Spectral computations for birth and death chains
aa r X i v : . [ m a t h . P R ] M a y SPECTRAL COMPUTATIONS FOR BIRTH AND DEATHCHAINS
GUAN-YU CHEN AND LAURENT SALOFF-COSTE Abstract.
We consider the spectrum of birth and death chains on a n -path.An iterative scheme is proposed to compute any eigenvalue with exponentialconvergence rate independent of n . This allows one to determine the wholespectrum in order n elementary operations. Using the same idea, we alsoprovide a lower bound on the spectral gap, which is of the correct order onsome classes of examples. Introduction
Let G = ( V, E ) be the undirected finite path with vertex set V = { , , ...n } andedge set E = {{ i, i + 1 } : i = 1 , , ..., n − } . Given two positive measures π, ν on V, E with π ( V ) = 1, the Dirichlet form and variance associated with ν and π aredefined by E ν ( f, g ) := n − X i =1 [ f ( i ) − f ( i + 1)][ g ( i ) − g ( i + 1)] ν ( i, i + 1)and Var π ( f ) := π ( f ) − π ( f ) , where f, g are functions on V . When convenient, we set ν (0 ,
1) = ν ( n, n + 1) = 0.The spectral gap of G with respect to π, ν is defined as λ Gπ,ν := min (cid:26) E ν ( f, f )Var π ( f ) (cid:12)(cid:12)(cid:12)(cid:12) f is non-constant (cid:27) . Let M Gπ,ν be a matrix given by M Gπ,ν ( i, j ) = 0 for | i − j | > M Gπ,ν ( i, j ) = − ν ( i, j ) π ( i ) , ∀| i − j | = 1 , M Gπ,ν ( i, i ) = ν ( i − , i ) + ν ( i, i + 1) π ( i ) . Obviously, λ Gπ,ν is the smallest non-zero eigenvalue of M Gπ,ν .Undirected paths equipped with measures π, ν are closely related to birth anddeath chains. A birth and death chain on { , , , ..., n } with birth rate p i , deathrate q i and holding rate r i is a Markov chain with transition matrix K given by(1.1) K ( i, i + 1) = p i , K ( i, i −
1) = q i , K ( i, i ) = r i , ∀ ≤ i ≤ n, where p i + q i + r i = 1 and p n = q = 0. Under the assumption of irreducibility,that is, p i q i +1 > ≤ i < n , K has a unique stationary distribution π given by π ( i ) = c ( p · · · p i − ) / ( q · · · q i ), where c is the positive constant such that Mathematics Subject Classification.
Key words and phrases.
Birth and death chains, spectrum. Partially supported by NSC grant NSC100-2115-M-009-003-MY2 and NCTS, Taiwan. Partially supported by NSF grant DMS-1004771. P ni =0 π ( i ) = 1. The smallest non-zero eigenvalue of I − K is exactly the spectralgap of the path on { , , ..., n } with measures π, ν , where ν ( i, i + 1) = π ( i ) p i = π ( i + 1) q i +1 for 0 ≤ i < n .Note that if is the constant function of value 1 and ψ is a minimizer for λ Gπ,ν ,then ψ − π ( ψ ) is an eigenvector of M Gπ,ν . This implies that any minimizer ψ for λ Gπ,ν satisfying π ( ψ ) = 0 satisfies the Euler-Lagrange equation,(1.2) λ Gπ,ν π ( i ) ψ ( i ) = [ ψ ( i ) − ψ ( i − ν ( i − , i ) + [ ψ ( i ) − ψ ( i + 1)] ν ( i, i + 1) , for all 1 ≤ i ≤ n . Assuming the connectedness of G (i.e., the superdiagonal andsubdiagonal entries of M Gπ,ν are positive), the rank of M Gπ,ν − λI is at least n −
1. Thisimplies that all eigenvalues of M Gπ,ν are simple. See Lemma A.3 for an illustration.Observe that, by (1.2), any non-trivial eigenvector of M Gπ,ν has mean 0 under π .This implies that all minimizers for the spectral gap are of the form aψ + b , where a, b are constants and ψ is a nontrivial solution of (1.2). In 2009, Miclo obtainedimplicitly the following result. Theorem 1.1. [15, Proposition 1] If ψ is a minimizer for λ Gπ,ν , then ψ must bemonotonic, that is, either ψ ( i ) ≤ ψ ( i + 1) for all ≤ i < n or ψ ( i ) ≥ ψ ( i + 1) forall ≤ i < n . One aim of this paper is to provide a scheme to compute the spectrum of M Gπ,ν , inparticular, the spectral gap. Based on Miclo’s observation, it is natural to considerthe following algorithm.(A1) Choose two positive reals λ , a in advance and set, for k = 0 , , ... ,1 . ψ k (1) = − a, . ψ k ( i + 1) = ψ k ( i ) + { [ ψ k ( i ) − ψ k ( i − ν ( i − , i ) − λ k π ( i ) ψ k ( i ) } + ν ( i, i + 1) , for 1 ≤ i < n, where t + = max { t, } , . λ k +1 = E ν ( ψ k , ψ k )Var π ( ψ k ) . The following theorems discuss the behavior of λ k . Theorem 1.2 (Convergence to the exact value) . Referring to (A1) , if n = 2 , then λ k = λ Gπ,ν for all k ≥ . If n ≥ , then the sequence ( λ k , ψ k ) satisfies (1) If λ = λ Gπ,ν , then λ k = λ Gπ,ν for all k ≥ . (2) If λ = λ Gπ,ν , then λ k > λ k +1 > λ Gπ,ν for k ≥ . (3) Set ( λ ∗ , ψ ∗ ) = lim k →∞ ( λ k , ψ k ) . Then, λ ∗ = E ν ( ψ ∗ , ψ ∗ ) / Var π ( ψ ∗ ) = λ Gπ,ν and π ( ψ ∗ ) = 0 . Theorem 1.3 (Rate of convergence) . Referring to
Theorem 1.2 , there is a constant σ ∈ (0 , independent of the choice of ( λ , a ) such that ≤ λ k − λ Gπ,ν ≤ σ k − λ forall k ≥ . By Theorem 1.3, we know that the sequence λ k generated in (A1) converges tothe spectral gap exponentially but the rate ( − log σ ) is undetermined. The followingalternative scheme is based on using more information on the spectral gap and willprovide convergence at a constant rate. PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 3 (A2) Choose a > , L < λ Gπ,ν < U in advance and set, for k = 0 , , ... ,1 . ψ k (1) = − a, λ k = ( L k + U k )2 . ψ k ( i + 1) = ψ k ( i ) + { [ ψ k ( i ) − ψ k ( i − ν ( i − , i ) − λ k π ( i ) ψ k ( i ) } + ν ( i, i + 1) , for 1 ≤ i < n, where t + = max { t, } , . L k +1 = L k , U k +1 = λ k if π ( ψ k ) > L k +1 = λ k , U k +1 = U k if π ( ψ k ) < L k +1 = U k +1 = λ k if π ( ψ k ) = 0 . Theorem 1.4 (Dichotomy method) . Referring to (A2) , it holds true that ≤ max { U k − λ Gπ,ν , λ
Gπ,ν − L k } ≤ ( U − L )2 − k , ∀ k ≥ . In Theorem 1.4, the convergence to the spectral gap is exponentially fast withexplicit rate, log 2. See Remark 2.2 for a discussion on the choice of L and U . Forhigher order spectra, Miclo has a detailed description of the shape of eigenvectors in[14] and this will motivate the definition of similar algorithms for every eigenvaluein spectrum. See (D i ) and Theorem 3.4 for a generalization of (A2) and Theorem3.14 for a localized version of Theorem 1.3.The spectral gap is an important parameter in the quantitative analysis ofMarkov chains. The cutoff phenomenon, a sharp phase transition phenomenonfor Markov chains, was introduced by Aldous and Diaconis in early 1980s. It isof interest in many applications. A heuristic conjecture proposed by Peres in 2004says that the cutoff exists if and only if the product of the spectral gap and themixing time tends to infinity. Assuming reversibility, this has been proved to holdfor L p -convergence with 1 < p ≤ ∞ in [2]. For the L -convergence, Ding et al. [10] prove this conjecture for continuous time birth and death chains. In order touse Peres’ conjecture in practice, the orders of the magnitudes of spectral gap andmixing time are required. The second aspect of this paper is to derive a theoreticallower bound on the spectral gap using only the birth and death rates. This lowerbound is obtained using the same idea used to analyze the above algorithm. Forestimates on the mixing time of birth and death chains, we refer the readers tothe recent work [4] by Chen and Saloff-Coste. For illustration, we consider severalexamples of specific interest and show that the lower bound provided here is in factof the correct order in these examples.This article is organized as follows. In Section 2, the algorithms in (A1)-(A2)are explored and proofs for Theorems 1.2-1.4 are given. In Section 3, the spectrumof M Gπ,ν is discussed further and, based on Miclo’s work [14], Algorithm (A2) isgeneralized to any specified eigenvalue of M Gπ,ν . Our method is applicable for pathsof infinite length (one-sided) and this is described in Section 4. For illustration, weconsider some Metropolis chains and display numerical results of Algorithm (A2)in Section 5. In Section 6, we focus on uniform measures with bottlenecks anddetermine the correct order of the spectral gap using the theory in Sections 2-3. Itis worthwhile to remark that the assumptions in Section 6 can be relaxed using thecomparison technique in [7, 8]. As the work in this paper can also be regarded asa stochastic counterpart of theory of finite Jacobi matrices, we would like to referthe readers to [18, 19] for a complementary perspective.
G.-Y. CHEN AND L. SALOFF-COSTE Convergence to the spectral gap
This section is devoted to proving Theorems 1.2-1.4. First, we prove Theorem1.1 in the following form.
Lemma 2.1.
Let λ > and ψ be a non-constant function on V . Suppose ( λ, ψ ) solves (1.2) and ψ is monotonic. Then, ψ is strictly monotonic, that is, either ψ ( i ) < ψ ( i + 1) for ≤ i < n or ψ ( i ) > ψ ( i + 1) for ≤ i < n .Proof. Obviously, (1.2) implies that π ( ψ ) = 0. Without loss of generality, it sufficesto consider the case when ψ (1) < ψ ( n ) >
0. Since ψ is non-constant and λ Gπ,ν >
0, we have ψ (1) < ψ (2) and ψ ( n − < ψ ( n ). Note that if there are1 < i < j < n such that ψ ( i − < ψ ( i ), ψ ( j ) < ψ ( j + 1) and ψ ( k ) = ψ ( i ) = ψ ( j )for i ≤ k ≤ j , then (1.2) yields λ Gπ,ν π ( i ) ψ ( i ) = [ ψ ( i ) − ψ ( i − ν ( i − , i ) + [ ψ ( i ) − ψ ( i + 1)] ν ( i, i + 1) > λ Gπ,ν π ( j ) ψ ( j ) = [ ψ ( j ) − ψ ( j − ν ( j − , j ) + [ ψ ( j ) − ψ ( j + 1)] ν ( j, j + 1) < , a contradiction. Thus, ψ is strictly increasing. (cid:3) We note the following corollary.
Corollary 2.2.
Let ( λ, ψ ) be a pair satisfying (1.2) . Then, λ = λ Gπ,ν if and only if ψ is monotonic.Proof. One direction is obvious from Theorem 1.1. For the other direction, assumethat ψ is monotonic and let φ be a minimizer for λ Gπ,ν with π ( φ ) = 0. Since ( λ, ψ )and ( λ Gπ,ν , φ ) are solutions to (1.2), one has λπ ( ψφ ) = E ν ( ψ, φ ) = λ Gπ,ν π ( φψ ) . By Lemma 2.1, ψ and φ are strictly monotonic and this implies E ν ( ψ, φ ) = 0. As aconsequence of the above equations, we have λ = λ Gπ,ν . (cid:3) The following proposition is the key to Theorem 1.2.
Proposition 2.3.
Suppose that ( λ, ψ ) satisfies λ > , ψ (1) < and, for ≤ i < n , (2.1) ψ ( i + 1) = ψ ( i ) + { [ ψ ( i ) − ψ ( i − ν ( i − , i ) − λπ ( i ) ψ ( i ) } + ν ( i, i + 1) , where t + = max { t, } . Then, the following are equivalent. (1) E ν ( ψ, ψ ) = λ Var π ( ψ ) . (2) π ( ψ ) = 0 . (3) λ = λ Gπ,ν .Furthermore, if n ≥ , then any of the above is equivalent to (4) E ν ( ψ, ψ ) = λ Gπ,ν
Var( ψ ) Remark . For n = 2, it is an easy exercise to show that λ Gπ,ν = ν (1 , / ( π (1) π (2)).By following the formula in (2.1), one has ψ (2) = ψ (1)[1 − λπ (1) /ν (1 , E ν ( ψ, ψ ) / Var π ( ψ ) = λ Gπ,ν . PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 5
Proof of Proposition 2.3.
Set B = { ≤ i ≤ n | ψ ( i ) = ψ ( n ) } and B c = { , , ..., i } .Since ψ (1) < λ > ψ (1) < ψ (2) and B c is nonempty. According to (2.1), ψ is non-decreasing. Note that if ψ ( i ) = ψ ( i +1), then ψ ( i ) ≥ ψ ( i +2) = ψ ( i +1).This implies ψ is strictly increasing on { , , ..., i + 1 } and, for 1 ≤ i ≤ i , λπ ( i ) ψ ( i ) = [ ψ ( i ) − ψ ( i + 1)] ν ( i, i + 1) + [ ψ ( i ) − ψ ( i − ν ( i − , i ) . Multiplying ψ ( i ) on both sides and summing over all i in B c yields λ i X i =1 ψ ( i ) π ( i ) = i − X i =1 [ ψ ( i ) − ψ ( i + 1)] ν ( i, i + 1)+ ψ ( i )[ ψ ( i ) − ψ ( i + 1)] ν ( i , i + 1)= E ν ( ψ, ψ ) + ψ ( i + 1)[ ψ ( i ) − ψ ( i + 1)] ν ( i , i + 1)= E ν ( ψ, ψ ) + λψ ( n ) i X i =1 ψ ( i ) π ( i ) . This is equivalent to(2.2) E ν ( ψ, ψ ) = λ Var π ( ψ ) + λπ ( ψ )[ π ( ψ ) − ψ ( n )] , which proves (1) ⇔ (2).If λ = λ Gπ,ν , then ψ is an eigenvector for M Gπ,ν associated to λ Gπ,ν . This proves(3) ⇒ (2). For (2) ⇒ (3), assume that π ( ψ ) = 0. In this case, ψ must be strictlyincreasing. Otherwise, ψ ( i ) = ψ ( n ) > i ∈ B and, according to (2.1), thisimplies λ Var π ( ψ ) > λ n − X i =1 π ( i ) ψ ( i ) ≥ n − X i =1 [ ψ ( i ) − ψ ( i + 1)] ν ( i, i + 1) = E ( ψ, ψ ) , which contradicts (1). As ψ is strictly increasing and π ( ψ ) = 0, ( λ, ψ ) solves (1.2).By Corollary 2.2, λ = λ Gπ,ν .To finish the proof, it remains to show (4) ⇒ (3) ((3) ⇒ (4) is obvious from theequivalence among (1), (2) and (3)). Assume that E ν ( ψ, ψ ) = λ Gπ,ν
Var π ( ψ ). ByLemma 2.1, ψ is strictly monotonic and this implies, for 1 ≤ i < n , λπ ( i ) ψ ( i ) = [ ψ ( i ) − ψ ( i + 1)] ν ( i, i + 1) + [ ψ ( i ) − ψ ( i − ν ( i − , i ) . As ψ is a minimizer for λ Gπ,ν , one has, for 1 ≤ i ≤ n , λ Gπ,ν π ( i )[ ψ ( i ) − π ( ψ )] = [ ψ ( i ) − ψ ( i + 1)] ν ( i, i + 1) + [ ψ ( i ) − ψ ( i − ν ( i − , i ) . If λ = λ Gπ,ν , the comparison of both systems yields ψ ( i ) = λ Gπ,ν π ( ψ ) λ Gπ,ν − λ , ∀ ≤ i < n. As n ≥ ψ (1) = ψ (2), a contradiction! This forces λ = λ Gπ,ν , as desired. (cid:3)
The following is a simple corollary of Proposition 2.3, which plays an importantrole in proving Theorem 1.4.
Corollary 2.4.
Let n ≥ . For λ > , let φ λ be the vector generated by (2.1) with φ (1) < . Then, ( λ − λ Gπ,ν ) π ( φ λ ) > for λ > and λ = λ Gπ,ν . G.-Y. CHEN AND L. SALOFF-COSTE
Proof.
Without loss of generality, we fix φ λ (1) = − λ >
0. Set T ( λ ) = π ( φ λ ). To prove this corollary, it suffices to show that T ( λ ) ( < λ < λ Gπ,ν > λ > λ Gπ,ν . For λ >
0, define L ( λ ) := E ν ( φ λ , φ λ ) / Var π ( φ λ ). By (2.2), one has(2.3) L ( λ ) − λ = λT ( λ )[ π ( φ λ ) − φ λ ( n )]Var π ( φ λ ) . Since φ λ is non-constant, π ( φ λ ) < φ λ ( n ). This implies T ( λ ) < λ ∈ (0 , λ Gπ,ν ).For λ > λ
Gπ,ν , set I = ( λ Gπ,ν , ∞ ). By Proposition 2.3, T ( λ ) = 0 if and only if λ = λ Gπ,ν . By the continuity of T , this implies either T ( I ) ⊂ ( −∞ ,
0) or T ( I ) ⊂ (0 , ∞ ).In the case T ( I ) ⊂ ( −∞ , L ( λ ) > λ for λ ∈ I . As L ( I ) is bounded, L k ( λ ) is convergent with limit e λ > λ Gπ,ν and this yields0 = lim k →∞ [ L k +1 ( λ ) − L k ( λ )] = e λT ( e λ )[ π ( φ e λ ) − φ e λ ( n )]Var π ( φ e λ ) > , a contradiction. Hence, T ( λ ) > λ > λ Gπ,ν . (cid:3) Proof of Theorem 1.2.
The proof for n = 2 is obvious from a direct computationand we deal with the case n ≥
3, here. By the equivalence of Proposition 2.3 (3)-(4), if λ = λ Gπ,ν , then λ k = λ Gπ,ν for all k ≥
1. If λ = λ Gπ,ν , then λ k > λ Gπ,ν for k ≥
1. Note that ( λ k , ψ k ) solves the system in (2.1). By (2.2), this implies λ k +1 − λ k = λ k π ( ψ k )[ π ( ψ k ) − ψ k ( n )]Var π ( ψ k ) , ∀ k ≥ . The strict monotonicity of λ k in (2) comes immediately from Corollary 2.4. In(3), the continuity of (2.1) in λ implies that ( λ ∗ , ψ ∗ ) is a solution to (2.1) and E ν ( ψ ∗ , ψ ∗ ) = λ ∗ Var( ψ ∗ ). By Proposition 2.3, λ ∗ = λ Gπ,ν and π ( ψ ∗ ) = 0, as desired. (cid:3) Proof of Theorem 1.3.
Recall the notation in the proof of Corollary 2.4: For λ > φ λ be the function defined by (2.1) and L ( λ ) = E ν ( φ λ , φ λ ) / Var π ( φ λ ). By (2.2)and Corollary 2.4, L ( λ ) ∈ ( λ Gπ,ν , λ ) for λ > λ
Gπ,ν . As L is bounded, Theorem 1.3follows from Lemma A.1. (cid:3) Proof of theorem 1.4.
Immediate from Corollary 2.4. (cid:3)
In the end of this section, we use the following proposition to find how the shapeof the function ψ in (2.1) evolves with λ . In Proposition 2.5, we set φ λ = ψ when ψ is given by (2.1). It is easy to see from (2.1) that φ λ is strictly increasing beforesome constant, say i = i ( λ ), and then stays constant equal to φ λ ( i ) after i . Theproposition shows how the constant i ( λ ) evolves. Proposition 2.5.
For λ > , let φ λ be the function generated by (2.1) with φ λ (1) = − and, for ≤ i ≤ n , set T i ( λ ) = P ij =1 φ λ ( i ) π ( i ) . For ≤ i < n , let a i ( λ ) = 1 + π ( i + 1) /π ( i ) − λπ ( i + 1) /ν ( i, i + 1) , PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 7 (2.4) A i ( λ ) = a ( λ ) 1 0 0 · · · π (3) π (2) a ( λ ) 1 0 ... π (4) π (3) a ( λ ) . . . . . . ... . . . . . . . . . ... . . . . . . a i − ( λ ) 10 · · · · · · π ( i +1) π ( i ) a i ( λ ) , and let λ ( i ) be the smallest root of det A i ( λ ) = 0 . Then, (1) λ Gπ,ν = λ ( n − < λ ( n − < · · · < λ (1) . (2) φ λ ( i ) < φ λ ( i + 1) = φ λ ( i + 2) for λ ∈ [ λ ( i ) , λ ( i − ) and ≤ i ≤ n − , where λ (0) := ∞ . (3) φ λ ( n − < φ λ ( n ) for λ ∈ (0 , λ ( n − ) .In particular, T i +1 ( λ ) = − π (1) det A i ( λ ) for λ ∈ (0 , λ ( i − ) and ( λ − λ ( i ) ) T i +1 ( λ ) > for λ ∈ (0 , λ ( i ) ) ∪ ( λ ( i ) , ∞ ) with ≤ i ≤ n − .Proof. By Lemma A.2, λ (1) > λ (2) > · · · > λ ( n − > ≤ i ≤ n − A i ( λ ) ( > ∀ λ ∈ ( −∞ , λ ( i ) ) < ∀ λ ∈ ( λ ( i ) , λ ( i − ) , where λ (0) = ∞ . Note that if T i ( λ ) < ≤ i ≤ n −
1, then φ λ ( j + 1) = φ λ ( j ) + [ φ λ ( j ) − φ λ ( j − ν ( j − , j ) − λπ ( j ) φ λ ( j ) ν ( j, j + 1) , ∀ ≤ j ≤ i. This implies(2.6) φ λ ( ℓ + 1) = φ λ ( ℓ ) − λν ( ℓ, ℓ + 1) ℓ X j =1 π ( j ) φ λ ( j ) , ∀ ≤ ℓ ≤ i. Multiplying π ( ℓ + 1) and adding up T ℓ ( λ ) yields T ℓ +1 ( λ ) = a ℓ ( λ ) T ℓ ( λ ) − π ( ℓ + 1) π ( ℓ ) T ℓ − ( λ ) , ∀ ≤ ℓ ≤ i. From the above discussion, we conclude that if T i ( λ ) <
0, then(2.7) T ℓ +1 ( λ ) = − π (1) det A ℓ ( λ ) , ∀ ≤ ℓ ≤ i. When ℓ = i −
1, (2.5) implies det A i − ( λ ) > λ < λ ( i − . By the continuityof T i and det A i − , if there is some λ < λ ( i − such that T i ( λ ) <
0, then T i ( λ ) = − π (1) det A i − ( λ ) for λ < λ ( i − . As a consequence of (2.7) with ℓ = i , thiswill imply T i +1 ( λ ) = − π (1) det A i ( λ ) for λ < λ ( i − . Hence, it remains to showthat T i ( λ ) < λ < λ ( i − . To see this, according to Corollary 2.4, onecan choose a constant e λ < min { λ Gπ,ν , λ ( i − } such that T n − ( e λ ) <
0. Since φ λ ( i )is non-decreasing in i , we obtain T i ( e λ ) <
0, as desired. This proves T i +1 ( λ ) = − π (1) det A i ( λ ) for λ < λ ( i − . In particular, T n ( λ ) = − π (1) det A n − ( λ ) for λ <λ ( n − . By Corollary 2.4, we have λ ( n − = λ Gπ,ν . This proves Proposition 2.5 (1).
G.-Y. CHEN AND L. SALOFF-COSTE
Next, observe that, for λ ∈ ( λ ( i ) , λ ( i − ), i +1 X j =1 π ( j ) φ λ ( j ) = T i +1 ( λ ) > , i X j =1 π ( j ) φ λ ( j ) = T i ( λ ) < . By (2.6), it is easy to see that [ φ λ ( i + 1) − φ λ ( i )] ν ( i, i + 1) = − λT i ( λ ) > φ λ ( i + 2) − φ λ ( i + 1)] ν ( i + 1 , i + 2)= { [ φ λ ( i + 1) − φ λ ( i )] ν ( i, i + 1) − λπ ( i + 1) φ λ ( i + 1) } + = {− λT i +1 ( λ ) } + = 0 . This proves Proposition 2.5 (2). To prove Proposition 2.5 (3), we use (1) to derive T n − ( λ ) = − π (1) det A n − ( λ ) < , ∀ λ ∈ (0 , λ ( n − ) . Using (2.6), this implies φ λ ( n − < φ λ ( n ). The last part of Proposition 2.5 followseasily from (2.5) and the fact that T i ( λ ) ≥ ⇒ T i +1 ( λ ) > T i ( λ ) ≤ ⇒ T i − ( λ ) < . (cid:3) Remark . In Proposition 2.5, if λ > λ (1) = ν (1 , π (1) − + π (2) − ], then φ λ ( i ) = φ λ (2) for i = 2 , ..., n . Note that, for λ ≥ λ (1) , φ λ (2) = − λπ (1) /ν (1 ,
2) and π ( φ λ ) = − λπ (1)(1 − π (1)) ν (1 , , Var π ( φ λ ) = λ π (1) (1 − π (1)) ν (1 , . By (2.3), this leads to L ( λ ) = ν (1 , / [ π (1)(1 − π (1)] for λ ≥ λ (1) . In the case n = 2,it is clear that ν (1 , / [ π (1)(1 − π (1)] = ν (1 , π (1) − + π (2) − ] = λ Gπ,ν .3.
Convergence to other eigenvalues
In this section, we generalize the algorithms (A1) and (A2) so that they can beapplied for the computation to any specified eigenvalue.3.1.
Basic setup and fundamental results.
Recall that G is a graph with vertexset V = { , , ..., n } and edge set E = {{ i, i + 1 }| i = 1 , , ..., n − } . Given twopositive measures π, ν on V, E with π ( V ) = 1, let M Gπ,ν be a n -by- n matrix definedin the introduction and given by(3.1) M Gπ,ν ( i, j ) = − ν ( i, j ) /π ( i ) if | i − j | = 1[ ν ( i − , i ) + ν ( i, i + 1)] /π ( i ) if j = i | i − j | > . Since ν is positive everywhere and M Gπ,ν is tridiagonal, all eigenvalues of M Gπ,ν havealgebraic multiplicity 1. Throughout this section, let { λ G < λ G < · · · < λ Gn − } denote the eigenvalues of M Gπ,ν with associated L ( π )-normalized eigenvectors ζ = , ζ , ..., ζ n − . Clearly, λ G = 0, λ G = λ Gπ,ν and, for 1 ≤ k ≤ n ,(3.2) λ Gi ζ i ( k ) π ( k ) = [ ζ i ( k ) − ζ i ( k − ν ( k − , k ) + [ ζ i ( k ) − ζ i ( k + 1)] ν ( k, k + 1) . Let 1 ≤ i ≤ n −
1. As ζ i is non-constant, it is clear that ζ i (1) = ζ i (2) and ζ i ( n − = ζ i ( n ). Moreover, if ζ i ( k ) = ζ i ( k + 1) for some 1 < k < n , then ζ i ( k ) = ζ i ( k −
1) and ζ i ( k + 1) = ζ i ( k + 2). Gantmacher and Krein [13] showed that PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 9 there are exactly i sign changes for ζ i with 1 ≤ i ≤ n . Miclo [14] gives a detaileddescription on the shape of ζ i as follows. Theorem 3.1.
For ≤ i ≤ n − , let ζ i be an eigenvector associated to the i thsmallest non-zero eigenvalue of the matrix in (3.1) with ζ i (1) < . Then, there are a < b ≤ a < b ≤ · · · ≤ a i < b i = n with a j +1 − b j ∈ { , } such that ζ i isstrictly increasing on [ a j , b j ] for odd j and is strictly decreasing on [ a j , b j ] for even j , and ζ i ( a j +1 ) = ζ i ( b j ) for ≤ j < i . In the following, we make some analysis related to the Euler-Lagrange equationsin (3.2).
Definition 3.1.
Fix n ≥ f be a function on { , , ..., n } . For 1 ≤ i ≤ n − f is called “Type i ” if there are 1 = a < b ≤ a < b ≤ · · · ≤ a i < b i ≤ n satisfying a j +1 − b j ∈ { , } such that(1) f is strictly monotonic on [ a j , b j ] for 1 ≤ j ≤ i .(2) [ f ( a j ) − f ( a j + 1)][ f ( a j +1 ) − f ( a j +1 + 1)] < ≤ j < i .(3) f ( a j +1 ) = f ( b j ), for 1 ≤ j < i , and f ( k ) = f ( b i ), for b i ≤ k ≤ n .The points a j , b j will be called “peak-valley points” in this paper. Remark . Note that the difference between Definition 3.1 and Theorem 3.1 is therequirement b i ≤ n , instead of b i = n . By Theorem 3.1, any eigenvector associatedto the i th smallest non-zero eigenvalue of the matrix in (3.1) must be of type i with b i = n . Definition 3.2.
Let π, ν be positive measures on
V, E with π ( V ) = 1. For λ ∈ R ,let ξ λ be a function on { , , ..., n } defined by ξ λ (1) = − ≤ k < n , ξ λ ( k + 1) = ξ λ ( k ) + [ ξ λ ( k ) − ξ λ ( k − ν ( k − , k ) − λπ ( k ) ξ λ ( k ) ν ( k, k + 1) . Remark . Note that ξ = − and, for λ < ξ λ is strictly decreasing and oftype 1. For λ >
0, if ξ λ ( k − < ξ λ ( k ) = ξ λ ( k + 1), then ξ λ ( k ) > ξ λ ( k + 2) < ξ λ ( k + 1). Similarly, if ξ λ ( k − > ξ λ ( k ) = ξ λ ( k + 1), then ξ λ ( k ) < ξ λ ( k + 2) > ξ λ ( k + 1). Thus, ξ λ must be of type i for some 1 ≤ i ≤ n − Lemma 3.2.
For λ > , let ξ λ be the function in Definition 3.2. Suppose that ξ λ is of type i with ≤ i ≤ n − . (1) If ξ λ ( n − = ξ λ ( n ) , then there is ǫ > such that ξ λ + δ is of type i for − ǫ < δ < ǫ . (2) If ξ λ ( n −
1) = ξ λ ( n ) , then there is ǫ > such that ξ λ + δ is of type i + 1 and ξ λ − δ is of type i for < δ < ǫ .Proof. Let a j , b j be the peak-valley points of ξ λ . By the continuity of ξ λ in λ andRemark 3.2, one can choose ǫ > δ ∈ ( − ǫ, ǫ ), ξ λ + δ remains strictlymonotonic on [ a j , b j ] for j = 1 , ..., i and[ ξ λ + δ ( b j − − ξ λ + δ ( b j )][ ξ λ + δ ( a j +1 + 1) − ξ λ + δ ( a j +1 )] > , for 1 ≤ j < i . In (1), b i = n . Fix δ ∈ ( − ǫ, ǫ ) and set a ′ = a = 1, b ′ i = b i = n . For1 < j < i , set b ′ j = a ′ j +1 = b j if [ ξ λ + δ ( b j − − ξ λ + δ ( b j )][ ξ λ + δ ( b j ) − ξ λ + δ ( a j +1 )] < b ′ j = a ′ j +1 = a j +1 if [ ξ λ + δ ( b j − − ξ λ + δ ( b j )][ ξ λ + δ ( b j ) − ξ λ + δ ( a j +1 )] > b ′ j = b j , a ′ j +1 = a j +1 if [ ξ λ + δ ( b j − − ξ λ + δ ( b j )][ ξ λ + δ ( b j ) − ξ λ + δ ( a j +1 )] = 0 . Clearly, ξ λ + δ is of type i with peak-valley points a ′ j , b ′ j . This proves Lemma 3.2 (1).For part (2), we consider i ≤ n − b i = n −
1. By similar argument asbefore, one can choose ǫ > ξ λ + δ to { , , ..., n − } isof type i for δ ∈ ( − ǫ, ǫ ). To finish the proof, it remains to compare ξ λ + δ ( n −
1) and ξ λ + δ ( n ). Recall that T j ( λ ) = P jk =1 ξ λ ( k ) π ( k ) as in the proof for Proposition 2.5.Using a similar reasoning as for (2.7), one shows that T i +1 ( λ ) = − π (1) det A i ( λ )for 1 ≤ i < n , where A i ( λ ) is the matrix in (2.4). This implies that the non-zero eigenvalues of M Gπ,ν , say λ G , ..., λ Gn − , are the roots of det A n − ( λ ) = 0. As aconsequence of Lemma A.2, det A n − ( λ ) = 0 has exactly n − α < α < · · · < α n − , and they satisfy the interlacing property λ Gj < α j < λ Gj +1 for1 ≤ j ≤ n −
2. Note that det A n − ( λ ) and det A n − ( λ ) tend to infinity as − λ tendsto infinity. This leads to the fact that if det A n − ( λ ) = 0 and det A n − ( λ ) < A n − ( · ) is strictly decreasing in a neighborhood of λ . If det A n − ( λ ) = 0and det A n − ( λ ) >
0, then det A n − ( · ) is strictly increasing in a neighborhood of λ . Back to the proof of (2). Suppose that ξ λ ( n − < ξ λ ( n − T n − ( λ ) = 0 and T n ( λ ) > A n − ( λ ) = 0and det A n − ( λ ) <
0. According to the conclusion in the previous paragraph, wecan find ǫ > A n − ( · ) is strictly decreasing on ( λ − ǫ, λ + ǫ ), whichyields ξ λ + δ ( n ) = ξ λ + δ ( n − − ( λ + δ ) T n − ( λ + δ ) ν ( n − , n ) ( < ξ λ + δ ( n −
1) if 0 < δ < ǫ> ξ λ + δ ( n −
1) if − ǫ < δ < . This gives the desired property in Lemma 3.2 (2). The other case, ξ λ ( n − >ξ λ ( n − (cid:3) The following proposition characterizes the shape of ξ λ for λ > Proposition 3.3.
For λ > , let ξ λ be the function in Definition 3.2 . Let λ G < · · · < λ Gn − be non-zero eigenvalues of M Gπ,ν in (3.1) and α < · · · < α n − be zerosof det A n − ( λ ) , where A n − ( · ) is the matrix in (2.4) . Then, (1) λ Gj < α j < λ Gj +1 , for ≤ j ≤ n − . (2) ξ λ is of type j for λ ∈ ( α j − , α j ] and ≤ j ≤ n − , where α := 0 and α n − := ∞ .Proof. (1) is immediate from Lemma A.2. For (2), note that α i is an eigenvalueof the submatix of M Gπ,ν obtained by removing the n th row and column. Thisimplies ξ α i ( n −
1) = ξ α i ( n ) for i = 1 , ..., n − ξ λ ( n − = ξ λ ( n ) for λ > λ / ∈ { α , ..., α n − } . By Lemma 3.2, ξ λ is of type i for α i − < λ ≤ α i . (cid:3) Given λ >
0, the above proposition provides a simple criterion to determine towhich of the intervals ( α j , α j +1 ] λ belongs to, that is, the type of ξ λ . However,knowing the type of ξ λ is not sufficient to determine whether λ is bigger or smallerthan λ Gi . We need the following remark. Remark . Using the same argument as the proof of Proposition 2.5, one can showthat π ( ξ λ ) = − π (1) det A n − ( λ ), where A n − ( λ ) is the matrix in (2.4). Clearly, π ( ξ λ ) has zeros λ G , ..., λ Gn − and tends to minus infinity as λ tends to minus infinity.This implies that π ( ξ λ ) <
0, for λ < λ G , and π ( ξ λ ) > ∀ λ ∈ ( λ G i − , λ G i ) , π ( ξ λ ) < ∀ λ ∈ ( λ G i , λ G i +1 ) , PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 11 for i ≥
1, where λ Gn := ∞ .As a consequence of Proposition 3.3 and Remark 3.3, we obtain the followingdichotomy algorithm, which is a generalization of (A2). Let 1 ≤ i ≤ n − i ) Choose positive reals L < λ Gi < U and set, for ℓ = 0 , , ... ,1 . ξ λ ℓ be the function generated by λ ℓ = ( L ℓ + U ℓ ) / , . According to Definition 3.1, set L ℓ +1 = L ℓ , U ℓ +1 = λ ℓ if ξ λ ℓ is of type j with j > i ,or if ξ λ ℓ is of type i and ( − i − π ( ξ λ ℓ ) > U ℓ +1 = U ℓ , L ℓ +1 = λ ℓ if ξ λ ℓ is of type j with j < i ,or if ξ λ ℓ is of type i and ( − i − π ( ξ λ ℓ ) < L ℓ +1 = U ℓ +1 = λ ℓ if ξ λ ℓ is of type i and π ( ξ λ ℓ ) > . Theorem 3.4.
Referring to (D i ) , ≤ max { U ℓ − λ Gi , λ Gi − L ℓ } ≤ ( U − L )2 − ℓ , ∀ ℓ ≥ . Proof.
Immediate from Proposition 3.3 and Remark 3.3. (cid:3)
Proposition 3.3 (2) bounds the eigenvalues using the shape of ξ λ generated fromone end point. We now introduce some other criteria to bound eigenvalues usingthe shape of ξ λ from either boundary point. Those results will be used to proveTheorem 6.1. Proposition 3.5.
For λ > , let ξ λ be the function in Definition 3.2 and e ξ λ be afunction given by e ξ λ ( k −
1) = e ξ λ ( k ) + [ e ξ λ ( k ) − e ξ λ ( k + 1)] ν ( k, k + 1) − λπ ( k ) e ξ λ ( k ) ν ( k − , k ) , for k = n, n − , ..., with e ξ λ ( n ) = − . Let λ G < · · · < λ Gn − be eigenvalues of M Gπ,ν in (3.1) and let f | B be the restriction of f to a subset B of V . Suppose ≤ k ≤ n . (1) If ξ λ | { ,...,k } is of type i with ( − i ξ λ ( k ) > and e ξ λ | { k ,...,n } is of type j with ( − j e ξ λ ( k ) > , then λ Gi + j − < λ < λ Gi + j − . (2) If ξ λ | { ,...,k } is of type i with ( − i ξ λ ( k ) < and e ξ λ | { k ,...,n } is of type j with ( − j e ξ λ ( k ) < , then λ Gi + j − < λ < λ Gi + j +1 . (3) If ξ λ | { ,...,k } is of type i with ( − i ξ λ ( k ) > and e ξ λ | { k ,...,n } is of type j with ( − j e ξ λ ( k ) < , then λ Gi + j − < λ < λ Gi + j .Proof. By Proposition 3.3, ξ λ ( n ) is a polynomial of degree n − − i +1 ξ λ Gi ( n ) > , ∀ ≤ i < n, ( − i +1 ξ β i ( n ) > , ∀ ≤ i < n − . This implies that there are w i ∈ ( β i , λ Gi +1 ), 0 ≤ i ≤ n −
2, such that ( − i +1 ξ λ ( n ) > λ ∈ ( w i − , w i ) and 0 ≤ i ≤ n − w − = −∞ and w n − = ∞ .The proofs for (1)-(3) in Proposition 3.5 are similar and we deal with (1) only.By the Euler-Lagrange equations in (3.2), it is easy to see that, for 1 ≤ l < n , ξ λ Gl and e ξ λ Gl are eigenvectors of M Gπ,ν in (3.1) associated with λ Gl , which implies ξ λ Gl = − ξ λ Gl ( n ) e ξ λ Gl . First, assume that λ ≤ λ Gi + j − . By Proposition 3.3, ξ λ Gi + j − | { ,...,k } is of type at least i and e ξ λ Gi + j − | { k ,...,n } is of type at least j . This implies thatthe patching of ξ λ Gi + j − | { ,...,k } and − ξ λ Gi + j − ( n ) e ξ λ Gi + j − | { k ,...,n } , which equals to ξ λ Gi + j − , is of type at least i + j −
1. This is a contradiction.Next, assume that λ ≥ λ Gi + j − . By Proposition 3.3, we may choose a < λ (resp. a < λ ) such that ξ λ | { ,...,k } (resp. e ξ λ | { k ,...,n } ) changes the type at a (resp. a ).If λ Gi + j − ≤ min { a , a } , then a similar reasoning as before implies that ξ λ Gi + j − isof type at most i + j −
2, a contradiction. If min { a , a } < λ Gi + j − < max { a , a } ,then exactly one of ξ λ Gi + j − | { ,...,k } and e ξ λ Gi + j − | { k ,...,n } does not change its type.This implies that the gluing point k can not be a local extremum and, thus, thepatching function is of type at most i + j −
2, another contradiction! According tothe discussion in the first paragraph of this proof, if λ Gi + j − ≥ max { a , a } , thennone of ξ λ Gi + j − | { ,...,k } and e ξ λ Gi + j − | { k ,...,n } changes type nor, of course, the sign at k . Consequently, we obtain ( − i + j ξ λ Gi + j − ( k ) e ξ λ Gi + j − ( k ) >
0, which contradictsthe fact ξ λ Gi + j − = − ξ λ Gi + j − ( n ) e ξ λ Gi + j − . (cid:3) Proposition 3.6.
For λ > and ≤ k ≤ n − , let s k ( λ ) be the k th sign change of ξ λ defined by s := 0 and s k +1 ( λ ) := inf { l > s k ( λ ) | ξ λ ( l ) ξ λ ( l − < or ξ λ ( l ) = 0 } ,where inf ∅ := n + 1 . Then, for < λ < λ , s k ( λ ) ≥ s k ( λ ) for all ≤ k ≤ n − .Proof. Let 1 ≤ k ≤ n −
1. If s k ( λ ) = n + 1, then it is clear that s k ( λ ) ≥ s k ( λ ).Suppose that s k ( λ ) = ℓ ≤ n . Obviously, ξ λ | { ,...,ℓ } is of type k . Referring to(2.4), let λ ℓ , ..., λ ℓℓ − be the roots of det A ℓ − ( λ ) = 0 and α ℓ , ..., α ℓℓ − be roots ofdet A ℓ − ( λ ) = 0. According to the first paragraph of the proof for Proposition3.5, there are w ℓi ∈ ( α ℓi − , λ ℓi ) with 1 ≤ i ≤ ℓ − − i +1 ξ λ ( ℓ ) > λ ∈ ( w ℓi , w ℓi +1 ) and 1 ≤ i ≤ ℓ −
1, where α ℓ := 0. Since ξ λ ( ℓ ) ξ λ ℓk ( ℓ ) ≥
0, one has w ℓk ≤ λ < α ℓk . As it is assumed that λ > λ , if λ > α ℓk , then ξ λ | { ,...,ℓ } is of typeat least k + 1 and, consequently, s k ( λ ) < ℓ = s k ( λ ). If λ < α ℓk , then ξ λ | { ,...,ℓ } is type k and ξ λ ( ℓ ) <
0. This implies s k ( λ ) ≤ ℓ = s k ( λ ), as desired. (cid:3) Bounding eigenvalues from below.
Motivated by Theorem 3.1, we intro-duce another scheme generalizing (2.1) to bound the other eigenvalues of M Gπ,ν frombelow.
Definition 3.3.
For λ >
0, let ξ λ be a function in Definition 3.2. If ξ λ is of type i ,1 ≤ i ≤ n −
1, with peak-valley points 1 = a < b ≤ a < b ≤ · · · ≤ a i < b i ≤ n ,then define ξ ( j ) λ ( k ) = ( ξ λ ( k ) for k ≤ b j ξ λ ( k ) = ξ λ ( b j ) for k > b j , ∀ ≤ j < i and set ξ ( j ) λ = ξ λ for i ≤ j ≤ n − PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 13
Remark . For λ >
0, if ξ λ is of type i , then ξ ( j ) λ is of type j for j < i . Moreover,for k < b j , ξ ( j ) λ ( k + 1) = ξ ( j ) λ ( k ) + [ ξ ( j ) λ ( k ) − ξ ( j ) λ ( k − ν ( k − , k ) − λπ ( k ) ξ ( j ) λ ( k ) ν ( k, k + 1)= ξ ( j ) λ ( k ) − λ [ π (1) ξ ( j ) λ (1) + · · · + π ( k ) ξ ( j ) λ ( k )] ν ( k, k + 1) , and, for b j ≤ k < n , ξ ( j ) λ ( k + 1) = ξ ( j ) λ ( k ) + F j ([ ξ ( j ) λ ( k ) − ξ ( j ) λ ( k − ν ( k − , k ) − λπ ( k ) ξ ( j ) λ ( k )) ν ( k, k + 1) , where F j ( t ) = max { t, } if j is odd, and F j ( t ) = min { t, } if j is even. Note that ξ (1) λ is exactly φ λ in Proposition 2.5.Thereafter, let L and L ( i ) be functions on (0 , ∞ ) defined by(3.3) L ( λ ) = E ν ( ξ λ , ξ λ )Var π ( ξ λ ) , L ( i ) ( λ ) = E ν ( ξ ( i ) λ , ξ ( i ) λ )Var π ( ξ ( i ) λ ) , ∀ ≤ i ≤ n − , where ξ λ and ξ ( i ) λ are functions in Definitions 3.2-3.3. Remark . Note that L = L ( n − . By a similar reasoning as in the proof for (2.2),one can show that, for λ > L ( λ ) = λ + λπ ( ξ λ )[ π ( ξ λ ) − ξ λ ( n )]Var π ( ξ λ ) , L ( i ) ( λ ) = λ + λπ ( ξ ( i ) λ )[ π ( ξ ( i ) λ ) − ξ ( i ) λ ( n )]Var π ( ξ ( i ) λ ) . From Proposition 3.3, it followss immediately that L ( λ ) = L ( i ) ( λ ) for λ ∈ (0 , α i ].To explore further L and L ( i ) , we need more information of π ( ξ λ ), π ( ξ ( i ) λ ), π ( ξ λ ) − ξ λ ( n ) and π ( ξ ( i ) λ ) − ξ ( i ) λ ( n ). Lemma 3.7.
Let ξ λ be the function in Definition 3.2 and λ Gi , α i be constants in Proposition 3.3 . Then, π ( ξ λ ) − ξ λ ( n ) = 0 has n − distinct roots, say β < β < · · · < β n − , which satisfy β = 0 and α i < β i < λ Gi +1 for ≤ i ≤ n − . Furthermore, π ( ξ λ ) − ξ λ ( n ) > for λ ∈ ( β i − , β i ) and π ( ξ λ ) − ξ λ ( n ) < for λ ∈ ( β i , β i +1 ) ,with β − = −∞ and β n − = ∞ .Proof. Set u ( λ ) := π ( ξ λ ) − ξ λ ( n ). According to Definition 3.2, u ( λ ) is a polynomialof degree n − u (0) = 0. Note that π ( ξ λ ) = 0 for λ ∈ { λ G , ..., λ Gn − } . If i is odd, then ξ λ Gi ( n − < ξ λ Gi ( n ). This implies ξ λ Gi ( n ) > u ( λ Gi ) < i is even, then u ( λ Gi ) > λ = α i with odd i , then ξ α i is of type i with ξ α i ( n −
1) = ξ α i ( n ). This implies ξ α i ( n ) > π ( ξ α i ) = π ( n ) ξ α i ( n ), whichyields u ( α i ) <
0. Similarly, one can show that u ( α i ) > i is even. (cid:3) Remark . We consider the sign of π ( ξ ( i ) λ ) and π ( ξ ( i ) λ ) − ξ ( i ) λ ( n ) in this remark.By Proposition 3.3, ξ ( i ) λ = ξ λ for λ ≤ α i . If λ > α i with 1 ≤ i ≤ n − ξ λ is of type j with j > i . Fix 1 ≤ i ≤ n − k = k ( λ ) =min { k | ξ ( i ) λ ( j ) = ξ ( i ) λ ( n ) , ∀ k ≤ j ≤ n } . Clearly, k ( λ ) ≤ n − λ > α i . Ob-serve that, for λ > α i with odd i , ξ λ ( k − < ξ λ ( k ) ≥ ξ λ ( k + 1), which implies P k − k =1 π ( k ) ξ λ ( k ) < P k k =1 π ( k ) ξ λ ( k ) ≥
0. A similar reasoning for the caseof even i gives P k − k =1 π ( k ) ξ λ ( k ) > P k k =1 π ( k ) ξ λ ( k ) ≤
0. Consequently, weobtain(3.4) ( − i − π ( ξ ( i ) λ ) > , ( − i [ π ( ξ ( i ) λ ) − ξ ( i ) λ ( n )] > , for λ > α i and 1 ≤ i ≤ n −
2. Note that, by Proposition 3.3, ξ ( i ) λ = ξ λ for λ ≤ α i . Inaddition with Remark 3.3, Lemma 3.7 and the continuity of ξ ( i ) λ , the first inequalityof (3.4) holds for λ > λ Gi and the second inequalities of (3.4) hold for λ > β i − .According to Lemma 3.7 and Remark 3.6, we derive a generalized version ofProposition 2.3 in the following. Proposition 3.8.
Let n ≥ and ≤ i ≤ n − . For λ > , let ξ λ , ξ ( i ) λ be thefunctions in Definition 3.2 and β i be the constants in Lemma 3.7 . (1) For λ > β i − , the following are equivalent. (1-1) E ν ( ξ ( i ) λ , ξ ( i ) λ ) = λ Var π ( ξ ( i ) λ ) . (1-2) π ( ξ ( i ) λ ) = 0 . (1-3) λ = λ Gi . (2) For β i − < λ < β i , the following are equivalent. (2-1) E ν ( ξ λ , ξ λ ) = λ Var π ( ξ λ ) . (2-2) π ( ξ λ ) = 0 . (2-3) λ = λ Gi .Proof. The proof for Proposition 3.8 (2) is similar to the proof for Proposition 3.8(1) and we deal only with the latter. By Lemma 3.7 and Remark 3.6, one has π ( ξ ( i ) λ )[ π ( ξ ( i ) λ ) − ξ ( i ) λ ( n )] ( < λ > λ Gi > β i − < λ < λ Gi . This proves the equivalence of (1-1) and (1-2). Under the assumption of (1-2) andusing Remark 3.3, one has λ ≤ α i . This implies ξ ( i ) λ = ξ λ is an eigenvector for M Gπ,ν with associated eigenvalue λ . As λ ∈ ( β i − , α i ], it must be the case λ = λ Gi . Thisgives (1-3), while (1-3) ⇒ (1-2) is obvious and omitted. (cid:3) Remark . It is worthwhile to note that if (1-1) and (2-1) of Proposition 3.8 areremoved, then the equivalence in (1) holds for λ > λ Gi − and the equivalence in (2)holds for λ ∈ ( λ Gi − , λ Gi +1 ). Once λ Gi − is known, we can determine λ Gi using thesign of π ( ξ ( i ) λ ). See Theorem 3.9 for details. Remark . Note that condition (4) of Proposition 2.3 is not included in Propo-sition 3.8. In fact, the equivalence may fail, that is, there may exist some λ ∈ ( β i − , β i ) \ { λ Gi } such that E ν ( ξ λ , ξ λ ) / Var π ( ξ λ ) = λ Gi . See Example 3.2 for a coun-terexample.As Proposition 3.8 focuses on the characterization of zeros of L ( λ ) − λ , thefollowing theorem concerns the sign of L ( λ ) − λ . Theorem 3.9.
Let λ Gi , α i , β i be the constants in Proposition 3.3 and
Lemma 3.7 ,and L be the function in (3.3) . Then, λ G , ..., λ Gn − , β , ..., β n − are fixed points of L and, for ≤ i ≤ n − , (1) L ( λ ) < λ for λ ∈ ( λ Gi , β i ) . PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 15 (2) L ( λ ) > λ for λ ∈ ( β i , λ Gi +1 ) . (3) L ( i ) ( λ ) < λ for λ ∈ ( λ Gi , ∞ ) .Proof. Immediate from Lemma 3.7 and Remarks 3.5-3.6. (cid:3)
By Theorem 3.9, we obtain a lower bound on any specified eigenvalue of M Gπ,ν . Corollary 3.10.
Let ≤ i ≤ n − and λ > λ Gi . Consider the sequence λ ℓ +1 = L ( i ) ( λ ℓ ) with ℓ ≥ and set λ ∗ = ( lim ℓ →∞ λ ℓ if λ ℓ converges sup ℓ ∈ I λ ℓ if λ ℓ diverges , where I = { ℓ | λ ℓ − > λ ℓ < λ ℓ +1 } . Then, λ ∗ ≤ λ Gi . It is not clear yet whether the sequence λ ℓ in Corollary 3.10 is convergent, evenlocally. This subject will be discussed in the next subsection. Now, we establishsome relations between the roots of det A i ( λ ) = 0 and the shape of ξ ( i ) λ . This is ageneralization of Proposition 2.5. Proposition 3.11.
For ≤ i ≤ n − , let A i ( λ ) be the matrix in (2.4) , θ ( i )1 < · · · < θ ( i ) i be zeros of det A i ( λ ) = 0 and set θ ( i − i := ∞ . Referring to the notationin Proposition 3.3 , it holds true that, for ≤ i ≤ n − , (1) λ Gi = θ ( n − i < α i = θ ( n − i < · · · < θ ( i ) i . (2) ξ ( i ) λ ( j ) = ξ ( i ) λ ( j + 1) = · · · = ξ ( i ) λ ( n ) for λ ∈ [ θ ( j ) i , θ ( j − i ) and i ≤ j ≤ n − . (3) ξ ( i ) λ ( n − = ξ ( i ) λ ( n ) for λ ∈ ( θ ( n − i − , θ ( n − i ) and i ≤ n − .Proof. The order in (1) is a simple application of Lemma A.3. For (2), fix 1 ≤ i ≤ n − γ ( λ ) = min { j | ξ ( i ) λ ( k ) = ξ ( i ) λ ( n ) , ∀ j ≤ k ≤ n } and B ( λ ) = { , , ..., γ ( λ ) } , B + ( λ ) = B ( λ ) ∪ { γ ( λ ) + 1 } . Clearly, i + 1 ≤ γ ( λ ) ≤ n . We use thenotation ξ λ | C to denote the restriction of ξ λ to a set C . Suppose that i is odd. ByRemark 3.4, ξ ( i ) λ = ξ λ on B ( λ ) and ξ λ | B ( λ ) is of type i with ξ λ ( γ ( λ ) − < ξ λ ( γ ( λ )) ≥ ξ λ ( γ ( λ ) + 1) . By Lemma 3.2(1), if ξ λ ( γ ( λ ) + 1) < ξ λ ( γ ( λ )), then there is ǫ > | δ | < ǫ , ξ λ + δ | B ( λ ) is of type i and ξ λ + δ ( γ ( λ ) − < ξ λ + δ ( γ ( λ )) > ξ λ + δ ( γ ( λ ) + 1) . This implies γ ( λ + δ ) = γ ( λ ) for δ ∈ ( − ǫ, ǫ ). By Lemma 3.2(2), if ξ λ ( γ ( λ ) + 1) = ξ λ ( γ ( λ )), then there is ǫ > δ ∈ ( − ǫ, ξ λ + δ | B + ( λ ) is of type i with ξ λ + δ ( γ ( λ ) − < ξ λ + δ ( γ ( λ )) < ξ λ + δ ( γ ( λ ) + 1) , and, for δ ∈ (0 , ǫ ), ξ λ + δ | B + ( λ ) is of type i + 1 with ξ λ + δ ( γ ( λ ) − < ξ λ + δ ( γ ( λ )) > ξ λ + δ ( γ ( λ ) + 1) . This yields γ ( λ + δ ) = γ ( λ ) for δ ∈ (0 , ǫ ) and γ ( λ + δ ) = γ ( λ ) + 1 for δ ∈ ( − ǫ, i is similar and we conclude from the above that γ ( λ ) is a non-increasing and right-continuous function taking values on { i +1 , ..., n } .Let c i +1 > · · · > c n − be the discontinuous points of γ ( λ ) such that γ ( c j ) = j for i + 1 ≤ j ≤ n −
1. As a consequence of the above discussion, ξ c j | { ,...,j } is of type i with ξ c j ( j ) = ξ c j ( j + 1) and this implies P jk =1 π ( k ) ξ c j ( k ) = 0. That means c j is a root of det A j − ( λ ) = 0 for j = i + 1 , ..., n −
1. By Proposition 3.3 and the second equality in (1), γ ( λ ) = n for θ ( n − i − < λ < θ ( n − i and, thus, c j ≥ θ ( n − i for j ≥ i + 1. As a consequence of the interlacing relationship θ ( ℓ ) i < θ ( ℓ − i < θ ( ℓ ) i +1 , itmust be c j = θ ( j +1) i for i + 1 ≤ j ≤ n −
1. This finishes the proof. (cid:3)
Remark . For 1 ≤ i ≤ n − θ ( i )1 , ..., θ ( i ) i are also non-zero eigenvalues of the( i + 1) × ( i + 1) principal submatrix of (3.1) indexed by 1 , ..., i + 1. Remark . In fact, by Proposition 2.5, ξ (1) λ ( n − = ξ (1) λ ( n ) for λ ∈ (0 , θ ( n − ),which is better than Proposition 3.11(3).3.3. Local convergence of L . This subsection is dedicated to the local conver-gence of L in (3.3). Let α i , β i , λ Gi be the constants in Proposition 3.3 and Lemma3.7. As before, let ζ = , ..., ζ n − denote the L ( π )-normalized eigenvectors of M Gπ,ν associated with λ G , ..., λ Gn − . Clearly, ξ λ Gi = − ζ i /ζ i (1) and ξ λ = P n − i =0 ρ i ( λ ) ζ i ,where ρ i ( λ ) = π ( ξ λ ζ i ) for 0 ≤ i ≤ n −
1. Note that ρ i ( λ ) is a polynomial of degree n − ρ i ( λ j ) = − δ i ( j ) /ζ i (1) for i, j ∈ { , , ..., n − } . This implies(3.5) ρ ( λ ) = − n − Y j =1 λ Gj − λλ Gj , ρ i ( λ ) = − λζ i (1) λ Gi n − Y j =1 ,j = i λ Gj − λλ Gj − λ Gi , for all 1 ≤ i ≤ n −
1. Moreover, by multiplying (3.2) with ξ λ ( k ) and summingup k , we obtain E ν ( ξ λ , ζ i ) = λ Gi ρ i ( λ ). In the same spirit, one can show that E ν ( ξ λ , ζ i ) = λ [ ρ i ( λ ) − ζ i ( n ) ρ ( λ )] using Definition 3.2. Putting both equationstogether yields(3.6) ρ i ( λ ) = λζ i ( n ) λ − λ Gi ρ ( λ ) , ∀ ≤ i ≤ n − . As a consequence of Remark 3.5, this gives(3.7) L ( λ ) = P n − i =1 λ Gi ρ i ( λ ) P n − i =1 ρ i ( λ ) = λ + P n − i =1 ( λ Gi − λ ) − ζ i ( n ) P n − i =1 ( λ Gi − λ ) − ζ i ( n ) , for λ / ∈ { λ G , ..., λ Gn − } . The next proposition follows immediately from the secondequation in (3.5) and (3.6). Proposition 3.12.
Let λ G , ..., λ Gn − be the non-zero eigenvalues of M Gπ,ν in (3.1) and ζ , ..., ζ n − be the corresponding L ( π ) -normalized eigenvectors. Then, ζ i (1) ζ i ( n ) = − n − Y j =1 ,j = i λ Gj λ Gj − λ Gi , ∀ ≤ i ≤ n − . Set u ( λ ) = P n − j =1 ( λ Gj − λ ) − ζ j ( n ). By Theorem 3.9, β , ..., β n − are zeros of u ( λ ) Q n − j =1 ( λ Gj − λ ), which is a polynomial of degree n −
2. This implies u ( λ ) = C n − Y j =1 λ Gj − λ n − Y j =1 ( β j − λ ) , where C = λ ··· λ n − β ··· β n − P n − j =1 ζ j ( n ) /λ Gj . Putting this back to L yields(3.8) 1 L ( λ ) − λ = u ′ ( λ ) u ( λ ) = n − X j =1 λ Gj − λ − n − X j =1 β j − λ , PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 17 for λ / ∈ { λ G , ..., λ Gn − , β , ..., β n − } . Proposition 3.13.
Let L be the function in (3.3) , λ Gi be the eigenvalue of M Gπ,ν and β i be the constant in Lemma 3.7 . Let D i = P n − j =1 ( β j − λ Gi ) − − P n − j =1 ,j = i ( λ Gj − λ Gi ) − with ≤ i ≤ n − . Then, for ≤ i ≤ n − , (1) If D i < , then there is τ ∈ ( λ Gi , β i ) such that L is strictly increasing on ( β i − , λ Gi ) ∪ ( τ, β i ) and strictly decreasing on ( λ Gi , τ ) . (2) If D i > , then there is η ∈ ( β i − , λ Gi ) such that L is strictly increasing on ( β i − , η ) ∪ ( λ Gi , β i ) and strictly increasing on ( η, λ Gi ) . (3) If D i = 0 , then L is strictly increasing on ( β i − , β i ) .Proof. Using (3.7) and (3.8), one can show that L ′ ( λ Gi ) = 0 and(3.9) L ′′ ( λ Gi ) = n − X j =1 ,j = i ζ i ( n ) λ Gj − λ Gi = 2 n − X j =1 β j − λ Gi − n − X j =1 ,j = i λ Gj − λ Gi = 2 D i . To prove (1) and (2), it suffices to show that if L ′ ( τ ) = 0 for some τ ∈ ( λ Gi , β i ),then τ is a local minimum of L , and if L ′ ( η ) = 0 for some η ∈ ( β i − , λ Gi ), then η isa local maximum of L . We discuss the first case, whereas the second case is similarand is omitted. Recall that u ( λ ) = P n − j =1 ( λ Gj − λ ) − ζ j ( n ). As τ is a critical pointfor L , one has 2( u ′ ( τ )) = u ( τ ) u ′′ ( τ ). This implies L ′′ ( τ ) = u ( τ )[3( u ′′ ( τ )) − u ′ ( τ ) u ′′′ ( τ )]2( u ′ ( τ )) > , where the last inequality uses the fact that u ( λ ) <
0, for λ ∈ ( λ Gi , β i ), and3( u ′′ ( λ )) − u ′ ( λ ) u ′′′ ( λ ) = − X ≤ i
0. Using the same proof as above,this implies that L ( λ ) is strictly increasing on ( λ G , β ) ∪ ( β n − , λ Gn − ). Moreover,by (3.7), one may compute( u ′ ( λ )) L ′ ( λ ) = − X i Theorem 3.14 (Local convergence) . Let λ > and set λ ℓ +1 = L ( λ ℓ ) for ℓ ≥ .Then, there is ǫ > such that the sequence ( λ ℓ ) ∞ ℓ =1 is monotonic and converges to λ Gi for λ ∈ ( λ Gi − ǫ, λ Gi + ǫ ) and ≤ i ≤ n − . We use the following examples to illustrate the different cases in Proposition3.13. Example . Let n > 1. A simple random walk on { , , ..., n } with reflecting probability 1 / K ( i, j ) = K (1 , 1) = K ( n, n ) = 1 / | i − j | = 1. It is easy to see that the uniform probability is the stationary distri-bution of K . In the setting of graph, we have ν ( i, i + 1) = 1 / (2 n ) and π ( i ) = 1 /n .One may apply the method in [11] to obtain the following spectral information. λ Gj = 1 − cos jπn , ζ j ( k ) = 1 q λ Gj (cid:18) sin jkπn − sin j ( k − πn (cid:19) , ∀ ≤ j < n. See, e.g., [3, Section 7]. By (3.9), we get D i = 12 n − X j =1 ,j = i sin ( jπ/n ) λ Gj ( λ Gj − λ Gi ) = n − X j =1 ,j = i jπ/n )cos( iπ/n ) − cos( jπ/n ) . Clearly, D > D n − < 0. If n is even, then D n/ < Example . An Ehrenfest chain on V = { , , ..., n } is a Markovchain with transition matrix K given by K ( i, i + 1) = 1 − i/n and K ( i + 1 , i ) =( i + 1) /n for i = 0 , ..., n − 1. The associated stationary distribution is the unbiasedbinomial distribution on V , that is, π ( i ) = (cid:0) ni (cid:1) − n for i ∈ V . To the Ehrenfestchain, the measure ν is defined by ν ( i, i + 1) = (cid:0) n − i (cid:1) − n for i = 0 , ..., n − 1. Usingthe group representation for the binary group { , } n , one may compute λ j = 2 jn , ζ j ( k ) = (cid:18) nj (cid:19) − / j X ℓ =0 ( − ℓ (cid:18) kℓ (cid:19)(cid:18) n − kj − ℓ (cid:19) , ∀ ≤ j ≤ n. Plugging this back into (3.9) yields D i = n n X j =1 ,j = i (cid:0) nj (cid:1) j − i > i < n/ 2= 0 for i = n/ < i > n/ . . This example points out the possibility of different signs in { D i | i = 1 , ..., n − } including 0. PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 19 A remark on the separation for birth and death chains. In this subsec-tion, we give a new proof of a result, Theorem 3.15, which deals with convergence inseparation distance for birth and death chains. Let ( X m ) ∞ m =0 be a birth and deathchain with transition matrix K given by (1.1). In the continuous time setting, weconsider the process Y t = X N t , where N t is a Poisson process with parameter 1independent of X m . Given the initial distribution µ , which is the distribution of X , the distributions of X m and Y t are respectively µK m and µe − t ( I − K ) , where e A := P ∞ l =0 A l /l !. Briefly, we write H t = e − t ( I − K ) . It is well-known that if K isirreducible, then µH t converges to π as t → ∞ . If K is irreducible and r i > i , then µK m converges to π as m → ∞ . Concerning the convergence, weconsider the separations of X m , Y t with respect to π , which are defined by d sep ( µ, m ) = max ≤ x ≤ n (cid:26) − µK m ( x ) π ( x ) (cid:27) , d c sep ( µ, t ) = max ≤ x ≤ n (cid:26) − µH t ( x ) π ( x ) (cid:27) . The following theorem is from [9]. Theorem 3.15. Let K be an irreducible birth and death chain on { , , ..., n } witheigenvalues λ = 0 < λ < · · · < λ n . (1) For the discrete time chain, if p i + q i +1 ≤ for all ≤ i < n , then d sep (0 , m ) = d sep ( n, m ) = n X j =1 n Y i =1 ,i = j λ i λ i − λ j (1 − λ j ) m . (2) For the continuous time chain, it holds true that d c sep (0 , t ) = d c sep ( n, t ) = n X j =1 n Y i =1 ,i = j λ i λ i − λ j e − λ j t . Diaconis and Fill [6, 12] introduce the concept of dual chain to express theseparations in Theorem 3.15 as the probability of the first passage time. Brownand Shao [1] characterize the first passage time using the eigenvalues of K for aspecial class of continuous time Markov chains including birth and death chains.The idea in [1] is also applicable for discrete time chains and this leads to theformula above. See [9] for further discussions. Here, we use Proposition 3.12 andLemma 3.16 to prove this result directly. Lemma 3.16. Let K be the transition matrix in (1.1) with stationary distribution π . Suppose that µ is a probability distribution satisfying µ ( i ) /π ( i ) ≤ µ ( i +1) /π ( i +1) for all ≤ i ≤ n − . (1) For the discrete time chain, if p i + q i +1 ≤ for all ≤ i < n , then µK m ( i ) /π ( i ) ≤ µK m ( i + 1) /π ( i + 1) for all ≤ i < n and m ≥ . (2) For the continuous time chain, µH t ( i ) /π ( i ) ≤ µH t ( i + 1) /π ( i + 1) for all ≤ i < n and t ≥ .Proof. Note that (2) follows from (1) if we write H t = exp {− t ( I − I + K ) } . For theproof of (1), observe that µK m +1 ( i ) π ( i ) = µK m ( i − π ( i − q i + µK m ( i ) π ( i ) r i + µK m ( i + 1) π ( i + 1) p i , ∀ i. By induction, if µK m ( i ) /π ( i ) ≤ µK m ( i + 1) /π ( i + 1) for 0 ≤ i < n , then µK m +1 ( i + 1) π ( i + 1) = µK m ( i ) π ( i ) q i +1 + µK m ( i + 1) π ( i + 1) r i +1 + µK m ( i + 2) π ( i + 2) p i +1 ≥ µK m ( i ) π ( i ) q i +1 + µK m ( i + 1) π ( i + 1) (1 − q i +1 ) ≥ µK m ( i ) π ( i ) (1 − p i ) + µK m ( i + 1) π ( i + 1) p i ≥ µK m +1 ( i ) π ( i ) . (cid:3) Remark . Lemma 3.16 is also developed in [10] in which it is shown that, forany non-negative function f , K m f is non-decreasing if f is non-decreasing for all m ≥ 0. Consider the adjoint chain K ∗ of K in L ( π ). As birth and death chainsare reversible, one has K ∗ = K . Using the identity µK/π = K ∗ ( µ/π ), it is easy tosee that the above proof is consistent with the proof in [10]. Proof of Theorem 3.15. Assume that K is irreducible and let λ = 0 < λ < · · · <λ n be the eigenvalues of I − K with L ( π )-normalized eigenvector ζ = , ..., ζ n .By Lemma 3.16, if µ satisfies µ ( i ) /π ( i ) ≥ µ ( i + 1) /π ( i + 1) for 0 ≤ i < n , then d c sep ( µ, t ) = 1 − µH t ( n ) π ( n ) = n X j =1 µ ( ζ j ) ζ j ( n ) e − λ j t , where µ ( ζ j ) = P ni =0 ζ j ( i ) µ ( i ). If K satisfies p i + q i +1 ≤ ≤ i < n , then d sep ( µ, m ) = 1 − µK m ( n ) π ( n ) = n X j =1 µ ( ζ j ) ζ j ( n )(1 − λ j ) m . By Proposition 3.12, setting µ to be one of the dirac measure δ , δ n leads to thedesired identities. (cid:3) Paths of infinite length In this section, the graph G = ( V, E ) under consideration is infinite with V = { , , ... } and E = {{ i, i + 1 }| i = 1 , , ... } . As before, let π, ν be positive measureson V, E satisfying π ( V ) = 1. The Dirichlet form and the variance are defined in asimilar way as in the introduction and the spectral gap of G with respect to π, ν isgiven by λ Gπ,ν = inf (cid:26) E ν ( f, f )Var π ( f ) (cid:12)(cid:12)(cid:12)(cid:12) f is non-constant and π ( f ) < ∞ (cid:27) . For n ≥ 2, let G n = ( V n , E n ) be the subgraph of G with V n = { , , ..., n } , E n = {{ i, i + 1 }| ≤ i < n } and let π n , ν n be normalized restrictions of π, ν to V n , E n .That is, π n ( i ) = c n π ( i ), ν n ( i, i + 1) = c n ν ( i, i + 1) with c n = 1 / [ π (1) + · · · + π ( n )].As before, let M Gπ,ν be an infinite matrix indexed by V and defined by(4.1) M Gπ,ν ( i, j ) = − ν ( i, j ) π ( i ) , ∀| i − j | = 1 , M Gπ,ν ( i, i ) = ν ( i − , i ) + ν ( i, i + 1) π ( i ) . Clearly, M G n π n ,ν n is the principal submatrix of M Gπ,ν indexed by V n × V n . Lemma 4.1. Referring to the above setting, λ G n +1 π n +1 ,ν n +1 < λ G n π n ,ν n for n > and λ Gπ,ν = lim n →∞ λ G n π n ,ν n . PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 21 Proof. Briefly, we write λ for λ Gπ,ν and λ n for λ G n π n ,ν n . Note that λ n is the smallestnon-zero eigenvalue of the principal submatrix of M Gπ,ν indexed by V n × V n . Asa consequence of Proposition 3.11(1) and Remark 3.9, λ n +1 < λ n . For n > φ n be a minimizer for λ n and define ψ n ( i ) = V n ( i ) φ n ( i ) for i ≥ 1. Clearly,one has E ν n ( φ n , φ n ) = c n E ν ( ψ n , ψ n ) and Var π n ( φ n ) = c n Var π ( ψ n ). This implies λ ≤ λ n for n ≥ 2. Let λ ∗ = lim n →∞ λ n . Note that it remains to show λ ∗ = λ .For ǫ > 0, choose a function f on V such that E ν ( f, f ) < ( λ + ǫ/ π ( f ) with π ( f ) < ∞ . For δ > 0, we choose N > π N ( g ) > (1 − δ )Var π ( f )and E ν N ( g, g ) < (1 + δ ) E ν ( f, f ), where g = f | V N , the restriction of f to V N . Thisimplies λ ∗ ≤ λ N ≤ E ν N ( g, g )Var π N ( g ) ≤ (1 + δ ) E ν ( f, f )(1 − δ )Var π ( f ) . Letting δ → ǫ → λ ∗ ≤ λ , as desired. (cid:3) Remark . Silver [17] contains a discussion of the (weak*) convergence of thespectral measure for G n to the spectral measure for G in a very general setting.Lemma 4.1 can also be proved using Theorem 4.3.4 in [17]. Proposition 4.2. For λ > , let φ λ (1) = − and φ λ ( i + 1) = φ λ ( i ) + { [ φ λ ( i ) − φ λ ( i − ν ( i − , i ) − λπ ( i ) φ λ ( i ) } + ν ( i, i + 1) , ∀ i ≥ . Set λ = ∞ and λ n = λ G n π n ,ν n for n ≥ . (1) For i ≥ and λ ∈ [ λ i , λ i − ) , φ λ ( i − < φ λ ( i ) = φ λ ( i + 1) . (2) For λ ∈ (0 , λ Gπ,ν ] , φ λ ( i ) < φ λ ( i + 1) for all i ≥ .Proof. Immediate from Proposition 3.11 and Remarks 3.9-3.10. (cid:3) Remark . By Proposition 4.2, one may generate a dichotomy algorithm for λ Gπ,ν using the shape of φ λ . See (D i ).The following theorem extends Theorem 1.1 to infinite paths. Theorem 4.3. If λ Gπ,ν > and E ν ( ψ, ψ ) / Var π ( ψ ) = λ Gπ,ν for some function ψ on V with π ( ψ ) = 0 , then ψ is strictly monotonic and satisfies λ Gπ,ν π ( i ) ψ ( i ) = [ ψ ( i ) − ψ ( i + 1)] ν ( i, i + 1) + [ ψ ( i ) − ψ ( i − ν ( i − , i ) , ∀ i ≥ . Theorem 4.4. For λ > , let φ λ be the function in Proposition 4.2 and set L ( λ ) = E π ( φ λ , φ λ ) / Var π ( φ λ ) . Then, (1) λ Gπ,ν < L ( λ ) < λ for λ ∈ ( λ Gπ,ν , ∞ ) . (2) L n ( λ ) → λ Gπ,ν as n → ∞ for λ ∈ ( λ Gπ,ν , ∞ ) .Proof. Let λ > λ Gπ,ν . By Lemma 4.1, λ i ≤ λ < λ i − for some i ≥ 2. By Proposition4.2 (1), one has φ λ ( i − < φ λ ( i ) = φ λ ( i + 1). As in (2.2), we obtain L ( λ ) = λ + λ π ( φ λ )[ π ( φ λ ) − φ λ ( i )]Var π ( φ λ ) , i X j =1 φ λ ( j ) π ( j ) ≥ . This leads to π ( φ λ ) > π ( φ λ ) < φ λ ( i ), which implies L ( λ ) < λ . That means L has no fixed point on ( λ Gπ,ν , ∞ ). The lower bound of (1) follows immediatelyfrom Theorem 4.3. For (2), set λ ∗ = lim n →∞ L n ( λ ) ≥ λ Gπ,ν . As a consequence of(1), L is continuous on ( λ Gπ,ν , ∞ ). If λ ∗ > λ Gπ,ν , then λ ∗ is a fixed point of L , acontradiction! Hence, λ ∗ = λ Gπ,ν . (cid:3) A numerical experiment In this section, we illustrate the algorithm (A2) on a specific Metropolis chain.The Metropolis algorithm introduced by Metropolis et al. in 1953 is a widely usedconstruction that produces a Markov chain with a given stationary distribution π .Let π be a positive probability measure on V and K be an irreducible Markovtransition matrix on V . For simplicity, we assume that K ( x, y ) = K ( y, x ) for all x, y ∈ V . The Metropolis chain evolves in the following way. Given the initialstate x , select a state, say y , according to K ( x, · ) and compute the ratio A ( x, y ) = π ( y ) /π ( x ). If A ( x, y ) ≥ 1, then move to y . If A ( x, y ) < 1, then flip a coin withprobability A ( x, y ) on heads and move to y if the head appears. If the coin landson tails, stay at x . Accordingly, if M is the transition matrix of the Metropolischain, then M ( x, y ) = K ( x, y ) if A ( x, y ) ≥ , x = yK ( x, y ) A ( x, y ) if A ( x, y ) < K ( x, x ) + P z : A ( x,z ) < K ( x, z )(1 − A ( x, z )) if x = y . It is easy to check π ( x ) M ( x, y ) = π ( y ) M ( y, x ). As K is irreducible, M is irreducible.Moreover, if π is not uniform, then M ( x, x ) > x ∈ V . This implies that M is aperiodic and, consequently, M t ( x, y ) → π ( y ) and e − t ( I − M ) ( x, y ) → π ( y ) as t → ∞ . For further information on Metropolis chains, see [5] and the referencestherein.For n ≥ 1, let G n = ( V n , E n ) be a graph with V n = { , ± , ..., ± n } and E n = {{ i, i +1 } : i = − n, ..., n − } . Suppose that K n is the transition matrix of the simplerandom walk on V n , that is, K n ( − n, − n ) = K n ( n, n ) = 1 / K n ( i, i + 1) = K n ( i + 1 , i ) = 1 / − n ≤ i < n . For a > 0, let ˇ π n,a , ˆ π n,a be probabilities on V n = { , ± , ..., ± n } given byˇ π n,a ( i ) = ˇ c n,a ( | i | + 1) a , ˆ π n,a ( i ) = ˆ c n,a ( n − | i | + 1) a , where ˇ c n,a and ˆ c n,a are normalizing constants. It is easy to compute that(5.1) c n,a / ≤ / ˆ c n,a < / ˇ c n,a ≤ c n,a , where c n,a = ( n + 1) a +1 a + 1 + ( n + 1) a . The Metropolis chains, ˇ K n,a and ˆ K n,a , for ˇ π n,a and ˆ π n,a based on the simple randomwalk K n have transition matrices given byˇ K n,a ( i, j ) = ˇ K n,a ( − i, − j ) , ˆ K n,a ( i, j ) = ˆ K n,a ( − i, − j )and ˇ K n,a ( i, j ) = if j = i + 1 , i ∈ [0 , n − i a i +1) a if j = i − , i ∈ [1 , n ] ( i +1) a − i a i +1) a if j = i, i / ∈ { , n } − n a n +1) a if i = j = n PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 23 and ˆ K n,a ( i, j ) = if j = i − , i ∈ [1 , n ] ( n − i ) a n − i +1) a if j = i + 1 , i ∈ [0 , n − ( n − i +1) a − ( n − i ) a n − i +1) a if j = i = 01 − n a ( n +1) a if i = j = 0 . Saloff-Coste [16] discussed the above chains and obtained the correct order ofthe spectral gaps. Let ˇ λ n,a , ˆ λ n,a denote the spectral gaps of ˇ K n,a , ˆ K n,a . Referringto the recent work in [4], one has1 / (4 C ) ≤ λ ≤ /C, where ( λ, C ) is any of (ˇ λ n,a , ˇ C n ( a )) and (ˆ λ n,a , ˆ C n ( a )), andˇ C n ( a ) = 2 max ≤ i ≤ n i − X j =0 ( j + 1) − a n X j = i ( j + 1) a , and ˆ C n ( a ) = 2 max ≤ i ≤ n i − X j =0 ( j + 1) a n − X j = i − ( j + 1) − a . Theorem 5.1. Let ˇ λ n,a , ˆ λ n,a be spectral gaps for ˇ K n,a , ˆ K n,a . Then, η − a (1 , n ) η a (2 , n + 1) ≤ ˇ λ n,a ≤ η − a (1 , n ) η a (2 , n + 1) , and η a (1 , ⌈ n/ ⌉ ) η − a ( ⌈ n/ ⌉ , n ) ≤ ˆ λ n,a ≤ η a (1 , ⌈ n/ ⌉ ) η − a ( ⌈ n/ ⌉ , n ) . where η a ( k, l ) = P li = k i a .Proof of Theorem 5.1. The bound for ˇ λ n,a follows immediately from the fact η − a (1 , n ) η a (2 , n + 1)2 ≤ ˇ C n ( a ) ≤ η − a (1 , n ) η a (2 , n + 1) . For ˆ λ n,a , note thatˆ C n ( a ) = 2 max n/ ≤ i ≤ n i − X j =0 ( j + 1) a n − X j = i − ( j + 1) − a . Taking i = ⌈ n/ ⌉ yields the upper bound. For the lower bound, we writeˆ C n ( a ) = 2 max n/ ≤ i ≤ n i − X j =0 (cid:18) − ji (cid:19) a n − i X j =0 (cid:18) − ji + j (cid:19) a . For i ≥ n/ 2, it is clear that i − X j =0 (cid:18) − ji (cid:19) a ≥ i − X j =0 (cid:18) − jn (cid:19) a ≥ n − X j =0 (cid:18) − jn (cid:19) a . Observe that, for a > C ′ i,n ( a )2 ≤ n − i X j =0 (cid:18) − ji + j (cid:19) a ≤ C ′ i,n ( a ) , where C ′ i,n ( a ) = 1 + ( i ( i/n ) a − − − a if a = 1 i log ni if a = 1 . It is clear that, for i ≥ n/ C ′ i,n ( a ) ≤ C ′⌈ n/ ⌉ ,n ( a ) and this leads to n − i X j =0 (cid:18) − ji + j (cid:19) a ≤ n −⌈ n/ ⌉ X j =0 (cid:18) − j ⌈ n/ ⌉ + j (cid:19) a . Summarizing all above gives the desired lower bound. (cid:3) Figure 1. These curves display the mapping m ˇ λ m,a η − a (1 , m ) η a (2 , m + 1) in Theorem 5.1 in orderfrom the top a = 0 . , . , . , . . 2. The right most pointcorresponds to a path of length n = 5000. Table 1. These numbers denote ˇ λ n,a η − a (1 , n ) η a (2 , n + 1) in The-orem 5.1. n 10000 20000 30000 40000 50000a=0.8 0.5983 0.5960 0.5948 0.5941 0.5935a=0.9 0.5652 0.5625 0.5610 0.5601 0.5594a=1.0 0.5405 0.5377 0.5362 0.5353 0.5345a=1.1 0.5235 0.5210 0.5197 0.5189 0.5183a=1.2 0.5128 0.5109 0.5099 0.5093 0.5088 PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 25 Remark . Comparing with [16, Theorem 9.5], the bounds for ˇ λ n,a given in The-orem 5.1 have a similar lower bound and an improved upper bound by a multipleof about 1 / 4. For ˆ λ n,a , observe that C ′′ i ( a )2 ≤ i − X j =0 (cid:18) − ji (cid:19) a ≤ C ′′ i ( a ) , where C ′′ i ( a ) = 1 + i − i − a a . Recall the constant C ′ i,n ( a ) in the proof of Theorem 5.1. Note that n + a a ) ≤ C ′′⌈ n/ ⌉ ( a ) ≤ n + a )(1 + a ) , and, for a > a = 1 and n ≥ C ′⌈ n/ ⌉ ,n ( a ) ≤ n + 12(1 + a ) sup a> ,a =1 (2 − a − a )1 − a ≤ n + a )1 + a , where the last inequality is obtained by considering the subcases a < a ≥ a = 1 and n ∈ { , } . In the same spirit,one can show that C ′⌈ n/ ⌉ ,n ( a ) ≥ n + a a ) . This yields(5.3) ( n + a ) a ) ≤ ˆ C n,a ≤ n + a ) (1 + a ) , ∀ n ≥ . Hence, we have ˆ λ n,a ≍ (1 + a ) / ( n + a ) . As a consequence of (5.1) and (5.2), weobtain that, uniformly for a > / ˇ λ n,a ≍ n a (cid:18)(cid:18) n (cid:19) a + n a (cid:19) (1 + v ( n, a )) as n → ∞ , where v ( n, 1) = log n and v ( n, a ) = ( n − a − / (1 − a ) for a = 1. Remark . Note that the lower bound in Theorem 6.1 provides the correct orderof the spectral gap for the chain ˇ K n,a uniformly in a but not for ˆ K n,a . For instance,if a grows with n , say a = n , then Theorem 6.1 implies 1 / ˆ λ n,n = O ( n ), while (5.3)gives 1 / ˆ λ n,n ≍ Remark . Consider the chain in Theorem 5.1. A numerical experiment of Algo-rithm (A2) is implemented and the data is collected in Figure 1 and Table 1. Onemay conjecture that ˇ λ n,a η − a (1 , n ) η a (2 , n + 1) → c ( a ) as n → ∞ , where c ( a ) is aconstant depending on a .6. Spectral gaps for uniform measures with bottlenecks In this section, we discuss some examples of special interests and show how thetheory developed in the previous sections can be used to bound the spectral gap. Inthe first subsection, we develop a lower bound on the spectral gap in a very generalsetting using the theory in Section 3. In the second subsection, we focuses on thecase of one bottleneck, where a precise estimation on the spectral gap is presented.Those computations are based on the theoretical work in Section 2. In the thirdsubsection, we consider the case of multiple bottlenecks in which the exact order ofthe spectral gap is determined for some special classes of chains. In what follows, we will use the notation π ( A ) to represent the summation P i ∈ A π ( i ) for any measure π on V and any set A ⊂ V . Given two sequencesof positive reals a n , b n , we write a n = O ( b n ) if a n /b n is bounded. If a n = O ( b n )and b n = O ( a n ), we write a n ≍ b n . If a n /b n → 1, we write a n ∼ b n .6.1. A lower bound on the spectral gap. In this subsection, we give a lowerbound on the spectral gap in the general case. Theorem 6.1. Let G = ( V, E ) be a graph with vertex set V = { , , ..., n } andedge set E = {{ i, i + 1 }| i = 0 , ..., n − } . Let π, ν be positive measures on V, E with π ( V ) = 1 . Then, λ Gπ,ν ≥ max ≤ i ≤ n i − X j =0 π ([0 , j ]) ν ( j, j + 1) − ∧ n X j = i +1 π ([ j, n ]) ν ( j − , j ) − , where a ∧ b := min { a, b } .Remark . Let C be the lower of the spectral gap in Theorem 6.1. Note that, forany positive reals, ( a + b ) / ≤ max { a, b } ≤ a + b . Using this fact, it is easy to seethat C ′ ≤ C ≤ C ′ , where C ′ = max ≤ i ≤ n i − X j =0 π ([0 , j ]) ν ( j, j + 1) + n X j = i +1 π ([ j, n ]) ν ( j − , j ) − . In particular, if i is the median of π , that is, π ([0 , i ]) ≥ / π ([ i , n ]) ≥ / C ′ = i − X j =0 π ([0 , j ]) ν ( j, j + 1) + n X j = i +1 π ([ j, n ]) ν ( j − , j ) − . Remark . Let ( X m ) ∞ m =0 be an irreducible birth and death chain on { , , ..., n } with birth rate p i , death rate q i and holding rate r i as in (1.1). For 0 ≤ i ≤ n ,set τ i = min { m ≥ | X m = i } as the first passage time to state i . By the strongMarkov property, the expected hitting time to i started at 0 can be expressed as E τ i = i − X j =0 π ([0 , j ]) p j π ( j ) , E n τ i = n X j = i +1 π ([ j, n ]) q j π ( j ) , where π is the stationary distribution of ( X m ) ∞ m =0 . Let λ be the spectral gap for( X m ) ∞ m =0 . Then, λ = λ Gπ,ν , where G is the path with vertex set { , ..., n } and ν ( i, i + 1) = p i π ( i ) = q i +1 π ( i + 1) for 0 ≤ i < n . The conclusion of Theorem 6.1can be written as 1 /λ ≤ min ≤ i ≤ n { E τ i ∨ E n τ i } . Remark . The lower bound in Theorem 6.1 is not necessary the right order ofthe spectral gap. See Remark 5.2. Proof of Theorem 6.1. For λ > 0, let ξ λ be the function in Definition 3.2. That is, ξ λ (0) = − i ≥ ξ λ ( i + 1) − ξ λ ( i )] ν ( i, i + 1) = [ ξ λ ( i ) − ξ λ ( i − ν ( i − , i ) − λπ ( i ) ξ λ ( i ) . PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 27 Inductively, one can show that if 1 /λ > P ℓ − j =0 [ π ([0 , j ]) /ν ( j, j + 1)], then ( < ξ λ ( i + 1) − ξ λ ( i ) ≤ λπ ([0 , i ]) /ν ( i, i + 1) , − ≤ ξ λ ( i + 1) ≤ − λ P ij =1 [ π ([0 , j ]) /ν ( j, j + 1)] < , for 0 ≤ i ≤ ℓ − 1. One may do a similar computation from the other end point and,by Proposition 3.5, this implies1 /λ G n π n ,ν n ≤ max ℓ − X j =0 π ([0 , j ]) ν ( j, j + 1) , n − ℓ X j =1 π ([ n − j + 1 , n ]) ν ( n − j, n − j + 1) . Taking the minimum over 1 ≤ ℓ ≤ n gives the desired inequality. (cid:3) One bottleneck. For n ≥ 1, let G n = ( V n , E n ) be the path on { , , ..., n } and set π n ≡ / ( n + 1) and ν n ≡ / ( n + 1) with C > 0. Using Feller’s method in[11, Chapter XVI.3], one can show that the eigenvalues of M G n π n ,ν n are 2(1 − cos iπn +1 )for 0 ≤ i ≤ n . Theorem 6.2. For n ≥ , let ǫ n > , ≤ x n ≤ ⌈ n/ ⌉ and set π n ≡ / ( n + 1) , (6.1) ν x n n ( x n − , x n ) = ǫ n n + 1 , ν x n n ( i − , i ) = 1 n + 1 , ∀ i = x n . Then, the spectral gap are bounded by n / x n /ǫ n ≤ λ Gπ n ,ν xnn ≤ min (cid:26) (cid:18) − cos πn − x n + 1 (cid:19) , ǫ n x n (cid:27) . In particular, λ G n π n ,ν xnn ≍ min { /n , ǫ n /x n } .Proof of Theorem 6.2. The lower bound is immediate from Theorem 6.1 by choos-ing i = ⌈ n/ ⌉ in the computation of the maximum. For the upper bound, we set λ n = 1 − cos πn +1 and let f n be the function on V n − x n defined by f n (0) = − ≤ i ≤ n − x n − f n ( i + 1) = f n ( i ) + [ f n ( i ) − f n ( i − ν n − x n ( i − , i ) − λ n − x n π n − x n ( i ) f n ( i ) ν n − x n ( i, i + 1) . By Proposition 2.3, E ν n − xn ( f n , f n ) = 2 λ n − x n Var π n − xn ( f n ) and π n − x n ( f n ) = 0. Let g n be the function on V n defined by g n ( n − i ) = f n ( i ) for 0 ≤ i ≤ n − x n and g n ( i ) = f n ( n − x n ) for 0 ≤ i < x n . A direct computation shows that( n + 1) E ν xnn ( g n , g n ) = ( n − x n + 1) E ν n − xn ( f n , f n )and( n + 1)Var π n ( g n , g n ) = ( n − x n + 1)Var π n − xn ( f n ) + x n ( n − x n + 1) n + 1 f n ( n − x n ) . This implies λ G n π n ,ν xnn ≤ λ n − x n . On the other hand, using the test function, h n ( i ) = n − x n + 1 for 0 ≤ i < x n and h n ( i ) = − x n for x n ≤ i ≤ n , one has E ν xnn ( h n , h n ) / Var π n ( h n ) = ǫ n ( n + 1) / [ x n ( n − x n + 1)] ≤ ǫ n /x n . This finishes theproof. (cid:3) The next theorem has a detailed description on the coefficient of the spectralgap. The proof is based on Section 3, particularly Proposition 3.11 and Remark3.10, and is given in the appendix. Theorem 6.3. For n ≥ , let x n , ǫ n , π n , ν x n n be as in Theorem 6.2. Suppose x n / ( ǫ n n ) → a ∈ [0 , ∞ ] and x n /n → b ∈ [0 , / . (1) If a < ∞ and b = 0 , then λ G n π n ,ν xnn ∼ min { π , a − } n − . (2) If a < ∞ and b ∈ (0 , / , then λ G n π n ,ν xnn ∼ Cn − , where C is the uniquepositive solution of the following equation. − π − π aC − b − bC ∞ X i =1 (1 − b ) i − bC ( i − C )[(1 − b ) i − b C ] = 0 . (3) If a = ∞ , then λ G n π n ,ν xnn ∼ ǫ n /x n . Multiple bottlenecks. In this subsection, we consider paths with multiplebottlenecks. As before, G n = ( V n , E n ) with V n = { , , ..., n } and E n = {{ i, i +1 }| i = 0 , ..., n − } . Let k be a positive integer and x n = ( x n, , ..., x n,k ) be a k -vector satisfying x n,i ∈ V n and x n, ≥ x n,i < x n,i +1 for 1 ≤ i < k . Let ǫ n = ( ǫ n, , ..., ǫ n,k ) be a vector with positive entries and ν x n n be the measure on E n given by(6.2) ν x n n ( i − , i ) = ( / ( n + 1) if i / ∈ { x n, , ..., x n,k } ǫ n,j / ( n + 1) if i = x n,j , ≤ j ≤ k . Theorem 6.4. Let G n = ( V n , E n ) be the path on { , ..., n } . For ≤ k ≤ n , let π n be the uniform probability on V n and ν x n n be the measure on E n given by (6.2) .Then, min { / (4 n ) , C n, / } ≤ λ G n π n ,ν xnn ≤ min (cid:26) (cid:18) − cos πn − k + 1 (cid:19) , C n, (cid:27) , where C n, = n k X i =1 min { x n,i , n − x n,i + 1 } (cid:18) ǫ n,i − (cid:19)! − and C n, = min ≤ m ≤ m ≤ n ( n + 1) m P i = m /ǫ n,i P m ≤ i ≤ j ≤ m x n,i ( n − x n,j + 1) / ( ǫ n,i ǫ n,j ) . Remark . Observe that, in Theorem 6.4, 1 − cos πn − k +1 ≍ n − and C n, ≤ min ≤ j ≤ k (cid:26) ǫ n,j min { x n,j , n − x n,j + 1 } (cid:27) = min (cid:26) min j : x n,j ≤ n ǫ n,j x n,j , min j : x n,j > n ǫ n,j n − x n,j + 1 (cid:27) . Proof of Theorem 6.4. We first prove the upper bound. Let f be a function on { , , ..., n } satisfying f ( x n,j − 1) = f ( x n,j ) for 1 ≤ i ≤ k and f be a function on { , ..., n − k } obtained by identifying points x n,i − x n,i for 1 ≤ i ≤ k . By PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 29 setting f as a minimizer for λ G n − k π n − k ,ν n − k with π n ( f ) = 0, we obtain2 (cid:18) − cos 2 πn − k + 1 (cid:19) = E ν n − k ( f , f )Var π n − k ( f ) ≥ E ν n − k ( f , f ) π n − k ( f ) ≥ E ν n ( f , f ) π n ( f ) = E ν n ( f , f )Var π n ( f ) . To see the other upper bound, let f j be the function on V n satisfying g j ( i ) = − ( n − x n,j + 1) for 0 ≤ i ≤ x n,j − g j ( i ) = x n,j for x n,j ≤ i ≤ n . Computations showthat π n ( g j ) = 0, π n ( g i g j ) = x n,i ( n − x n,j +1) for i ≤ j , and E ν n ( g j , g j ) = ǫ n,j ( n +1).Set g = P kj =1 a j g j . As a consequence of the above discussion, we obtain E ν n ( g, g )Var π n ( g ) = ( n + 1) P ki =1 a i ǫ n,i P i Figure 2. The dashed lines denote the weak edges of ν in Theo-rem 6.5. r r r r r r r r♣ ♣ ♣ ♣r r r r r r r r r r♣ ♣ ♣ ♣ ✛ ✲ aJ n ✛ ✲ aJ n ✛ ✲ J n /a ✛ ✲ J n /a ✁✁☛ ❆❆❯ Most weak edges Theorem 6.5. For n ≥ , let π n ≡ / ( n + 1) and ν n be the measure in (6.2) with k n bottlenecks satisfying n − k n ≍ n . Suppose there are I n ⊂ { , ..., k n } , a ∈ (0 , and J n > such that | I n | is bounded and, for i / ∈ I n , aJ n ≤ min { x n,i , n − x n,i +1 } ≤ J n /a . Then, λ G n π n ,ν n ≍ min n , min i ∈ I n ǫ n,i min { x n,i , n − x n,i + 1 } , (cid:16)P k n i =1 ,i/ ∈ I n /ǫ n,i (cid:17) − J n . Proof. It is easy to get the lower bound from Theorem 6.4, while the upper boundis the minimum of C n, over all connected components of { , ..., ℓ } \ I n and { ℓ +1 , ..., k n } \ I n . (cid:3) See Figure 2 for a reference on the bottlenecks. The following are immediatecorollaries of Theorems 6.4-6.5. Corollary 6.6 (Finitely many bottlenecks) . Referring to Theorem 6.5 , if k n isbounded, then λ G n π n ,ν n ≍ min (cid:26) n , min ≤ i ≤ k n ǫ n,i min { x n,i , n − x n,i + i } (cid:27) . Corollary 6.7 (Bottlenecks far away the boundary) . Referring to Theorem 6.5 , if n − k n ≍ n and there are a ∈ (0 , and J n > such that aJ n < min { x n,i , n − x n,i +1 } < J n /a for ≤ i ≤ k n , then λ G n π n ,ν n ≍ min n , (cid:16)P k n j =1 /ǫ n,i (cid:17) − J n . Corollary 6.8 (Uniformly distributed bottlenecks) . Referring to Theorem 6.5 , if min i ǫ n,i ≍ max i ǫ n,i and x n,i = ⌊ in/k n ⌋ with k n ≤ n/ , then λ G n π n ,ν n ≍ min (cid:26) n , ǫ n, nk n (cid:27) . Remark . Note that the assumption of the uniformity of π and ν , except at thebottlenecks, can be relaxed by using a comparison argument. Appendix A. Techniques and proofs We start with an elementary lemma. Lemma A.1. Let a > and f : [ a, ∞ ) → R be a continuous function satisfying f ( a ) = a and f ( x ) ∈ [ a, x ) for x > a . For b > a , set C b = sup a ≤ x ≤ b { ( f ( x ) − a ) / ( x − a ) } . Then, C b < and a ≤ f n ( b ) ≤ a + C nb ( b − a ) for n ≥ . Moreover, if f is bounded on [ a, ∞ ) , then a ≤ f n ( x ) ≤ a + C n ( x − a ) for n ≥ and x ≥ a with C = sup a ≤ t< ∞ { ( f ( t ) − a ) / ( t − a ) } < . Lemma A.2. Let ( a i , b i , c i ) ∞ i =1 be sequences of reals with b i > and c i > . For n ≥ and t ∈ R , let M n ( t ) = a − c t · · · b a − c t ... b . . . . . . . . . ... . . . . . . . . . ... . . . . . . a n − − c n − t · · · · · · b n − a n − c n t . Then, there are n distinct real roots for det M n ( t ) = 0 , say t ( n )1 < · · · < t ( n ) n , and t ( n +1) j < t ( n ) j < t ( n +1) j +1 , ∀ ≤ j ≤ n, n ≥ . Furthermore, if a ≥ and a i +1 ≥ b i , then t ( n )1 > for all n ≥ . To prove Lemma A.2, we need the following statement. PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 31 Lemma A.3. Fix n > and, for i ≤ ≤ n , let a i , b i , d i be reals with b i > and d i = 0 . Consider the following matrix (A.1) M = a d · · · d − b a d ... d − b a . . . . . . ... . . . . . . . . . ... . . . . . . a n − d n − · · · · · · d − n − b n − a n . Then, the eigenvalues of M are distinct reals and independent of d , ..., d n − . Fur-thermore, if a ≥ and a i +1 ≥ b i , then all eigenvalues of M are positive.Proof of Lemma A.3. Let X, Y be diagonal matrices with X = Y = 1, X ii = d d · · · d i − and Y ii = ( b b · · · b i − ) − / ( d d · · · d i − ) for i > 1. One can showthat XM X − = a · · · b a b a . . . . . . ...0 0 . . . . . . . . . 0... . . . . . . a n − · · · · · · b n − a n . Since XM X − is independent of the choice of d , ..., d n − , the eigenvalues of M are independent of d , ..., d n − . Note that Y M Y − is Hermitian. This implies thatthe eigenvalues of M are all real. As M is tridiagonal with non-zero entries inthe superdiagonal, the rank of M − λI is either n − n . This implies that theeigenvalues of M are all distinct.Next, assume that a ≥ a i +1 ≥ b i . Let ( Y M Y − ) i be the leading i × i principal matrices of Y M Y − . By induction, one can prove that det( Y M Y − ) i = Q ij =1 ℓ j , where ℓ = a and ℓ j +1 = a j +1 − b j /ℓ j for 1 ≤ j < n . By the assumptionat the beginning of this paragraph, ℓ j ≥ ≤ j < n and det( Y M Y − ) i > ≤ i ≤ n . As the leading principal matrices have positive determinants,( Y M Y − ) is positive definite. This proves that all eigenvalues of M are positive. (cid:3) Proof of Lemma A.2. We prove this lemma by induction. For n = 1, it is clear that t (1)1 = a /c is the root for det M ( t ). For n = 2, note that det M ( t ) is a quadraticfunction that tends to infinity as | t | → ∞ . Since det M ( t (1)1 ) = − b < 0, thepolynomial, det M ( t ), has two real roots, say t (2)1 < t (2)2 , satisfying t (2)1 < t (1)1 < t (2)2 .Now, we assume that, for some n ≥ 1, det M n ( t ) and det M n +1 ( t ) have reals roots( t ( n ) i ) ni =1 and ( t ( n +1) i ) n +1 i =1 satisfying t ( n +1) i < t ( n ) i < t ( n +1) i +1 for 1 ≤ i ≤ n . Clearly,det M n ( t ) → ∞ as t → −∞ . This impliesdet M n ( t ( n +1)2 k +2 ) < < det M n ( t ( n +1)2 k +1 ) , ∀ k ≥ . Observe that det M n +2 ( t ) = ( a n +2 − c n +2 t ) det M n +1 ( t ) − b n +1 det M n ( t ). Replacing t with t ( n +1) i yieldsdet M n +2 ( t ( n +1)2 k +2 ) > > det M n +2 ( t ( n +1)2 k +1 ) , ∀ k ≥ . This proves that det M n +2 ( t ) has ( n + 2) distinct real roots with the desired inter-lacing property.For the second part, assume that a ≥ a i +1 ≥ b i for all i ≥ 1. For n = 1, it is obvious that t (1)1 > 0. Suppose t ( n )1 > 0. According to the first part, wehave t ( n +1)2 > t ( n )1 > 0. By Lemma A.3, det M n +1 (0) > 0, which implies t ( n +1)1 = 0.As it is known that det M n +1 ( t ) < t ∈ ( t ( n +1)1 , t ( n +1)2 ), it must be the case t ( n +1)1 > 0. Otherwise, there will be another root for det M n +1 ( t ) between t ( n +1)1 and 0, which is a contradiction. (cid:3) Proof of Theorem 6.3. For convenience, we set λ mn = 1 − cos mπn +1 for 1 ≤ m ≤ n andlet A i ( λ ) be the i -by- i tridiagonal matrix with entries ( A i ( λ )) kl = 1 for | k − l | = 1and ( A i ( λ )) kk = 2 − λ . For 1 ≤ j ≤ i , let B ji ( λ ) be the matrix equal to A i exceptthe ( j, j )-entry, which is defined by ( B ji ( λ, ǫ )) jj = 2 − λ/ǫ . By Remark 3.9, λ G n π n ,ν xnn is the smallest root of det B x n n ( λ, ǫ n ) = 0 and ( λ n,m ) nm =1 are roots of det A n ( λ ) = 0.Note that, for 1 ≤ j ≤ n ,det B jn ( λ, ǫ )det A j − ( λ ) det A n − j ( λ ) = ∆ jn ( λ, ǫ ) = 2 − λ/ǫ − R j − ( λ ) − R n − j ( λ ) , where det A ( λ ) := 1, det A − ( λ ) := 0 and R j ( λ ) = det A j − ( λ )det A j ( λ ) = Q j − i =1 (2 λ ij − − λ ) Q ji =1 (2 λ ij − λ ) . To prove this theorem, one has to determine the sign of ∆ jn ( λ, ǫ ).Let ℓ n = δ n /n with δ n → 0. As n → ∞ ,log 2 λ in − ℓ n λ in = − δ n λ in n (1 + o (1)) , where o (1) is uniform for 1 ≤ i ≤ n . Note that Q ji =1 (2 λ ij ) = det A j (0) = j + 1.This implieslog R n ( ℓ n ) = log nn + 1 + n X i =1 λ in n − n − X i =1 λ in − ( n − ! δ n (1 + o (1))2= log nn + 1 + O (cid:18) δ n n (cid:19) . By a similar reasoning, one can prove that log R j ( ℓ n ) = log jj +1 + O ( δ n /n ) forbounded j . This shows that, for j n ∈ { , ..., n } and ℓ n = o ( j − n ),(A.2) R j n ( ℓ n ) = 1 − j n + 1 + O ( j n ℓ n ) , as n → ∞ . PECTRAL COMPUTATIONS FOR BIRTH AND DEATH CHAINS 33 Next, we compute R j n (2 C n λ j n ) with C n → C ∈ (0 , 1) and j n → ∞ . Note that, for n large enough,(A.3) log R j n (2 C n λ j n ) = j n − X i =1 λ ij n − − λ ij n λ ij n − j n − X i =1 λ ij n − − λ ij n λ ij n ! + C n j n − X i =1 λ j n ( λ ij n − − λ ij n )( λ ij n − C n λ j n ) λ ij n − log 4 + O ( j − n ) . Calculus shows that j n − X i =1 λ ij n − − λ ij n λ ij n ! = 1 πj n Z π θ sin θ (1 − cos θ ) dθ + O ( j − n )= 8 log 2 − π / j n + O ( j − n )and j n − X i =1 λ j n ( λ ij n − − λ ij n )( λ ij n − Cλ j n ) λ ij n = 2 j n ∞ X i =1 i − C + O ( j − n ) . Observe that, as n → ∞ ,log j n j n + 1 = log R j n (0) = j n − X i =1 λ ij n − − λ ij n λ j n − log 4 + O ( j − n ) . Putting this back into (A.3) implies(A.4) R j n (2 C n λ j n ) = 1 + − − π C n ∞ X i =1 i − C n ! j n + O ( j − n ) . We consider the following two cases. Case 1: x n = O ( ǫ n n ) . In this case, Theorem 6.2 implies that λ G n π n ,ν xnn ≍ n − . Weassume further that x n / ( ǫ n n ) → a and x n /n → b with a ∈ [0 , ∞ ) and b ∈ [0 , / C n → C ∈ (0 , j n with x n − n − x n in (A.4)yields that, for b = 0,∆ x n n (2 C n λ n − x n , ǫ n ) = (1 − π aC )(1 + o (1)) x n and, for b ∈ (0 , / x n n (2 C n λ n − x n , ǫ n ) = (cid:18) − π − π aC − b − bCκ b ( C ) (cid:19) (1 + o (1)) b (1 − b ) n , where κ t ( c ) = P ∞ i =1 (1 − t ) i − tc ( i − c )[(1 − t ) i − t c ] . This proves (1) and (2). Case 2: ǫn = o ( x n ) . This is exactly (3) and the result is immediate from Theorem6.2. (cid:3) References [1] M. Brown and Y.-S. Shao. Identifying coefficients in the spectral representation for firstpassage time distributions. Probab. Engrg. Inform. Sci. , 1:69–74, 1987.[2] Guan-Yu Chen and Laurent Saloff-Coste. The cutoff phenomenon for ergodic markov pro-cesses. Electron. J. Probab. , 13:26–78, 2008. 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Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300,Taiwan E-mail address : [email protected] Malott Hall, Department of Mathematics, Cornell University, Ithaca, NY 14853-4201 E-mail address ::