Cyclic products and optimal traps in cyclic birth and death chains

A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.


Introduction and main results
Birth and death chains are (discrete-time, time-homogeneous) Markov chains on Z with transition probabilities (p i,j ) i,j∈Z satisfying p i,i+1 + p i,i−1 = 1 for each i ∈ Z. Often "birth and death chain" allows p i,i > 0, but here for simplicity we assume that p i,i = 0.
In this paper we consider cyclic birth and death chains X = (X n ) n 0 on Z, by which we mean that there exist m ∈ N and p m = (p i ) m−1 i=0 ∈ (0, 1) m such that for each i ∈ Z, (p i,i = 0 and) p i,i+1 = p i mod m . Such models have been studied in general dimensions in e.g. [9,13,7], and have been called random walks in periodic environment. To avoid any confusion with periodicity of a Markov chain we will refer to them as random walks in cyclic environment, or cyclic birth and death chains (CBD). In the 1-dimensional setting there is an elementary criterion for transience (|X n | → ∞) and recurrence (X n = 0 infinitely often), in which the crucial quantity is The following result can be proved using standard Markov chain techniques. Each conclusion holds with probability 1.
Proof. Observe the chain X first at time 0, and thereafter observe the chain X at times at which its displacement is ±m from the previous observation. This new walk is (m×) a simple random walk that is symmetric (hence recurrent, with velocity 0) in the third case above and biased to the right or left otherwise (see Lemma 16 and its proof below for more details). Since the expected time for X to reach ±m is finite, this proves the claim for the original chain X as well.
Motivated by trapping behaviour prevalent in random walk in random environment on Z (where (p i,i+1 ) i∈Z are chosen to be i.i.d. random variables), we are interested in how the velocity v depends on the order of the p i for fixed m. According to Proposition 1, the sign of v (or equivalently, whether or not X n → ±∞) does not depend on the order of the p i . If the velocity is 0 then it can't be changed by changing the order of the p i , but in this case the variance may be of interest. Therefore we are primarily interested in the case where X n → ∞ (and v > 0) with probability 1. In particular, given a sequence p m = (p i ) m−1 i=0 ∈ (0, 1) m for which γ(p m ) < 1, (so all velocities arising from permutations will have positive sign), here are two natural questions that one can ask: Q1: What is the number N (p m ) of distinct speeds achievable via permutations of p m ? Q2: In which order one should arrange these values to achieve the minimum speed, or indeed the maximum speed?
The second of these questions has been considered elsewhere in some special cases. After the conclusion of our work, we learned that in [3] Q2 is analysed in the case where p 0 = 1. The authors therein note the possible relevance of these kinds of combinatorial optimisation problems to e.g. constructing intruder-resilient networks. In [8], a problem similar to Q2 is considered in the case of vectors p m where each p i is either 1/2 or p for some fixed p. Naturally each of these papers has features in common with our work, but we shall see that e.g. the assumption p 0 = 1 fundamentally changes the nature of the optimisation problem. In particular, the main tool used in [3] fails to work in our setting. Both Q1 and Q2 turn out to be interesting. In this paper we state some conjectures and provide partial answers to these questions, with our main results being Theorems 2 and 6 below. There is trivially only 1 possible speed when m = 1, 2. Theorem 2 states that for m 3 and Lebesgue a.e. p m the answer to Q1 is (m − 1)!/2. This value arises from the fact that the velocity is typically only invariant to rotations and reversal of the elements of p m . Note that invariance under rotations is trivial, while invariance under reversal seems to be a new (and we think surprising) result.
Obviously the equality in Theorem 2 cannot be satisfied (for m > 3) for every p m = (p 0 , . . . , p m−1 ) ∈ (0, 1) m , since e.g. if p i = p j then the permutation that simply switches i and j also preserves the speed. Theorem 2 immediately implies that when m = 3 all rearrangements of p m give the same velocity, while for m 4 and typical p m , multiple different velocities are achievable via rearrangement.
