Throughout the paper, n ≥ 2 is an integer, T n is the full transformation semigroup on X n = { 1 , ⋯ , n } , S n ⊂ T n is the symmetric group (group of units of T n ), and X denotes the lazy simple random walk on Γ α (defined precisely in Section 2.4).

Theorem1.1 (Characterisation of Eulerian functional digraphs).Let α ∈ T n .Thefunctionaldigraph Γ α isEulerian (i.e. d + ( v ) = d − ( v ) forevery v ∈ X n ) ifandonlyif α ∈ S n .Consequently:

(1) Thenumberof α ∈ T n forwhich Γ α isEulerianis n ! .

(2) Thenumberof α ∈ T n forwhich Γ α isconnectedandEulerianis ( n − 1 ) ! .

EveryEulerianfunctionaldigraph Γ α isself-converse (isomorphictoitsarc-reversal).ThenumberofisomorphismclassesofconnectedEulerianfunctionaldigraphson X n equalsthenumberofintegerpartitionsof n .

Theorem1.2 (Mixing-time bounds). There exist absolute constants c , C > 0 such that for all n ≥ 2 and any α ∈ S n which is a single n -cycle (so that Γ α is a directed n -cycle),

Theorem1.3 (Exploration times). Let α ∈ S n and suppose Γ α is connected. There exist absolute constants c 1 , c 2 > 0 such that for all k ≤ n and all starting vertices v ∈ X n :

(1) (Equidistributed cycle type.) If every cycle of α has the same length, then E v [ T k ] ≤ c 1   k 2 .

(2) (General.) E v [ T k ] ≤ c 2   k 3 .

Here T k = inf { t ≥ 0 : | { X 0 , X 1 , ⋯ , X t } | = k } isthefirsttimethewalkhasvisited k distinctvertices.

Remark1.4. The bounds in Theorems 1.2 and 1.3 are consequences of the general O ( m n ) mixing-time theorem and O ( k 2 ) / O ( k 3 ) exploration-time theorems of [23] applied with m = n (one directed arc per vertex in a functional digraph). The novelty of the present work lies in 1) identifying exactly which elements of T n give rise to connected Eulerian functional digraphs and computing their count, 2) establishing tight bounds for the single-cycle case, 3) demonstrating sensitivity for a natural two-cycle family arising from T n , and 4) the explicit connection between the algebraic cycle structure of S n and the mixing behaviour of the associated random walk.

Theorem1.5 (Sensitivity of mixing). Let n be even and let α n ∈ S n be a permutation of cycle type ( n / 2 , n / 2 ) , so that Γ α n consists of two directed cycles C 1 , C 2 of length n / 2 sharing a common base vertex 0. Define a modified laziness assignment: vertices in the segment [ n / 4 , 3 n / 4 ] (indices modulo n / 2 ) of C 1 have laziness α * = 2 / ( 5 + 1 ) ; all other vertices have laziness 1/2. Then there exist positive constants c 1 , c 2 (independent of n ) such that for all n :

By contrast, with laziness 1/2 uniformly at all vertices, t m i x ≍ n 2 .

Section 2 collects background on T n , functional digraphs, and Markov chain mixing. Section 3 proves Theorem 1.1. Section 4 establishes Theorem 1.2, including the lower bound and a cover-time corollary. Section 5 proves Theorem 1.3. Section 6 proves Theorem 1.5, the sensitivity result.

Let X n = { 1 , 2 , ⋯ , n } . The fulltransformationsemigroup T n is the set of all functions α : X n → X n , equipped with the composition ( α β ) ( x ) = β ( α ( x ) ) . It satisfies | T n | = n n . The image and rank of α ∈ T n are

The symmetricgroup S n ⊂ T n consists of all bijections α : X n → X n ; it has n ! elements and is the (unique) group of units of T n .

Definition2.1. The functionaldigraph of α ∈ T n is Γ α = ( X n , A α ) with A α = { ( x , α ( x ) ) : x ∈ X n } .

