � = 0 . � L ( f ) The Poincar´ e inequality is � f − π ( f ) � 2 ≤ C E ( f , f ) ∀ f . The spectral gap λ is the inverse of the smallest C such that the Poincar´ e inequality holds. In general, 2 α ≤ λ ≤ ω . G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 13 / 25
L p mixing If K is reversible, i.e. P t is self-adjoint in L 2 ( π ), then For 2 ≤ p ≤ ∞ , 2 α ≤ T p (1 / e ) ≤ 1 1 2 α (3 + log + log π − 1 ∗ ) . For 1 < p < 2, 2 m p α ≤ T p (1 / e ) ≤ 1 1 4 α (4 + log + log π − 1 ∗ ) , where log + t = 0 ∨ log t and m p = 1 + ⌈ (2 − p ) / (2 p − 2) ⌉ . Remark: For non-reversible cases, the upper bound is similar. (multiply the right side with 2) but, however, the lower is unknown. G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 14 / 25
The L p cutoff L p cutoff Consider a family of ergodic Markov transition functions F = { p n ( t , · , · ) | n = 1 , 2 , ... } . For p ∈ [1 , ∞ ], F is said to present a L p cutoff if there is a sequence t n such that � ∞ if 0 < c < 1 n →∞ D n , p ( ct n ) = lim . 0 if c > 1 t n is called the L p cutoff sequence (or cutoff time). Equivalently, F presents a L p cutoff if and only if T n , p ( ǫ ) lim T n , p ( δ ) = 1 ∀ 0 < ǫ < δ < ∞ . n →∞ In particular, if t n is a L p cutoff sequence, then t n ∼ T n , p ( ǫ ) for all ǫ > 0. G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 15 / 25
The L p cutoff Theorem Fix p ∈ (1 , ∞ ). Consider a family F of Markov transition functions p n ( t , · , · ) with t ∈ T = [0 , ∞ ) and invariant probability π n . Let λ n be the spectral gap and T n , p ( ǫ ) be the L p mixing time of the n th transition function. (i) If there is ǫ > 0 such that n →∞ λ n T n , p ( ǫ ) = ∞ , lim then F has a L p cutoff. (ii) If p n ( t , · , · ) is normal (that is, P n , t P ∗ n , t = P ∗ n , t P n , t in L 2 ( π n )) and F has a L p cutoff, then n →∞ λ n T n , p ( ǫ ) = ∞ ∀ ǫ > 0 . lim Remark: Similar result holds true for the discrete time case. G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 16 / 25
The L p cutoff Random transposition Random transposition is a random walk on the symmetric group S n of degree n driven by Q , a probability satisfying Q ( id ) = 1 / n and Q ( i , j ) = 2 / n 2 . Diaconis & Shahshahani (1981) showed the existence of L p cutoff for 1 ≤ p ≤ 2 at time n log n . Representation theory yields all eigenvalues so that λ n ∼ 2 / n . Random transpositions present the L p cutoff with 2 < p < ∞ . (In fact, L ∞ cutoff exists.) But, however, the cutoff sequence is unknown. G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 17 / 25
The L p cutoff Regular expander graphs A lazy simple random walk on a graph is a Markov chain whose transit evolves in the way that (i) Remain at current state with probability 1 / 2. (ii) Move to neighbors uniformly at random with probability 1 / 2. Fix k > 0 and consider a family of k -regular graphs G n = ( V n , E n ) satisfying for all n , � |{ edges between A and A c }| � � � min � A � = ∅ , | A | ≤ | V n | / 2 ≥ ǫ > 0 . � | A | The simple random walk on G n has T n , p ( ǫ ) ≥ c k ( ǫ ) log | V n | , inf n λ n > 0 . If | V n | → ∞ , then the family has a L p cutoff for 1 < p ≤ ∞ . But, the L 1 cutoff is open. G.-Y. Chen (DAM, NCTU) Log-Sobolev constant August 10, 2010 18 / 25
Recommend
More recommend