To simplify discussions about "optimal" permutations, it is convenient (and loses no generality) to restrict attention henceforth to p m for which the elements are nonincreasing (so p 0 p 1 . . . p m−1 ). In this case we believe that for fixed m there exists a permutation σ greedy that is the universal minimiser of the speed for all such p m . That is, for each m there is a unique (up to rotations and reversals) permutation that minimises the speed no matter what the values of the p 0 p 1 . . . p m−1 .
Definition 3. Given a vector a m = (a 0 , a 1 , . . . , a m−1 ) ∈ (0, ∞) m with non-increasing entries, define the circular symmetric ordering to be (a 0 , a 2 , a 4 , . . . , a 5 , a 3 , a 1 ). This ordering has been called the pendulum arrangement in [3]. We have adopted the terminology of [1], which is the earliest paper that we are aware of dealing with problems of this type.
Let S m be the set of permutations of (0, 1, . . . , m−1). Of course, S m can be considered as the set of bijections from {0, 1, . . . , m − 1} to itself. We will use standard () notation for permutations, e.g. if m = 4 and σ = (0231) then σ(0) = 0, σ(1) = 2 etc.. For a vector We call the permutation corresponding to the circular symmetric ordering σ greedy , because it groups large values of a m with each other, and small values of a m with each other in a circular way.
Definition 4. The greedy permutation σ greedy is given by By definition, the greedy permutation depends on m but not on the actual values a m , e.g. if m = 9 then σ greedy = (081726354). Let a m ∈ (0, ∞) m with decreasing entries. For r ∈ [m] and a permutation σ ∈ S m define with indices interpreted mod m. This quantity has been considered elsewhere (e.g. [1]) in the case r = 2. A version of this quantity involving non-cyclic products has been analysed in [3]. The non-cyclic version (which corresponds to setting some a m−1 = 0) appears to be easier to analyse.
The following conjecture says that the greedy permutation maximises P r for each r. We think that it is an interesting standalone open problem. It also immediately implies that the greedy permutation minimises the speed (see Proposition 7 and Conjecture 8 below).
The cases r = 1 and r = m in Conjecture 5 are trivially true. The case r = 2 is not difficult to prove, and appears as early as [1]. Such facts are termed circular rearrangement inequalities in [14] (see also [2]). We present a simple proof for the case r = 2 and also give a (non-trivial) proof in the case r = 3. Observe that the electronic journal of combinatorics 30(2) (2023), #P2.52 from which we conclude that if a is the product of all elements of a m then where a −1 m = (a −1 0 , . . . , a −1 m−1 ). This observation together with the aforementioned results for r 3 gives rise to the following. Theorem 6. The conclusion of Conjecture 5 holds for (m, r) such that r 3 or m − 3 r m.
It is immediate from Theorem 6 that the conclusion of Conjecture 5 holds for all r ∈ [m] when m 7. A version of Conjecture 5 for non-cyclic products (which arise in the cyclic case precisely when some a i = 0) is known to hold [3,Theorem 6]. The proof therein does not work for cyclic products.
The relevance of Conjecture 5 and Theorem 6 to Q2 can be seen from the following explicit formula for the velocity, in which I denotes the identity permutation and ρ m = (ρ 0 , ρ 1 , . . . , ρ m−1 ). . ( Since for any r, sums over starting indices k of consecutive products of ρ terms are invariant under rotations and reversals (reversing the order of p m ), we can immediately conclude from (2) that v(p m ) is invariant under rotations and reversals of the elements of p m , as claimed earlier. In particular, when m = 3 there is only one possible velocity, since all permutations are combinations of rotations and reversal.
As noted earlier, we find the fact that the speed is invariant under reversals to be somewhat surprising, and is not at all obvious from other expressions for the velocity. For example, as in Lemma 14 in Section 2.1 below, the velocity can also be written as v(p m ) = m−1 i=0 π i (2p i − 1), where π = (π 0 , . . . , π m−1 ) is the stationary distribution of the chain X • n = X n mod m. This stationary distribution behaves "nicely" under rotations but not under reversal of the elements of p m -see Example 15 in Section 2.1 below.