Every vertex of Γ α satisfies d + ( v ) = 1 . The in-degree of a vertex y equals | α − 1 ( y ) | . The following structural fact is standard; see, for instance, ([1], Chapter 1).

Proposition2.1 (Structure theorem for functional digraphs).Eachweaklyconnectedcomponentof Γ α containsexactlyonedirectedcycle, togetherwithdirectedtreesrootedatthecyclevertices (withedgesdirectedtowardtheroot).

Example2.2.For n = 4 and α = ( 1 2 3 4 2 3 1 2 ) (so α ( 1 ) = 2 , α ( 2 ) = 3 , α ( 3 ) = 1 , α ( 4 ) = 2 ), thedigraph Γ α hasthedirected 3-cycle 1 → 2 → 3 → 1 withvertex 4 feedingintovertex 2 viathearc (4, 2).Here rank ( α ) = 3 .

The following characterisation is proved in the work [12]; we recall it here for the reader’s convenience.

Proposition2.2 ([12]). Let α ∈ T n . Then:

(1) Γ α is stronglyconnected if and only if rank ( α ) = n , equivalently α ∈ S n .

(2) Γ α is strictlyunilaterallyconnected if and only if rank ( α ) = n − 1 .

(3) Γ α is strictlyweaklyconnected if and only if rank ( α ) ∈ { 2 , ⋯ , n − 2 } and Γ α has exactly one directed cycle.

(4) Γ α is disconnected if and only if Γ α has at least two directed cycles.

Moreover, the number of α ∈ T n for which Γ α is strongly connected is ( n − 1 ) ! .

Definition2.3. Let G = ( V , E ) be a strongly connected directed graph with d + ( v ) ≥ 1 for all v . The lazysimplerandomwalk on G is the Markov chain X = ( X t ) t ≥ 0 on V with transition probabilities

When G is Eulerian (and hence strongly connected for connected G ), the uniform distribution π ( v ) = 1 / n is stationary for P . For an irreducible Markov chain with transition matrix P and stationary distribution π , we define the ℒ ∞ mixingtime (also called the uniformmixingtime)

and the totalvariationmixingtime

These are related by t m i x ≤ t u n i f ; see ([29], Chapter 4).

The main analytic tool is the spectral profile introduced by Goel, Montenegro and Tetali [30]. Let P be the transition matrix of the lazy walk on a connected Eulerian digraph with stationary distribution π . The Dirichletform of a function f : V → R is

For ∅ ≠ S ⊆ V define

and the spectralprofile Λ : [ π * , 1 ] → [ 0 , ∞ ) by

Theorem2.4 {Spectral profile theorem, ([30], Theorem 1.1)}. Let P be an irreducible transition matrix with P ( x , x ) ≥ δ > 0 for all x . Then for every a > 0 ,

A crucial tool for analysing non-reversible Eulerian chains is the following. Let P ^ be the time-reversal of P , defined by π ( v ) P ^ ( v , u ) = π ( u ) P ( u , v ) . On an Eulerian digraph, reversing time is equivalent to reversing all arc directions. The symmetrisation is Q = ( P + P ^ ) / 2 .

Lemma2.1 ([23]).Foranyfunction f : V → R ,

Moreover, Q isreversiblewithstationarydistribution π , sostandardcommute-timeestimatesapplyto Q .

Proof. 1) Since Γ α is functional, d + ( v ) = 1 for all x ∈ X n . The digraph is Eulerian if and only if also d − ( v ) = 1 for all x , i.e. every vertex has exactly one pre-image under α . This is precisely the condition that α is a bijection, so α ∈ S n . Since | S n | = n ! , there are n ! Eulerian functional digraphs.

2) By Proposition 2.2(1), Γ α is strongly connected (equivalently, connected and Eulerian) if and only if α ∈ S n and rank ( α ) = n , which forces Γ α to be a single directed cycle. The number of n -cycles in S n is ( n − 1 ) ! , giving the second count.

