Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility with R.Fernandez, F .Manzo, F .R.Nardi (in progress) Dipartimento di Matematica, Universit` a di Roma Tre Bielefeld, 5-10-2012
Plan of the talk -1- Metastability and first hitting -2- Known results and tools -3- The non reversible case [FMNS] -4- Recurrence as a robust tool
-1- Metastability and first hitting Physical systems near a phase transition (e.g. ferromagnet or saturated gas) - trapped for an abnormal long time in a state —the metastable phase— different from the eventual equilibrium state consistent with the thermodynamical potentials; - subsequently, undergoing a sudden transition at a random time from the metastable to the stable state. Statistical mechanics model: - space state X , e.g. X = {− 1 , + 1 } Λ ; interaction, e.g. Ising hamiltonian; - evolution: Markov chain on X , reversible w.r.t. Gibbs measure e − β H ; Z - decay of the metastable state: convergence to equilibrium of the chain, (equilibrium state), e.g. configuration of minimal energy in the limit of small temperature β → ∞ .
An example to introduce the problem Random walk on X = { 1 , ..., n } reversible w.r.t. the following hamiltonians, i.e., with stationary measure π ( x ) = e − β H ( x ) : Z Let δ = H ( x ) − H ( x − 1 ) for x = 2 , ..., n − 1 and define
e − δβ p x := P ( x , x + 1 ) = 1 + e − δβ , x = 1 , .., n − 1 , r 1 := P ( 1 , 1 ) = 1 − p 1 , q n = e − β [ H ( n − 1 ) − H ( n )] q x := P ( x , x − 1 ) = 1 − p x , x = 2 , .., n − 1 , , r n = 1 − q n 1 + e δβ X t n 1 t / exp ( !" ) For large β (similarly large n [Barrera,Bertoncini,Fernandez]): after many unsuccessful attempts there is a fast transition to n .
Metastability is characterized by: ◮ Exit from a well (valley) of H with a motion against the drift: large deviation regime. ◮ Many visits to the metastable state (bottom of the well, point 1 in the ex.) before the transition to the stable one ( n ), large tunneling time with exponential distribution if properly rescaled. ◮ The existence of critical configurations separating the metastable state from the stable one, ( n − 1 ), first hitting to rare events.
Our goal Metastability / first hitting to rare events is usually studied in the literature for reversible Markov chain. Our goal: prove asymptotic exponential behavior in a non reversible context, when |X| → ∞ . Example: deck of n cards, |X| = n ! Markov chain: top-in-at-random shuffling invariant maesure = uniform distribution first hitting to a particular configuration G . ◮ non reversible case ◮ entropic barrier
People: The “first hitting” community: [K] Keilson (1979) (FM) [AB], [B] Aldous, Brown (1982-92)... Day (1983), Galves, Schmitt (1990), [IMcD] Iscoe, McDonald (1994), Asselah, Dai Pra (1997), Abadi, Galves (2001)... [FL] Fill, Lyzinski (2012) ... The “metastable” community: [LP] Lebowitz, Penrose (1966) (FM) [FW] Freidlin, Wentzell (1984) (FM) [CGOV] Cassandro, Galves, Olivieri, Vares (1984) [it] Martinelli, Olivieri, S. (1989)... [fr] Catoni, Cerf (1995)... [B.et al] Bovier, Eckhoff, Gayrard, Klein (2001)... [BL] Bertrand, Landim (2011/12) Bianchi, Gaudilli` [BG] ere (2012) ... (FM) = founding member
-2- Known results and tools: first hitting community The model: X t ; t ≥ 0 irreducible, finite-state, reversible Markov chain in continuous time, with transition rate matrix Q and stationary distribution π , so that π Q = 0 and DBC : π i Q ij = π j Q ji P = ✶ + Q 0 = λ 0 < λ 1 ≤ λ 2 ≤ ... real eigenvalues of the matrix − Q , R = 1 /λ 1 : relaxation time of the chain. If the set A is such that R / ❊ π τ A is small then is possible [AB] to obtain estimate like : R / ❊ π τ A | P π ( τ A / ❊ π τ A > t ) − e − t | ≤ ∀ t > 0 (1) 1 + R / ❊ π τ A R t P π ( τ A > t ) ≥ ( 1 − ) exp {− } (2) ❊ α τ A ❊ α τ A Moreover in the regime R ≪ t ≪ ❊ π τ A bounds on the density function are given [AB] in order to obtain a control on the distribution of τ A also on scale smaller than ❊ π τ A .
Results by metastability community ( X ( n ) , n ≥ 1 ) sequence of finite state spaces, with |X ( n ) | = n � � X ( n ) t ∈ ❘ sequence of continuous time, irreducible, reversible Markov t chains on X ( n ) Q ( n ) transition rate matrix generating the chain X ( n ) t ( π ( n ) , n ≥ 1 ) invariant measures asymptotics n → ∞ . starting at x ∈ X ( n ) , and the hitting time to a set G ( n ) ⊂ X ( n ) : � ∈ G ( n ) � τ ( n ) , x t ≥ 0 : X ( n ) , x = inf (3) t G ( n ) x ( n ) ∈ X ( n ) = metastable state 0 G ( n ) ⊂ X ( n ) = critical configurations (or stable state) x ( n ) τ 0 G ( n ) → ( d ) − n →∞ Y ∼ Exp ( 1 ) x ( n ) ❊ τ 0 G ( n )
Hypotheses for metastability i) pathwise approach: ([CGOV], [it], [fr]) ( n ) , x ( n ) recurrence to x ( n ) in a time R n ≪ ❊ τ with large probability. 0 0 G ( n ) ii) potential theoretical approach: ([B. et al], [BL]) Hp . A : n →∞ n ρ A ( n ) = 0 lim (4) Hp . B : n →∞ ρ B ( n ) = 0 lim (5) � � ( n ) , x ( n ) ( n ) , x ( n ) � τ 0 < � τ 0 P G ( n ) x ( n ) � � ρ A ( n ) := sup 0 τ ( n ) , z τ ( n ) , z 0 , G ( n ) } < � z ∈X ( n ) \{ x ( n ) P � 0 , G ( n ) } z { x ( n ) ❊ τ ( n ) , z { x ( n ) 0 , G ( n ) } ρ B ( n ) := sup . ( n ) , x ( n ) z ∈X ( n ) \{ x ( n ) ❊ τ 0 0 , G ( n ) } G ( n ) � � τ ( n ) , x t > 0 : X ( n ) , x := min ∈ A � A t
Some tools (fh) - collapsed chain technique: hitting to a single state A ≡ j ([K], [AB]) - For reversible chains for each j , by using spectral representation and Laplace transform, τ π j is a geometric convolution of suitable i.i.d.r.v. W i : N � τ π j = W i (6) i = 1 with N a geometric random variable of parameter π j , approximately exponential in the sup norm [B] when π ( j ) is small. - [FL] generalize (6) to non reversible chains under additional hypotheses. - τ π j is completely monotone [K]: m � P π ( τ j > t ) = p i exp {− γ i t } (7) I = 1 with p i ≥ 0 and 0 < γ 1 < γ 2 < ... < γ m the distinct eigenvalues of − Q j . Complete monotonicity is a powerful tool and exponential behavior follows from it [AB].
- interlacing between eigenvalues of − Q j and − Q 0 = λ 0 < γ 1 ≤ λ 1 ≤ γ 2 ≤ λ 2 ≤ ... Canceling out common pairs of eigenvalues from the two spectra and renumbering them we obtain 0 = λ 0 < γ 1 < λ 1 < γ 2 < λ 2 < ... < γ m < λ m again by Laplace transform [B] m � τ π j ∼ Y i i − 1 with � � 1 − γ i e − γ i t , P ( Y i > t ) = t > 0 , i = 1 , ..., m λ i Moreover � � 1 − γ 1 e − γ 1 t ≤ P π ( τ j > t ) ≤ ( 1 − π ( j )) e − γ 1 t . λ 1 - [IMcD] exponential behavior in the non reversible case under additional implicit hypotheses (related to auxiliary processes involved in the proof) by studying the smallest real eigenvalue of a suitable Dirichlet problem.
Some tools (m) i) Different strategies in different regimes. Freidlin Wentzell techniques, cycle decomposition, cycle paths... In FW theory reversibility is not required, cycles and cycle path are easily defined in the reversible case. ii) Spectral characteristic of the generator, tools from potential theory, variational principles,... 1 c ( i , j ) = r ( i , j ) = π i P ij r ( i , j ) = r ( j , i ) ⇐ ⇒ DBC Extension to non reversible chains by Eckhoff (not published) (??) [BL] non reversible case under additional implicit hypotheses, which are not easy to verify in the non reversible case.
-3-The non reversible case [FMNS1] ( X ( n ) , n ≥ 1 ) sequence of finite state spaces, with |X ( n ) | = n � � X ( n ) t ∈ ❘ continuous time, irreducible Markov chains on X ( n ) t = metastable state, G ( n ) = critical configurations (or stable state) x ( n ) 0 in the following sense: Hp. G ( T n ) : there exist sequences r n → 0 and R n ≪ T n such that � � τ ( n ) , x sup 0 , G ( n ) } > R n ≤ r n . P { x ( n ) x ∈X asymptotic results: the following are equivalent x ( n ) τ 0 → ( d ) G ( n ) − n →∞ Y ∼ Exp ( 1 ) T n � � ( n ) , x ( n ) ∃ ℓ ≥ 1 , ξ < 1 : P τ 0 > ℓ T n ≤ ξ uniformly in n G ( n ) quantitative results: R | P x ( τ G / ❊ x 0 τ G > t ) − e − t | ≤ f ( , r ) E x 0 τ G
Comparison of hypotheses [FMNS] T LT = n mean local time spent in x ( n ) before reaching G ( n ) starting from x ( n ) 0 0 � � � � ( n ) , x ( n ) T Q ξ := inf t : P τ 0 > t ≤ ξ n G ( n ) � � ( n ) , x ( n ) T E n = ❊ τ 0 G ( n ) The following implications hold: ⇒ Hp . ( T Q ξ ⇒ Hp . G ( T LT ⇒ Hp . G ( T E Hp . A = n ) = n ) ⇐ n ) ⇐ ⇒ Hp . B for any ξ < 1. Furthermore, the missing implications are false.
-4-Recurrence as a robust tool a) Factorization property: If S > R with � � τ x sup P { x 0 , G } > R ≤ r . x ∈X then for any t , s > R S � � � � � � τ x 0 τ x 0 � � ≥ P G > tS + R − r P G > sS τ x 0 � � � � � � P G > ( t + s ) S τ x 0 τ x 0 ≤ P G > tS − R + r P G > sS . τ ( tS ) = inf { T > tS ; X T ∈ { x 0 , G }} P ( τ ( tS ) − tS > R ) ≤ r
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