As noted above, the following is an immediate corollary of Conjecture 5 (and Proposition 7). It says that the greedy permutation minimises the speed.
Conjecture 8 (Greedy is least speedy). For any p m with non-increasing entries, such that γ < 1 (so all possible speeds will be positive), then for every For example, if m = 9 and the p i are decreasing in i with ρ(p m ) < 1 then according to Conjecture 8, for any permutation As a consequence of Theorems 2 and 6 above we obtain the following (we omit the proof).
Corollary 9. For m 7 and p m with non-increasing entries and ρ < 1, the speed is minimised by the greedy permutation (i.e. Conjecture 8 holds for m 7).
Remark 10. Notice that for m 7 the ordering that minimises the speed is also the one that minimises (interpret the following with indices mod m), (expand the square, and note that only the sum of mixed terms depends on the order). One might interpret this as saying that the speed is minimised by having a "smooth" ordering (a cyclic arrangement of the elements of p m that has no large jumps).
We stress that the maximiser of P r appears to be universal. In other words we believe that the greedy permutation maximises P r for every r and every a m with decreasing entries. In the language of [14] this says that the circular symmetrical order maximises P r . In [14] it was shown that the so-called circular alternating order minimises P 2 . As noted earlier, Conjectures 5 and 8 have been verified in the special case p 0 = 1 in [3]. In [3] a natural and key tool used is the utilisation of a special class of so-called "improving permutations". These permutations are p m -dependent permutations (that permute p 1 , . . . , p m−1 ) that are shown to increase the value of P r (when p 0 = 1). The case p 0 = 1 however means one is not dealing with cyclic permutations at all since terms involving ρ 0 are zero in this case. There are numerical counterexamples showing that the "improving permutations" of [3] do not in general increase P r when ρ 0 = 0, even when r = 3 (fixing p 0 to be the maximum of the p i and permuting the rest) or r = 4 (where we permute all p i ). It would be of interest to find a sufficiently rich class of "improving permutations" in the cyclic setting.
The following two examples (which can be verified by simply evaluating the cyclic products for all possible permutations) show that the minimal ordering is neither constant over r for fixed a m , nor constant over a m for fixed r.
Since the permutations which minimise products of r consecutive terms in the cycle are not constant over r, the above observations don't give any conclusion for permutations that maximise the speed. Nevertheless, we have the following.  Moreover, in case (i) above the speed is not maximised at v((p m ) σ ) ≈ 0.19787, and in case (ii) above the speed is not maximised at v((p m ) σ ) ≈ 0.04668.
The remainder of this paper is organised as follows: In Section 2 we further discuss the context of our results: we present some elementary (implicit) speed formulae, compare results about the velocity for the cyclic birth and death chain to that of a related model of random walk in random environment, and briefly discuss the central limit theorem. In Section 3 we prove Proposition 7. To do this we follow an approach that will be familiar to researchers in the area of Random Walk in Random Environment (RWRE), and then manipulate the resulting expression to get (2). Some understanding of discrete-time Markov chains (specifically birth and death chains) is required to understand Sections 2 and 3. The reader who is happy to start with (2) as given can proceed directly to Sections 4 and 5 where we prove Theorems 2 and 6 respectively. Theorem 2 will be proved by showing: (i) that the speed is invariant to rotations and reversal of the elements of p m (the former is trivial, while we find the latter to be rather surprising), and; (ii) for typical p m these kinds of permutations are the only ones which do not change the speed. Theorem 6 will be proved by induction on m for r = 2, 3. The cases 3 < r < m − 3 remain open.

Discussion
In this section we further discuss the context of our results. Let us begin with some simple (and standard) implicit formulas for the velocity.