3) An Eulerian functional digraph Γ α ( α ∈ S n ) consists of k disjoint directed cycles of lengths λ 1 ≥ ⋯ ≥ λ k with ∑ i λ i = n . Its arc-reversal Γ α rev has the same multiset of directed cycle lengths, so Γ α ≅ Γ α rev (both are unions of directed cycles of the same lengths), hence Γ α is self-converse. Isomorphism classes correspond to cycle-type partitions of n , of which there are p ( n ) (the number of partitions of n ).

Remark3.1. The self-converse property gives a clean semigroup-theoretic meaning to the Eulerian condition: the Eulerian elements of T n are precisely those whose functional digraph is invariant (up to isomorphism) under arc reversal.

Every α ∈ S n has a unique cycle-type partition λ ( α ) = ( λ 1 ≥ ⋯ ≥ λ k ) of n , and Γ α is the disjoint union of k directed cycles of lengths λ 1 , ⋯ , λ k . We record the following for later use.

Proposition3.1.For α ∈ S n , thelazywalkonanyconnectedcomponentof Γ α isirreducible, andtheuniformdistribution π ( v ) = 1 / n (overall v ∈ X n ) isstationaryforthelazywalkon Γ α .

Proof. Each component is a directed cycle, so the walk on it is irreducible, with the uniform distribution on the component being stationary. The global stationary distribution (over disconnected components) is uniform by the bi-stochastic property of the transition matrix.

Remark3.2. Since Theorem 1.2 concerns connectedEulerianfunctionaldigraphs ( Γ α asingle n -cycle, ormoregenerally α ∈ S n withconnected Γ α ), themixing-timeanalysisisrestrictedtothestronglyconnectedcasethroughout Sections 4 and 6.

Lemma4.1 (Spectral profile bound).Let α ∈ S n besuchthat Γ α isconnected (asingledirected n -cycle, m = n ).Forevery r ∈ [ π * , 1 ] ,

Proof. We consider the lazy simple random walk on a directed n -cycle Γ α , which represents the unique connected Eulerian functional digraph topology. Let P denote the transition matrix of this lazy walk, defined by:

The uniform distribution π ( v ) = 1 / n is stationary because the underlying digraph is Eulerian. We evaluate the spectral profile Λ ( r ) introduced in [30]:

where ℰ P ( f , f ) = 1 2 ∑ v , w π ( v ) P ( v , w ) ( f ( v ) − f ( w ) ) 2 is the associated Dirichlet form.

Step1—Symmetrisation. For a non-reversible Markov chain, the spectral profile is controlled by the reversible symmetrisation:

where P ^ is the time-reversal defined by π ( v ) P ( v , w ) = π ( w ) P ( w , v ) . By ([23], Equation (2.3)), the Dirichlet form satisfies the invariant identity:

Consequently, the localized eigenvalue λ ( S ) for P is identical to the corresponding quantity for Q , implying that Λ ( r ) is preserved under symmetrisation. The reversible chain Q corresponds exactly to a simple symmetric random walk on an undirected n -cycle with holding probability 1/2.

Step2—ExittimeestimateviatheDirichletform. For a reversible Markov chain with stationary distribution π , it holds for any set S ⊂ V {see e.g., ([30], Lemma 4.3) or [29]} that:

where τ S c = inf { t ≥ 0 : X t ∉ S } and E v denotes the expectation under the symmetrised chain Q initiated at v . This bound confirms that the worst-case expected exit time from S controls the localized spectral gap.

Step3—Boundingtheexittimebythecommutetime. Under the reversible chain Q , the commute time C ( v , w ) = E v [ τ w ] + E w [ τ v ] satisfies the commute-time identity {cf. ([29], Proposition 10.6)}:

where ℛ eff ( v , w ) is the effective resistance between v and w in the corresponding electrical network. For an undirected n -cycle, the effective resistance between two vertices at distance ℓ is exactly ℓ . Thus, for any v , w ∈ X n , we have:

Let S ⊂ V such that π ( S ) = | S | / n ≤ r . For any v ∈ S , the undirected distance from v to the complement S c is at most | S | , with equality occurring when S forms a contiguous arc. Choosing a vertex w ∈ S c that achieves this minimal boundary distance yields ℓ ( v , w ) ≤ | S | . By the commute-time identity and the fact that τ S c ≤ τ w , we obtain:

Step4—Relating | S | to π ( S ) . Substituting the cardinality relation | S | = n π ( S ) ≤ n r into the exit time bound yields:

Step5—Spectralprofilelowerbound. Applying the exit time maximization to the Dirichlet relation from Step 2 establishes that for every subset S satisfying π ( S ) ≤ r :

Taking the infimum over all such valid subsets yields the uniform lower bound:

as required to establish the uniform mixing time order.

ProofofTheorem1.2. On a single n -cycle, the minimal stationary mass is π * = 1 / n and the laziness parameter is δ = 1 / 2 . Substituting the spectral profile bound (4.1) into the general convergence framework of Theorem 2.4 yields:

Setting a = 1 / 4 delivers t u n i f ≤ 64 n 2 + 2 = O ( n 2 ) , which establishes the upper bound of Theorem 1.2. For the matching lower bound, let α ∈ S n be a single n -cycle. The lazy random walk on this directed cycle is equivalent to a lazy biased random walk on ℤ n that moves clockwise with probability 1/2 and remains stationary with probability 1/2. Starting from a fixed vertex v , the position of the walk at time t concentrates near the expected position v + t 2 ( mod n ) with a standard deviation of order O ( t ) . The distribution achieves total variation mixing only when this standard deviation scales to the order of the space circumference n , which requires t = Ω ( n 2 ) . More precisely, letting μ t denote the distribution of X t initialized at v , choosing t = c 0 n 2 for a sufficiently small absolute constant c 0 confines the probability mass of μ t to an arc of length O ( n ) in ℤ n . This yields the total variation distance inequality ‖ μ t − π ‖ T V ≥ 1 / 4 , which completes the proof.

Corollary4.1.ForthelazyrandomwalkonaconnectedEulerianfunctionaldigraph Γ α ( α ∈ S n , n -cycle),

foranabsoluteconstant c > 0 .

Proof. We use the DFS spanning-tree argument of ([23], Section 4). For the undirected version of Γ α (the n -cycle), the only spanning tree is a path, and its two directed commute times each satisfy Com ( v i , v i + 1 ) ≤ m ⋅ 1 = n by the commute-time bound of ([23], Lemma 4.1). Summing over the n − 1 edges of the spanning path gives E v [ τ c o v ] ≤ ( n − 1 ) ⋅ n ≤ n 2 .

Lemma5.1 (Commute times). Let α ∈ S n with Γ α connected, and let v , w ∈ X n be at undirected distance ℓ in Γ α . Then the commute time Com ( v , w ) = H ( v , w ) + H ( w , v ) satisfies

Proof.Since Γ α isaconnectedEulerianfunctionaldigraphwith α ∈ S n , itisasingledirected n -cycle (Theorem 1.1), which implies m = n . The result follows directly from ([23], Lemma 4.1) with m = n : on each excursion from v back to itself, w is hit with probability at least 1 / d + ( v ) = 1 , so H ( v , w ) ≤ n . Accounting for the path length ℓ along the cycle yields H ( w , v ) ≤ n ( ℓ − 1 ) + n = n ℓ . Both the Eulerian and strong connectivity conditions required by [23] hold uniformly for the directed n -cycle.

Proof.Wemayassume k ≤ n / 2 ; otherwise, theexplorationtimeisboundedbythetotalcovertime E [ τ c o v ] = O ( n 2 ) = O ( k 2 ) via Corollary 4.1. Set t = C k 2 for a constant C to be determined. Since the stationary distribution π ( v ) = 1 / n is uniform (Proposition 3.1) and the transition matrix P is bi-stochastic, it holds for every v ∈ X n that:

where N v ( t ) = | { s < t : X s = v } | counts the visits to v up to time t . By Markov’s inequality, the set of states A = { w : E w [ N v ( t ) ] > s } has cardinality bounded by | A | ≤ t / s .

For v ∈ A , we invoke the return-time bounding technique of ([23], Lemma 5.1). Because the transition matrix on our directed cycle is bi-stochastic, the uniform mass balance ensures that the green-function potential estimates hold without requiring a reversible step. Applying the commute-time bounds from Lemma 5.1 yields:

Optimizing over the threshold parameter by setting s = 10 t gives E v [ N v ( t ) ] ≤ 2 10 t ≤ 7 t . Let v 1 , v 2 , ⋯ be the distinctly visited vertices in chronological order. Applying Markov’s inequality to the total allocation sum ∑ i = 1 k − 1 N v i ( t ) = t , we obtain:

Setting t = 196 k 2 forces this exit probability to be ≤1/2. A standard geometric trials argument then confirms that the total expected exploration time satisfies E v [ T k ] ≤ 2 t = 392 k 2 = O ( k 2 ) .

To obtain the general O ( k 3 ) bound (Theorem 1.3(2)), we control the number of distinct vertices visited by the walk in the worst case, which may scale up to n . The strategy decomposes the walk into phases whose lengths are bounded by the expected hitting times to specific target sets. This decomposition relies on a Hamiltonian cycle in the undirected cube of the graph, a combinatorial property independent of edge orientation.

Lemma5.2 (Hamiltonian cycle in G 3 ).Foranyconnectedundirectedgraph G , thegraph G 3 (whereverticesareadjacentiftheirdistancein G isatmost 3) containsaHamiltoniancycle.

Proof. See [23]. The proof proceeds via induction on a spanning tree to construct an explicit vertex ordering.

Fix a Hamiltonian cycle v 1 , v 2 , ⋯ , v n , v n + 1 = v 1 in the undirected cube of Γ α . Because the underlying digraph is a directed n -cycle, the undirected distances between vertices are given by the minimal arc distance around the cycle, ensuring that d u n d i r ( v i , v i + 1 ) ≤ 3 for all i . Following the frameworks of [23] and [25], we implement a phase decomposition of the walk on the original directed n -cycle, utilizing the fixed ordering of the undirected cube to categorize vertices.

Phasedefinition. At the beginning of phase i , let Y i denote the set of previously visited vertices. The final vertex visited in phase i − 1 is s i , and its successor on the Hamiltonian cycle according to the fixed ordering is r i . A vertex v j ∉ Y i is defined as good in phase i if for every ℓ ≤ n :

where indices are taken modulo n . Let U i denote the set of good vertices, and B i = V ∖ ( U i ∪ Y i ) denote the set of bad vertices. Phase i terminates when the walk either reaches r i or hits any vertex in U i . This configuration ensures that each successive phase either advances along the cycle ordering or discovers a good vertex.

Lemma5.3 (Number of phases, [25]). At most 2 k phases are required before k distinct vertices are visited. Moreover, if | Y i | ≤ n / 2 , then | B i | ≤ | Y i | .

Proof. See [23] or the original combinatorial formulation in [25].

Lemma5.4 (Phase length bound). For a connected Eulerian functional digraph, let W ⊔ Z = X n be a partition such that W induces a connected subgraph, and let s ∈ W ∩ ∂ Z under undirected adjacency. Then

Proof. We specialize the proof of [23] to the functional case where m = n . Let

Because each vertex in the directed n -cycle satisfies d + ( v ) = 1 , the condition d + ( v ) ≤ 2 | W | holds automatically for all | W | ≥ 1 . Thus, W 2 = ∅ and W 1 = W . For any s ∈ W ∩ ∂ Z , since W 2 is empty, we apply the general Eulerian contraction bound from [23]:

The factor 2 | E ( W 1 , Z ∪ W 2 ) | is bounded by 4 | W | 2 because each vertex has degree 2 on the underlying cycle. To maintain structural consistency with universal bounds in the literature and account for boundary configurations where W 2 is non-empty, we retain the conservative assignment H ( s , Z ) ≤ 12 | W | 2 .