Elementary speed formulae
i=0 denote the stationary distribution of X • (which depends on p m ). Then we have the following.
Readers familiar with random walk in random environment might interpret Lemma 14 as a formula for the speed given in terms of the environment viewed from the particle.
One can find an explicit (albeit complicated) formula for π, and hence for v, by solving a recursion for mean return times, but we will not present this here. The following example however demonstrates that the invariance of the speed (under all rotations) in the case m = 3 is not at all trivial.
which is not equal to any of the π i (p 0 , p 1 , p 2 ) in general. E.g For j = 0, 1, . . . , m − 1, and n 1, let N n (j) = #{r < n : X r mod m = j}. Then Thus, Note that N n (j) is the number of visits by (the irreducible, finite-state discrete-time Markov chain) X • to j prior to time n. Therefore n −1 N n (j) → π j almost surely. Since ∆ j,i are independent this implies that Let T denote the first hitting time of {−m, m} by the chain X, and let h = P(X T = m) = 1 − P(X T = −m). Then a standard Markov chain calculation gives h = (1 + γ) −1 (see for example page 67 of [4]), and we have the following formula. .
Proof. Let T (0) = 0, and for i 1 let Note that E[T ] is trivially finite, so v > 0 as soon as h > 1/2, i.e. as soon as γ < 1. Let T + denote the first hitting time of m by the chain X. Then standard renewal arguments give the following.
Proof. Let T which is finite almost surely since γ < 1. Now proceed as in the proof of Lemma 16.
Each of the above representations for v is standard, but we would describe as implicit in the sense that π in Lemma 14 and the expectations in the denominators in Lemmas 16 and 17 are not explicit functions of p m . Nevertheless, we will use Lemma 17 to prove Proposition 7. It is intuitively obvious that for γ < 1 the denominator in Lemma 17 is strictly decreasing in each p i . This can be made rigorous via a simple coupling argument to obtain the following.
Proof. Let p m ∈ (0, 1) m be given. Symmetry arguments allow us to assume without loss of generality that γ = γ(p m ) 1. Let p m be equal to p m except that p i > p i . If γ = 1 then the claim holds by Proposition 1. Otherwise γ < 1 and E[T + ] < ∞ in Lemma 17. It is easy (see e.g. [5,6]) to define a probability space on which copies of the CBD p m and the CBD p m are both defined, and such that: (i) T + T + almost surely, and (ii) T + < T + with positive probability. This shows that E[T + ] < E[T + ] in Lemma 17 which completes the proof.
For u ∈ [0, 1), let P m (u) = {p m ∈ (0, 1) m : v(p m ) = u} denote the set of (ordered) vectors of length m that have speed u. According to Lemma 18, for each p 1 , . . . , p m−1 there is at most one value of p 0 for which v(p m ) = u. Therefore P m (u) is a subspace of dimension at most m − 1, and it has Lebesgue measure 0. in Ω. Let F be the power set of Ω, and µ be the uniform measure on Ω. Then (Ω, F, µ) is ergodic with respect to the shift operator θ((ω x ) x∈Z ) = (ω x+1 ) x∈Z 1 . As such, any result from the theory of random walk in ergodic random environment that holds for a.e. environment holds for the CBD with ω x = p x mod m etc. For example, a law of large numbers with an implicit formula for the speed, is known to hold for random walk in ergodic random environment, see e.g. [15].

Comparison with RWRE
It is natural to compare results for cyclic birth and death (CBD) processes to those for (uniformly elliptic) i.i.d. RWRE with right step probability from each site being uniformly selected from our set of probabilities {p 0 , p 1 , . . . , p m−1 } (counting multiplicities if there are any). The results of Solomon [10] in this special setting become: • The walker is transient to +∞ if and only if • If the walker is transient to +∞ then the velocity is strictly positive if and only if m −1 m−1 i=0 ρ i < 1, in which case the velocity is equal to In other words, the criteria for transience (for RW i.i.d. RE and for cyclic birth and death chains) "match", but the criteria for positivity of the speed do not. Both of these observations are to be expected -in the former case one can see the criteria as coming from a calculation involving the resistance to +∞ (and −∞) together with the LLN for the limiting proportion of time that each environment appears. In the latter case obviously the velocity of the RWRE above should be invariant to permutations of the elements of p m since choosing a uniform i.i.d. sample from p m ignores any ordering. Moreover, the disorder in the environment allows much stronger traps to be created. In view of the last observation, it is natural to ask whether the speed for this RWRE is always less than the CBD (when (3)  where the speed v(p m ) of the CBD is strictly positive, but smaller than the speed of the corresponding RWRE. We interpret this observation as saying that when m is large it is possible to create really bad traps in CBD by very specific orderings of p m and that traps as bad or worse occur extremely rarely in the i.i.d. RE. When m is small any particular ordering of the p m will appear fairly often in the i.i.d. RE, as will "even worse" traps.

CLT
Thus far we have only discussed how the deterministic limiting velocity behaves as a function of p m . One might also ask about the variance, and a central limit theorem. Let T 0 = 0 and T k = inf{n > T k−1 : |X n − X T k−1 | = m} and W k = m −1 X T k . Then (W k ) k∈N is a nearest-neighbour simple random walk on Z with P(W k = W k−1 + 1) = h (recall that h is the probability that the walk X hits m before −m). It follows immediately that k −1 W k → 2h − 1 almost surely. Moreover, We cannot apply the standard CLT for random walk in ergodic random environment (e.g. [15, Theorem 2.2.1]) because our environment is non-mixing (it is completely determined by its value in any interval of length m). Nevertheless one can use the Markov chain central limit theorem to obtain a CLT (see e.g. [9]): For each p m ∈ (0, 1) m there exists a deterministic σ 2 = σ 2 (p m ) > 0 such that The constant σ 2 can be expressed in terms of π and k-step transition probabilities for all k, but is not really tractable in this form. It would be of interest to find a more explicit expression in terms of p m . In the case v = 0, Takenami [13] has proved a local limit theorem for the walk.

Proof of Proposition 7
Fix m, p m , and recall Lemma 17. For i 0 let S i := min{n 0 : X n = i}.
Note that since the random walk is transient to the right, we have that S i < ∞ a.s. We will derive a set of m linear equations for , where E j denotes expectation with respect to the law of the chain X, starting from state j. Note that Therefore This set of equations can be written as M e = 1 where which is invariant under permutations on the sub-indices 0, 1, . . . , m−1. It is also non-zero since γ < 1. Hence, we can apply Cramer's rule to get that for i = 0, . . . , m − 1, where M (j) is the matrix obtained after replacing the j-th column of M by 1, and |A| denotes the determinant of A.
Also note that while the other |M (i+1) | are of the same form but with the vector (p i ) m−1 i=0 changed to (p σ i ) m−1 i=0 , where σ is a rotation. Let R m be the set of rotation permutations, that is, compositions of the permutation (123 . . . 0). It follows that the velocity can be written as .

The denominator is equal to
the electronic journal of combinatorics 30(2) (2023), #P2.52 Recalling that ρ i = (1 − p i )/p i , this can be written as Letting r = m − j and i = m − 1 − i 2 and using the fact that the sum over k is a sum over the whole cycle, we see that the second term in the square brackets in (4)  By separating off the term j = m − 1, and using the fact that an empty product is equal to 1, the first term in the square brackets in (4) is Now let r = m − j − 1 and i = m − 1 − i 2 to see that this is equal to Cancelling factors of m and using the definition of P r completes the proof.

Proof of Theorem 2
Given a subset E ⊂ R d and x ∈ R d , we write E + x = {y + x : y ∈ E} and define and for a sequence (x n ) n 1 we define E x 1 , and recursively In what follows we will denote the Lebesgue measure on R d by λ and for y ∈ R d , |y| 2 its l 2 -norm. We will need the following multi-point variant of Steinhaus's Theorem (see [11]) in R d . Although we expect that this is well-known, we have not found this particular statement in the literature, so we include a proof for completeness.
Proof. We may assume that E is bounded, since if not then there exists a bounded set A such that λ(A ∩ E) ∈ (0, ∞) and applying the theorem to E = A ∩ E yields the claim for E as well.
We claim that if λ(E) > 0 then for every ε > 0 there exists δ(ε) > 0 such that This follows from the proof of Weil's general version of Steinhaus's Theorem in [12]. We repeat parts of the argument here: Since Lebesgue measure is regular we may assume that E is compact and therefore there exists an open set U ⊃ E with λ(U ) < (1 + ε)λ(E). Then there exists δ > 0 such that {v + y : v ∈ E, |y| < δ} ⊂ U . This implies that for any Now we deviate very slightly from the proof in [12]. Note that This proves the claim. Now fix m ∈ N and ε > 0 let δ = δ(ε/m). Let u 1 , u 2 , . . . , u m be such that |u i | < δ for we can prove by induction that λ(∩ k i=1 E u i ) > (1 − kε/m)λ(E). Indeed, the case k = 1 holds, and assuming the result for all j k we have from (9) that Finally, let n be given, and let δ = δ(ε/2 n ). Let y 1 , y 2 , . . . , y n be such that |y i | < δ/n. All partial sums u I = i∈I y i (for I ⊂ [n]) satisfy |u I | < δ. By the result that we have already proved (with m = 2 n ), for such y i , we have In particular, the set on the left is non-empty. This proves the result. Proof of Theorem 2. The set of p m for which γ = 0 has Lebesgue measure 0, so we may assume that γ = 0. By symmetry (apply the result to 1 − p m when γ > 1) we may assume that γ < 1, so we can use the formula (2). The statement is trivial for m = 3 since there is exactly 1 speed for each p m in this case. We fix m 4 in what follows.
Let J m denote the set of permutations of {0, . . . , m − 1} that are not compositions of rotations and reversal. To prove the theorem, it is sufficient to show that for Lebesgue a.e. p m any permutation σ ∈ J m does not give the same velocity, i.e. v(p m ) = v((p m ) σ ). Given a permutation σ of {0, . . . , m − 1}, for i = j, 0 i, j m − 1, we will say that σ(i) is adjacent to σ(j) if σ(i) = σ(j) + 1 or σ(i) = σ(j) − 1, where the sum is mod m. Note that J m is precisely the set of permutations that do not preserve all adjacency relations, i.e. σ ∈ J m if and only if there exists a k ∈ {0, . . . , m − 1} such that σ(k) is not adjacent to σ(k + 1).
Step 2. Note that the terms in (10)  x σ(i+s) Step 3. For each 0 i m − 1, h > 0 and function g : Note that the operator ∆ i h is simply a discrete derivative. Let us describe how iterations of these operators act on products of ρ i , which is a central component of the proof. Consider a function G : R m → R of the form where A ⊂ {0, 1, . . . , m − 1}. It is easy to see that otherwise.
It follows that if #A then For 0 j m − 1 and a permutation σ let Note that x σ (j) and x σ (j+1) are "missing" from this product.

Proof of Theorem 6
In this section we prove Theorem 6. Recall that for r ∈ [m] and a permutation σ ∈ S m we have Suppose that we prove the result for r = k ∈ [m]. Since the entries of a m are decreasing, the reciprocals a −1 m of a m listed in reverse order (write this vector as a † m ) are also decreasing. So we know that the σ greedy maximises P k (·, a † m ). But each P r is trivially invariant to reversals so σ greedy maximises P k (·, a −1 m ). The observation (1) then shows that σ greedy maximises P m−k (·, a m ). It therefore suffices to prove the claim for r = 2, 3. We prove each of these results by induction on m.