Proof. If 2 k > n , the expectation E [ T k ] is bounded directly by the global cover time, which via Corollary 4.1 satisfies E [ T k ] ≤ c n 2 ≤ 8 c k 3 . We therefore assume for the remainder of the proof that 2 k ≤ n .

Let Φ i denote the length of phase i executed prior to visiting k distinct vertices. The phase starts at s i with target set Z i = { r i } ∪ U i , where r i is the successor of s i on the Hamiltonian cycle and U i represents the good vertices. The complement W i = Y i ∪ B i ∖ { r i } satisfies the conditions of Lemma 5.4 with Z = Z i . Because the Hamiltonian cycle lies in the undirected cube G 3 , the undirected distance satisfies d u n d i r ( s i , r i ) ≤ 3 . Mapping these tracking constraints to the general Eulerian framework of ([23], Theorem 1.8, proof), the phase lengths are governed by the quadratic volume of the transient subsets, yielding the conditional estimate:

The scalar factor of 3 accounts for the maximum path steps required to transition from s i to r i or to an active vertex in U i . By Lemma 5.3, at most 2 k phases are required to identify k distinct vertices. Summing over the active phases yields:

establishing the cubic exploration bound.

For reversible Markov chains, Peres and Sousi [26] proved that the mixing time is robust to changes of laziness parameters: if the laziness at each vertex is in [ c 1 , c 2 ] ⊂ ( 0 , 1 ) , the mixing time changes only by a constant multiplicative factor. In this section, we show this robustness fails for the Eulerian functional digraphs of T n .

Let n be even and fix α n ∈ S n with cycle type ( n / 2 , n / 2 ) . Label the two directed cycles C 1 = ( 0 , 1 , ⋯ , n / 2 − 1 , 0 ) and C 2 = ( 0 , n / 2 , n / 2 + 1 , ⋯ , n − 1 , 0 ) , both of length n / 2 , sharing vertex 0. Define the modifiedlaziness assignment:

At each vertex v , the modified lazy walk stays at v with probability p lazy ( v ) and moves to the unique out-neighbour of v on its cycle with probability 1 − p lazy ( v ) ; at vertex 0 it also chooses one of the two cycles uniformly at random.

Remark6.1. The choice α * = 2 / ( 5 + 1 ) is the reciprocal of the golden ratio ϕ = ( 5 + 1 ) / 2 . It satisfies 1 / ( 1 − α * ) = ( 5 + 1 ) / ( 5 − 1 ) and makes the ratio of expected excursion times r * = E [ T 1 ] / E [ T 2 ] = ( 7 + 5 ) / 8 an irrational number whose continued-fraction partial quotients are bounded (in fact by 4). The three-distance theorem (Lemma 6.2) quantifies the distribution of fractional parts { k ξ } for an irrational ξ : the maximal gap between consecutive points among { ξ , 2 ξ , ⋯ , n ξ } is at most c B / n , and any interval of length 1 / n contains at most B + 2 points, where B is an upper bound on the partial quotients of ξ . If the partial quotients were unbounded, the constants in these estimates would depend on n in an uncontrollable way (e.g., a huge partial quotient could create a much larger gap). Bounded partial quotients guarantee that the constants c and B are absolute (independent of n ), which is essential for the local limit estimates in Lemma 6.3 and ultimately for establishing the polynomial mixing lower bound t m i x ≥ c n 3 / 2 .

For the walk X on Γ α n with the modified laziness (6.1), define a round as the time elapsed from a visit to 0 to the next return to 0 after having first hit the midpoint ( n / 4 for C 1 , 3n/4 for C 2 ). Let T 1 (respectively T 2 ) denote the duration of a round on C 1 (respectively C 2 ).

Lemma6.1 (Expected excursion times). There is a function f ( n ) = 3 n 4 ( 1 + O ( 2 − n / 4 ) ) such that

In particular, the ratio r * : = E [ T 1 ] / E [ T 2 ] = ( 2 + 1 / ( 1 − α * ) ) / 4 is irrational with bounded continued-fraction coefficients.

Proof. The expected and variance formulae follow by decomposing each excursion into per-vertex residence times and applying Wald’s identity, exactly as in ([23], Lemma 4.4). The explicit computation gives

from which the asymptotic f ( n ) = 3 n 4 ( 1 + O ( 2 − n / 4 ) ) is immediate. The irrationality of r * follows from α * = 2 / ( 5 + 1 ) , which gives 1 / ( 1 − α * ) = ( 5 + 1 ) / ( 5 − 1 ) = ( 3 + 5 ) / 2 , so r * = ( 7 + 5 ) / 8 , an irrational algebraic number with bounded partial quotients.

Let ( ξ i ) i ≥ 1 be i.i.d. Bernoulli (1/2) random variables (indicating which cycle is visited in round i ), let ( T i 1 ) i ≥ 1 and ( T i 2 ) i ≥ 1 be independent i.i.d. copies of T 1 and T 2 respectively, all mutually independent. Define the cumulativeroundtime

Lemma6.2 (Three-distance theorem).Let ξ beanirrationalnumberwithallcontinued-fractionpartialquotientsboundedby B .Let x k = { k ξ } : = k ξ mod 1 .Then:

1) Themaximumgapofthesequence { x 1 , ⋯ , x n } in [ 0 , 1 ) isatmost c B / n forauniversalconstant c .

2) Everyintervaloflength 1 / n in [ 0 , 1 ) containsatmost B + 2 elementsofthesequence.

Proof. This is a quantitative form of the classical three-distance (Steinhaus) theorem; see ([21], inequality (3.17)).

Lemma6.3 (Return-time distribution). For the walk X with modifiedlaziness (6.1), for all t ∈ [ n 3 / 2 / 50 ,   10 n 3 / 2 ] ,

Proof. We work with the walk X on the two-cycle graph Γ α n with the modified laziness (5.1). Recall the round construction: between successive visits to the common vertex 0, the walk performs an excursion entirely on C 1 or C 2 , with durations T 1 and T 2 respectively. Let ( ξ i ) i ≥ 1 be i.i.d. Bernoulli(1/2) (choose C 1 if ξ i = 1 , C 2 if 0), independent of the excursion durations. Define

where ( T i 1 ) i ≥ 1 and ( T i 2 ) i ≥ 1 are i.i.d. copies of T 1 and T 2 respectively, independent of each other and of ( ξ i ) . The return time to 0 after k rounds is exactly S k . We aim to prove that for all t in the range [ n 3 / 2 / 50 ,   10 n 3 / 2 ] ,

Step1—Decomposeintomeanplusfluctuation. Write T i j = μ j + D i j , where μ j = E [ T j ] and D i j are zero-mean fluctuations. From Lemma 6.1,

μ 1 = ( 2 + 1 / ( 1 − α * ) ) f ( n ) , μ 2 = 4 f ( n ) , and V a r ( D i j ) ≍ n . Let L k = ∑ i = 1 k ξ i be the number of rounds on C 1 among the first k . Then

Step2—LocalCLTforthefluctuations. Because D i j have finite variance of order n and are independent, we apply a local central limit theorem for lattice distributions (see e.g. [31]). For any integers x , y with | x | , | y | = O ( n 1 / 4 ) and k ≍ n , the binomial distribution of L k and the conditional distribution of the fluctuation Y k yield:

The constants implicit in ≍ are absolute because the component distributions possess bounded third moments, ensuring uniform convergence bounds in the local limit regimes.

Step3—Reformulatetheevent S k = t . From the structural decomposition, the target configuration satisfies:

Let d = μ 1 − μ 2 . Note that d = μ 2 ( r * − 1 ) where r * = μ 1 / μ 2 = ( 7 + 5 ) / 8 , which is an algebraic irrational with bounded partial quotients. Set ι = d / ( μ 1 + μ 2 ) = ( r * − 1 ) / ( r * + 1 ) , which shares this Diophantine property. Isolating the excursion count yields: