Markov Chain

Why canonical paths matter

A finite Markov chain moves among states according to transition probabilities. Its second largest eigenvalue measures how slowly some variation in the starting state can disappear. Directly computing that eigenvalue is often difficult when the state space is large. A geometric bound can be more useful: choose a route between each ordered pair of states and measure how much weighted traffic the routes impose on each transition. Diaconis and Stroock present this route-based bound as Proposition 7 of their 1991 paper, following the canonical-path method of Jerrum and Sinclair (Diaconis–Stroock, §3B). It makes a spectral estimate depend on the geometry of chosen routes and the stationary weight of each edge.

The paper's purpose is to compare methods for bounding eigenvalues of reversible chains. Its earlier Poincaré estimate uses weighted path lengths. Proposition 7 instead passes through a bottleneck constant and Cheeger's inequality. These are different summaries of the same chain: one examines a system of paths, and the other examines flow across every cut. The comparison is useful on graphs where paths are easy to choose but the cut minimizing the bottleneck constant is not apparent (Diaconis–Stroock, pp. 52–55).

Finite reversible chains and directed path congestion

Let $V$ be a finite state space with at least two states. A transition matrix $P$ has entries $P(x,y)\geq 0$ and each row sums to one. It is irreducible when every state can be reached from every other state in some number of steps. A stationary distribution $\pi$ is a nonnegative probability vector satisfying $\pi P=\pi$. The chain is reversible when it satisfies detailed balance: $\pi(x)P(x,y)=\pi(y)P(y,x)$. Write $Q(x,y)=\pi(x)P(x,y)$ for the stationary edge flow. These are the standing assumptions of the paper's §1A, restated in §3B (Diaconis–Stroock, pp. 36, 54).

For every distinct ordered pair $(x,y)$, choose a finite walk $\gamma_{xy}$ from $x$ to $y$. Every consecutive step $(u,v)$ of the walk must have $Q(u,v)>0$. The walk may revisit a vertex or an edge; §3B places no no-repetition condition on these choices. An oriented edge is an ordered pair $(u,v)$ with positive flow. A walk contains it when one of its consecutive steps traverses it in that direction. The canonical-path congestion of the chosen walk system $\Gamma$ is the paper's quantity

$$\eta(\Gamma)=\max_{e:Q(e)>0}\frac{1}{Q(e)} \sum_{x\ne y:\,e\in\gamma_{xy}}\pi(x)\pi(y).$$

The sum assigns weight $\pi(x)\pi(y)$ to an ordered pair, once for each directed edge that its chosen walk uses. Irreducibility ensures that walks between different states exist. Under the mission's assumptions the positive-flow edge set is nonempty, so the maximum has its ordinary finite meaning (Diaconis–Stroock, (3.2), p. 54).

For a nonempty set $S\subseteq V$ with $\pi(S)\leq\tfrac12$, let $Q(S\times S^{\mathrm c})$ be the total stationary flow from $S$ to its complement. The bottleneck constant is

$$h=\min_{\varnothing\ne S\subseteq V,\,\pi(S)\leq 1/2} \frac{Q(S\times S^{\mathrm c})}{\pi(S)}.$$

The denominator is positive under irreducibility. Write $\beta_1$ for the largest eigenvalue of $P$ other than the simple eigenvalue $1$, and write $\lambda_1=1-\beta_1$ for the spectral gap. These conventions match §1A and equation (3.4) (Diaconis–Stroock, pp. 36, 56).

Formalization targets

The mission's goal is Proposition 7, for every valid choice of $\Gamma$:

$$\beta_1\leq 1-\frac{1}{8\eta(\Gamma)^2}.$$

Its milestones record the source's cut argument: a selected walk from $S$ to $S^{\mathrm c}$ crosses an oriented cut edge; the total pair weight obeys $\pi(S)\pi(S^{\mathrm c})\leq\eta Q(S\times S^{\mathrm c})$; and consequently $h\geq 1/(2\eta)$. The final milestone is the already formalized version of the paper's equation (3.4), $h^2/2\leq\lambda_1\leq 2h$ (Diaconis–Stroock, pp. 54, 56).

The companion result is the graph specialization (3.3). For simple random walk on a connected graph $G=(V,E)$, let $d_*$ be the maximum degree and let $b$ be the maximum number of ordered-pair walks traversing an oriented edge. It gives

$$\beta_1\leq 1-\frac12\left(\frac{|E|}{d_*^2b}\right)^2.$$

This statement uses the exact displayed bound of Remark 3.1; the earlier sentence in that remark uses a different normalization of $\eta$ (Diaconis–Stroock, p. 55).

What the bound supplies

The proposition converts any usable path system into an explicit spectral estimate. Once a graph or chain supplies paths whose edge loads can be bounded, the formula bounds $\beta_1$ without diagonalizing $P$ or finding the set that minimizes $h$. In the graph case, equation (3.3) expresses the estimate using only the edge count, maximum degree, and maximum directed path load. The paper compares that bound with its Poincaré bound on examples including cycles, cubes, and binary trees (Diaconis–Stroock, pp. 55–56).

Equation (3.4) already has a proved formalization in the published MarkovMixing library, under the same finite reversible-chain conventions. This mission adds the paper's path-congestion definition and cut estimates, then states the still-needed Proposition 7 and graph specialization against that library. The result is a reusable connection between a chosen family of walks and a chain's existing bottleneck and spectral objects.

Where the difficulty lies

The route load $\eta$ concerns individual directed edges, whereas $h$ takes a minimum over all admissible cuts. A single heavily loaded edge does not by itself identify the minimizing cut. The central difficulty is to control, for every cut at once, all weighted ordered pairs whose walks leave it. The direction of a crossing matters: $Q(S\times S^{\mathrm c})$ counts edges directed out of $S$, and $\eta$ must count traversals of those same orientations. Omitting an orientation or allowing negative unused edge flow would invalidate the intermediate cut estimate.

There is also a printed discrepancy. Proposition 6 on p. 52 displays a stronger upper bound on $\beta_1$ than the inequality established in §3C. The mission uses equation (3.4), whose lower gap bound yields the $1/(8\eta^2)$ constant actually stated in Proposition 7 (Diaconis–Stroock, pp. 52, 54, 56).

Formalization scope

Lean represents the state space as a finite type, the chain as a real matrix, and the stationary distribution as a real function. It imports the published MarkovMixing definitions of stochasticity, irreducibility, stationarity, detailed balance, $Q$, $h$, $\beta_1$, and simple random walk. A local definition represents each selected walk by its vertex list and derives directed edges from consecutive vertices. The hypothesis $2\leq |V|$ makes $\beta_1$ the paper's second eigenvalue and excludes the one-state junk value of the real supremum used in its definition. Positivity of $\eta$ follows from the chain and walk assumptions; it is not inserted as an extra assumption.

The walk system is universally quantified: the theorem concerns every valid selection, rather than one favorable selection. The local definition ignores equal-endpoint choices because they do not cross any $S$-to-$S^{\mathrm c}$ cut; §3B mentions all ordered pairs, so this convention gives a slightly stronger target and is stated explicitly. The cut lemma retains stochasticity and nonnegative stationary weights to keep the flow of unused cut edges nonnegative. The graph companion derives its Markov-chain properties from connectedness rather than assuming them.

The shared vertex-list representation and the cut crossing statement can serve later path-based bounds. Contributions toward positivity and finiteness of congestion, the uniform cut estimate, the bottleneck estimate, and the graph specialization all fit the mission. A definition that forces $\eta>0$ by fiat, or a goal that assumes the bottleneck estimate or Cheeger's inequality, would erase the mathematical content.

Selected references

• Persi Diaconis and Daniel Stroock, Geometric bounds for eigenvalues of Markov chains, Annals of Applied Probability 1(1):36–61, 1991. DOI.
• David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, 2009, Chapter 13. Book PDF.

Motivation

Markov chain Monte Carlo methods, card-shuffling schemes and randomized counting algorithms all run a Markov chain until its distribution is close to the stationary distribution. The quantity a user needs is a number of steps $n$ after which $P^n(x,\cdot)$ is within a prescribed variation distance of $\pi$. For reversible chains the standard way to get such a number is to bound the eigenvalues of the transition matrix and convert the eigenvalue bound into a variation distance bound.

P. Diaconis and D. Stroock, in Geometric bounds for eigenvalues of Markov chains (Ann. Appl. Probab. 1, 1991), prove geometric (path-counting) bounds on the second largest and the smallest eigenvalue of a reversible chain. Their §1D supplies the conversion step: Proposition 3 bounds the variation distance to stationarity, in discrete and in continuous time, by the eigenvalues. This mission formalizes that proposition. The eigenvalue bounds themselves are the subject of the other missions of this series.

Setting

Let $X$ be a finite set with $m = |X|$ elements and $P(x,y)$ the transition matrix of a Markov chain on $X$: $P(x,y)\ge 0$ and $\sum_y P(x,y) = 1$. The chain is irreducible if every state can be reached from every other state: for all $x,y$ there is $n$ with $P^n(x,y) > 0$. A probability distribution $\pi$ on $X$ is stationary if $\sum_x \pi(x)P(x,y) = \pi(y)$, and the chain is reversible with respect to $\pi$ if

$$\pi(x)P(x,y) = \pi(y)P(y,x)\qquad\text{for all } x,y\in X.$$

For an irreducible chain $\pi(x) > 0$ for every $x$. Reversibility makes $P$ a self-adjoint operator on $L^2(\pi)$, so its eigenvalues are real; they are written

$$1 = \beta_0 > \beta_1 \ge \beta_2 \ge \dots \ge \beta_{m-1} \ge -1,$$

the strict inequality $\beta_0 > \beta_1$ coming from irreducibility. Put $\beta_* = \max(\beta_1, |\beta_{m-1}|)$, the largest modulus of an eigenvalue other than $1$.

The variation distance between probability distributions $\mu$ and $\pi$ on $X$ is

$$\|\mu - \pi\|_{\mathrm{Var}} = \max_{A\subset X}|\mu(A) - \pi(A)| = \tfrac12\sum_{x\in X}|\mu(x)-\pi(x)|.$$

$P^n(x,\cdot)$ is the distribution of the chain after $n$ steps from $x$. The continuous-time chain jumps according to $P$ at the times of a rate-one Poisson process, and has kernel

$$P_t(x,y) = e^{-t}\sum_{n=0}^{\infty}\frac{t^n}{n!}P^n(x,y).$$

Formalization targets

Goal: Proposition 3, (1.9)

For every $x \in X$ and every $n\in\mathbb N$,

$$4\,\|P^n(x,\cdot)-\pi\|_{\mathrm{Var}}^2 \;\le\; \frac{1-\pi(x)}{\pi(x)}\,\beta_*^{2n}.$$

Milestones

The steps of the paper's proof (p. 42), in order:

1. the two expressions for $\|\cdot\|_{\mathrm{Var}}$ (p. 41), already proved on the platform;
2. the reversibility identity $\frac{1}{\pi(x)}P^{2n}(x,x) = \sum_y (P^n(x,y))^2/\pi(y)$;
3. the Cauchy–Schwarz bound $4\|P^n(x,\cdot)-\pi\|_{\mathrm{Var}}^2 \le \frac{1}{\pi(x)}P^{2n}(x,x) - 1$;
4. the orthogonal diagonalization of a reversible chain, already proved on the platform in the form $P^t(x,y)/\pi(y) = \sum_j f_j(x)f_j(y)\lambda_j^t$ with $(f_j)$ orthonormal in $L^2(\pi)$;
5. the diagonal bound $P^{2n}(x,x) \le \pi(x) + \beta_*^{2n}(1-\pi(x))$.

Companion: Proposition 3, (1.10)

For every $x\in X$ and every $t\in(0,\infty)$,

$$4\,\|P_t(x,\cdot)-\pi\|_{\mathrm{Var}}^2 \;\le\; \frac{1-\pi(x)}{\pi(x)}\,\exp\bigl[-2(1-\beta_1)t\bigr].$$

The paper gives no separate proof of (1.10); it says only that "the continuous time version can be proved by a virtually identical argument".

Significance

Inequality (1.9) is the step that turns an eigenvalue estimate into a mixing-time estimate: if $\beta_* \le 1 - \varepsilon$, then $\|P^n(x,\cdot)-\pi\|_{\mathrm{Var}} \le \delta$ as soon as $n \ge \frac{1}{2\varepsilon}\bigl(\log\frac{1}{4\delta^2} + \log\frac{1-\pi(x)}{\pi(x)}\bigr)$, up to the usual comparison $\log(1/(1-\varepsilon)) \ge \varepsilon$. The examples of the paper's Section 2 use Propositions 1 and 2 to bound $\beta_1$ and $\beta_{m-1}$, and this proposition to translate the bounds into a number of steps. Inequality (1.10) does the same for the continuous-time chain, where only $\beta_1$ is needed because the continuous-time chain has no periodicity.

The result is classical and its proof is short; it is the $\ell^2$ (chi-square) bound on variation distance found in many textbooks, e.g. Levin, Peres and Wilmer, Markov Chains and Mixing Times, Chapter 12. As far as the platform's catalog shows, neither (1.9) nor (1.10) has a machine-checked proof. The platform already has the spectral decomposition of a reversible chain and the identity between the two forms of total variation, both proved; this mission links them to the variation distance bound in the exact form used by the geometric-bounds literature. The diagonal bound (milestone 5) and the reversibility identity (milestone 2) are reusable statements about reversible chains in their own right.

Difficulty

The Cauchy–Schwarz step and the reversibility identity are finite-sum manipulations. The difficulty is in the spectral step. One has to show that, apart from the eigenvalue $1$ (whose eigenfunction is constant and contributes exactly $\pi(x)$ to $P^{2n}(x,x)$), every eigenvalue in the decomposition has modulus at most $\beta_*$, and that the remaining weights at $x$ sum to $1-\pi(x)$. This uses that $1$ is a simple eigenvalue, which needs irreducibility. Without irreducibility the bound fails. A second eigenvalue equal to $1$ would contribute $\Gamma_{xw}^2$ with no decay, while $\beta_*$, a maximum over eigenvalues different from $1$, does not see it.

A naive approach that bounds $|P^n(x,y)-\pi(y)|$ entry by entry and sums gives a factor depending on $|X|$ rather than on $\pi(x)$; the point of (1.9) is that the constant is $(1-\pi(x))/\pi(x)$ only. For (1.10) the matrix $P_t$ is an infinite series in $P$, and the decomposition has to be carried through the series, each eigenvalue $\beta$ contributing $e^{-2(1-\beta)t}$.

Formalization scope

The objects are the published MarkovMixing definitions (Levin–Peres–Wilmer layer): the chain is a real matrix P : Matrix V V ℝ on a Fintype V with IsStochastic P, MarkovMixing.Irreducible P, IsStationary P π (π is a probability vector with πP = π) and DetailedBalance P π. These four hypotheses are the paper's standing assumption of §1A; every statement of the mission carries all of them, and no other hypothesis.

• $\|P^n(x,\cdot)-\pi\|_{\mathrm{Var}}$ is tvDist (rowDist P n x) π, the supremum over subsets; $P_t(x,\cdot)$ is heatKernel P t x.
• $\beta_*$ is lambdaStar P, the supremum of $|\lambda|$ over real eigenvalues $\lambda \ne 1$; $\beta_1$ is lambdaTwo P, the supremum of real eigenvalues $\lambda\ne1$. For an irreducible reversible chain these are the paper's $\max(\beta_1,|\beta_{m-1}|)$ and $\beta_1$.
• On a one-point space the eigenvalue sets are empty and both quantities are the junk value $0$; both sides of (1.9) and (1.10) are then $0$, so no hypothesis $|X|\ge 2$ is added.
• $n\in\mathbb N$ includes $n = 0$, where (1.9) reads $4(1-\pi(x))^2 \le (1-\pi(x))/\pi(x)$, a true inequality. The time in (1.10) is strictly positive, as in the paper.
• Positivity of $\pi$ is not a hypothesis: it follows from irreducibility and stationarity.

A formalization that assumed milestone 3 or milestone 5 as a hypothesis of the goal, or that replaced $\beta_*$ by a free parameter bounded by hypothesis, would trivialize the goal and is excluded.

Needed infrastructure: finite-dimensional spectral theory for self-adjoint operators on $L^2(\pi)$ (available through the proved MarkovMixing.spectral_representation), simplicity of the eigenvalue $1$ for irreducible chains, Cauchy–Schwarz for finite sums, and termwise manipulation of the Poisson series for (1.10). Proofs of any milestone, and of the simplicity of the top eigenvalue as a separate lemma, are welcome.

Selected references

• P. Diaconis and D. Stroock, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab. 1(1):36–61, 1991. https://doi.org/10.1214/aoap/1177005980
• D. A. Levin, Y. Peres and E. L. Wilmer, Markov Chains and Mixing Times, 2nd ed., American Mathematical Society, 2017 (Lemma 12.2, Proposition 4.2). https://doi.org/10.1090/mbk/107

Why reversibility

A stochastic process is reversible when a film of it run backwards is statistically indistinguishable from the film run forwards. For Markov processes in equilibrium this distributional symmetry has an algebraic counterpart, the detailed balance conditions, and that counterpart is what makes large classes of queueing networks, migration processes, loss networks, clustering processes and population-genetics models solvable in closed form. F. P. Kelly's Reversibility and Stochastic Networks (Wiley, 1979) builds the whole theory of product-form equilibria on this link, and Chapter 1 sets it up.

The criterion that bears Kolmogorov's name goes back to A. Kolmogorov, "Zur Theorie der Markoffschen Ketten" (Math. Annalen 112, 1936), who showed that reversibility of a chain can be read off its transition probabilities around closed cycles. Kelly's Chapter 1 (§§1.1–1.7) states the criterion for chains (Theorem 1.7) and processes (Theorem 1.8), together with the detailed-balance characterization (Theorems 1.2, 1.3), the cut and tree lemmas (1.4, 1.5), the monotonicity of relative entropy (Theorem 1.6), truncation and rate alteration (Lemma 1.9, Corollary 1.10), and the description of the time-reversed process (Theorems 1.12–1.14). This mission is the first in a series covering the book.

Setting

Let $\mathcal S$ be a finite state space. A continuous-time Markov process $X(t)$, $t\in\mathbb R$, is specified by transition rates $q(j,k)\ge 0$, $j\ne k$, with Kelly's convention $q(j,j)=0$. Its generator is the matrix $Q$ with $Q(j,k)=q(j,k)$ off the diagonal and $Q(j,j)=-\sum_{k\ne j}q(j,k)$, and its transition matrices are $P(t)=e^{tQ}$. The rates are irreducible if every state can be reached from every other through transitions of positive rate. An equilibrium distribution is a collection of positive numbers $\pi(j)$ summing to one that satisfy the equilibrium equations

$$\pi(j)\sum_{k}q(j,k)=\sum_k\pi(k)q(k,j),\qquad j\in\mathcal S. \qquad (1.3)$$

The stationary process with rates $q$ and equilibrium distribution $\pi$ has finite-dimensional distributions

$$P\bigl(X(t_1)=j_1,\dots,X(t_n)=j_n\bigr)=\pi(j_1)\prod_{r=1}^{n-1}P(t_{r+1}-t_r)(j_r,j_{r+1}),\qquad t_1\le\dots\le t_n .$$

It is reversible (Kelly, p. 5) if $(X(t_1),\dots,X(t_n))$ has the same distribution as $(X(\tau-t_1),\dots,X(\tau-t_n))$ for all $t_1,\dots,t_n,\tau$. The rates satisfy detailed balance with $\pi$ if $\pi(j)q(j,k)=\pi(k)q(k,j)$ for all $j,k$, and Kolmogorov's cycle condition if

$$q(j_1,j_2)q(j_2,j_3)\cdots q(j_{n-1},j_n)q(j_n,j_1)=q(j_1,j_n)q(j_n,j_{n-1})\cdots q(j_3,j_2)q(j_2,j_1)\qquad (1.22)$$

for every finite sequence of states $j_1,\dots,j_n$. The discrete-time analogue replaces rates by a stochastic matrix $p(j,k)$ and $P(t)$ by the matrix power $P^t$, $t\in\mathbb Z_{\ge 0}$.

Formalization targets

Goal: Theorem 1.8 (p. 23)

For a stationary, irreducible Markov process on a finite state space,

$$X\ \text{is reversible}\iff q\ \text{satisfies (1.22) for every finite sequence of states}.$$

The left side is a statement about the joint laws of the process at all finite sets of times; the right side involves the rates alone, not even the equilibrium distribution.

Milestones

1. Theorems 1.2 and 1.3: reversibility of a stationary chain or process is equivalent to the existence of a positive, normalized solution of detailed balance, and that solution is the equilibrium distribution.
2. Theorem 1.7: Kolmogorov's criterion (1.21) for chains. (The platform's proved rate-level equivalence of (1.22) with detailed balance under two-way communication, Serfozo's Theorem 2.8, is included as a reference item but is not a milestone: it is not Kelly's statement.)
3. Lemmas 1.4 and 1.5: the flux across any cut balances, and a process whose graph is a tree is reversible.
4. Theorem 1.6: $H(t)=\sum_j\pi(j)h(u_j(t)/\pi(j))$ is strictly increasing for $t>0$ when $h$ is strictly concave and the initial distribution is not the equilibrium one.
5. Lemma 1.9 and Corollary 1.10: altering the rates across a cut by a factor $c>0$, or truncating to a subset, preserves reversibility with explicit equilibrium distributions.
6. Theorems 1.12–1.14: the reversed process $X(\tau-t)$ is stationary Markov with rates $q'(j,k)=\pi(k)q(k,j)/\pi(j)$; conditions (1.27)–(1.28) identify it; and dynamic reversibility is characterized by $\pi(j)=\pi(j^+)$ and $\pi(j)q(j,k)=\pi(k^+)q(k^+,j^+)$.

Significance

Theorem 1.8 is the working test for reversibility throughout the book and the queueing literature: a model's reversibility, and with it a product-form equilibrium obtained by solving detailed balance, can be decided by checking a finite list of cycles in its transition diagram. Theorem 1.3 converts the distributional property into equations that later chapters solve explicitly; Theorems 1.12 and 1.13 underlie the treatment of quasi-reversible queues and networks in Chapter 3; Lemma 1.9 and Corollary 1.10 produce the equilibria of loss systems and queues with shared buffers.

The algebraic content of several of these results is already proved on the platform at the level of rates: detailed balance implies the equilibrium equations, the reversed rates preserve $\pi$, truncation preserves detailed balance, and Kolmogorov's criterion is equivalent to detailed balance for two-way communicating rates. What is not formalized anywhere is the process-level statement: that these conditions are equivalent to the time-reversal symmetry of the finite-dimensional distributions built from $e^{tQ}$. This mission supplies that layer, which connects the rate identities to the probabilistic notion they are meant to capture.

Difficulty

The rate-level identities are short; the difficulty is the passage between them and the process. Reversibility constrains the joint law at every finite set of times, and that law is built from the matrix exponential $e^{tQ}$, whose entries are not explicit functions of the rates. Relating the two requires the analytic facts about $e^{tQ}$ for a generator (stationarity of $\pi$, the semigroup property, behaviour as $t\to 0$, strict positivity of the entries for $t>0$ under irreducibility) that Mathlib does not yet provide for Markov generators, together with bookkeeping for tuples of times given in arbitrary order, with ties. For Theorem 1.8 a positive, normalized equilibrium has to be produced from the cycle condition alone, on rates that may vanish in one direction only: two-way communication is not a hypothesis, so the platform's proved rate-level criterion does not apply directly. A first attempt that defines reversibility as detailed balance avoids all of this and proves nothing new; it is excluded below.

Formalization scope

The state space is a Fintype with decidable equality; Kelly allows a countable state space, and this restriction is stated in every item. Rates are q : S → S → ℝ with 0 ≤ q j k for j ≠ k and q j j = 0 as hypotheses. The transition matrices are NormedSpace.exp (t • generator q). Time sets are ℝ (processes) and ℤ (chains). Finite-dimensional distributions are defined for arbitrary finite tuples of time points by sorting them with Tuple.sort. Equilibrium means positive, summing to one, and satisfying (1.3) (resp. $\pi P=\pi$), and every theorem takes the equilibrium distribution of the stationary process as a hypothesis. Existing platform definitions are reused: FullBalance, DetailedBalance and reversedRates from KellyStochasticNetworks_Balance, the stochastic-matrix vocabulary of mm_basic, and truncatedRates from KellyStochasticNetworks_LossNetwork.

Reversibility is not defined as detailed balance or as equality of $q$ with its reversed rates: under such a definition Theorems 1.2 and 1.3 would be tautologies and Theorem 1.8 would be the already proved rate-level criterion. It is the distributional definition of p. 5. Kolmogorov's condition ranges over all sequences of all lengths, repetitions allowed, and two-way communication is not assumed.

A complete development needs the matrix exponential of a generator (positivity, the semigroup property, the derivative at $0$, and $\pi e^{tQ}=\pi$), which is reusable for any finite-state continuous-time Markov chain, and a lemma on reversing sorted tuples. Lemma 1.1 (a general stationary process) and Lemma 1.11 (a non-stationary reversal) are not included. Proofs of any milestone, and generalizations to countable state spaces, are welcome.

Selected references

• F. P. Kelly, Reversibility and Stochastic Networks, John Wiley & Sons, 1979, Chapter 1. https://www.statslab.cam.ac.uk/~frank/BOOKS/kelly_book.html
• A. Kolmogorov, "Zur Theorie der Markoffschen Ketten", Mathematische Annalen 112 (1936), 155–160. https://doi.org/10.1007/BF01565412
• R. Serfozo, Introduction to Stochastic Networks, Springer, 1999, Chapter 1. https://doi.org/10.1007/978-1-4612-1482-3
• F. P. Kelly and E. Yudovina, Stochastic Networks, Cambridge University Press, 2014, Chapter 1. https://doi.org/10.1017/CBO9781139565363

Motivation

For many natural chains, convergence to stationarity is not gradual: the distance stays near its maximum for a long time and then collapses to zero in a comparatively negligible window. A deck of cards under riffle shuffles is "not at all mixed" for six shuffles and "essentially mixed" after eight. This abrupt transition is the cutoff phenomenon, discovered by Aldous and Diaconis in the 1980s, and Chapter 18 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) develops its theory: precise definitions of cutoff and cutoff windows, the product criterion $t_{\mathrm{rel}}=o(t_{\mathrm{mix}})$ necessary for cutoff, and complete proofs for two model families — the biased walk on a segment and the lazy hypercube walk, the latter with the sharp $\tfrac12 n\log n$ location and window $n$, in both total variation and separation. Chapter 19 complements this with lamplighter walks: chains on the wreath-product state space of lamp configurations over a moving lamplighter, whose relaxation, mixing, and separation times are governed — beautifully — by the hitting and cover times of Missions VI. Both chapters are formalized in this mission.

Setting

A family of chains is a sequence $P^{(n)}$ on state spaces $V_n$ with stationary distributions $\pi_n$; all single-chain quantities acquire an index $n$. As before, $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$, $d_n(t)=\max_x\|P^{(n)t}(x,\cdot)-\pi_n\|_{TV}$, $t^{(n)}_{\mathrm{mix}}(\varepsilon)=\min\{t:d_n(t)\le\varepsilon\}$, and $t^{(n)}_{\mathrm{mix}}=t^{(n)}_{\mathrm{mix}}(1/4)$. The family has a cutoff when for every $0<\varepsilon<1$

$$\frac{t^{(n)}_{\mathrm{mix}}(\varepsilon)}{t^{(n)}_{\mathrm{mix}}(1-\varepsilon)}\;\longrightarrow\;1\qquad(n\to\infty),$$

and a cutoff at $t_n$ with window $w_n$ when $w_n=o(t_n)$ and the distance at time $t_n+\alpha w_n$ tends (in the appropriate limsup/liminf sense) to $1$ as $\alpha\to-\infty$ and to $0$ as $\alpha\to+\infty$. The separation distance from $x$ is $s_x(t)=\max_y\bigl(1-P^t(x,y)/\pi(y)\bigr)$ (Mission III), $s(t)=\max_xs_x(t)$, and a separation cutoff is defined by the same window template with $s$ in place of $d$. From Mission VII, $t_{\mathrm{rel}}=(1-\lambda_\star)^{-1}$ is the relaxation time; from Mission VI, $t_{\mathrm{hit}}=\max_{x,y}\mathbb E_x(\tau_y)$ and $t_{\mathrm{cov}}$ are the maximal hitting and cover times, and the pairwise distance $\bar d(t)=\max_{x,y}\|P^t(x,\cdot)-P^t(y,\cdot)\|_{TV}$ is from Mission II.

The concrete chains: the lazy biased walk on $\{0,\dots,n\}$ holds with probability $\tfrac12$ and otherwise steps up with probability $p>\tfrac12$, down with probability $1-p$ (reflecting at the endpoints); the lazy hypercube walk is the walk of Mission IV on $\{0,1\}^n$. The lamplighter chain $G^\ast$ over a graph $G$ has states (lamp configuration in $\{0,1\}^{V}$, lamplighter position in $V$); one step randomizes the current lamp, moves the lamplighter one step of the lazy walk on $G$, and randomizes the new lamp. Its stationary distribution is uniform lamps times the walk's stationary distribution.

Formalization targets

Goal

Theorem 18.3: the lazy hypercube walk has a cutoff at $\tfrac12\,n\log n$ with window $n$ — the family's total variation distance undergoes its full collapse in a window of size $\Theta(n)$ around $\tfrac12 n\log n$.

Milestones

• Lemma 18.1 — cutoff is equivalent to the step-function limit: $d_n(\lfloor c\,t^{(n)}_{\mathrm{mix}}\rfloor)\to1$ for every $c<1$ and $\to0$ for every $c>1$.
• Theorem 18.2 — the lazy biased walk on $\{0,\dots,n\}$ with bias $\beta=p-\tfrac12>0$ has a cutoff at $\beta^{-1}n$ with window $\sqrt n$.
• Proposition 18.4 (the product condition) — for a reversible family with $t^{(n)}_{\mathrm{mix}}\to\infty$, if $t^{(n)}_{\mathrm{mix}}\le C\,t^{(n)}_{\mathrm{rel}}$ for a fixed constant $C$, the family has no cutoff: $t_{\mathrm{rel}}=o(t_{\mathrm{mix}})$ is necessary.
• Theorem 18.8 — the lazy hypercube walk has a separation cutoff at $n\log n$ with window $n$ — at twice the total-variation cutoff time.
• Lemma 19.3 (Aldous–Diaconis) — the separation–total-variation relation $s(2t)\le1-\bigl(1-\bar d(t)\bigr)^2$ for reversible chains.
• Theorem 19.1 — for lamplighter chains over a growing family of connected graphs, $t_{\mathrm{rel}}(G_n^\ast)\asymp t_{\mathrm{hit}}(G_n)$: the relaxation time is comparable, with universal constants, to the maximal hitting time of the base walk.
• Theorem 19.2 — likewise $t_{\mathrm{mix}}(G_n^\ast)\asymp t_{\mathrm{cov}}(G_n)$: the lamplighter's mixing time is governed by the base walk's cover time.

Significance

The results. Cutoff is the deepest phenomenon in the quantitative theory of Markov chains: it says mixing is a phase transition in time. The hypercube is the fundamental example where everything can be computed — the eigenvalue structure of Mission VII delivers the upper bound and a refined distinguishing-statistic argument (Mission IV) the lower — and the $\tfrac12 n\log n$ location with window $n$ is the sharpest statement of the coupon-collector heuristic. The product condition 18.4 is the basic sanity criterion in the ongoing research program of characterizing cutoff. The lamplighter theorems tie together the entire series: hitting times (Mission VI), cover times (Mission VI), relaxation times (Mission VII), and separation (Missions III, XI) all meet in one family of chains that furnishes counterexamples — for instance, families with total-variation cutoff but no separation cutoff.

Formalizing them. Nothing about cutoff exists in any proof assistant. The definitions themselves (families of chains, windows, limsup/liminf in a real parameter) are a formalization contribution: they force precision about quantifier order that informal texts elide. The hypercube cutoff is a landmark target — a sharp two-sided asymptotic statement, not an inequality.

Difficulty

The upper half of the hypercube cutoff needs the full eigenvalue decomposition of the walk ($\lambda_j=1-j/n$ with multiplicity $\binom nj$, via Mission VII's spectral representation) and the $\ell^2$ bound summed over binomial coefficients; the lower half needs the Hamming-weight distinguishing statistic pushed to second-order precision (mean and variance at time $\tfrac12 n\log n+\alpha n$). Proposition 18.4 converts an eigenfunction with eigenvalue near $1$ into a quantitative anti-concentration statement — the formal content of "a bounded ratio forbids abrupt collapse". Theorem 18.2 rests on a central-limit-flavoured estimate for the biased walk's position, done with fourth-moment bounds rather than the CLT. The lamplighter theorems are the heaviest: the upper bounds couple lamp refreshment with the cover-time of the base walk, the lower bounds run separation-distance and eigenfunction arguments, and all four inequalities must hold with universal constants over an arbitrary growing graph family — the statements quantify over the family, so the proofs must too. The asymptotic language throughout (liminf/limsup over $n$, limits in the window parameter $\alpha$) exercises the filter library in earnest.

Formalization scope

Families are dependent functions ∀ n, Matrix (V n) (V n) ℝ over a sequence of finite state-space types. Cutoff and windows are rendered exactly by the book's Eq. (18.3) and §18.1: the window definition uses Filter.liminf/limsup over $n$ composed with limits in the real parameter $\alpha$ (through ⌊t n + α w n⌋₊, with the natural-floor convention on negative reals). The mixing-time ratio in the cutoff definition uses real division of the natural-valued mixing times (total division: the hypotheses keep denominators eventually positive). The biased walk's stationary distribution is passed as a hypothesis rather than a closed form. In the lamplighter theorems the comparability constants $c_1,c_2$ and the threshold $N$ are existentially quantified, with the graph family and its connectivity as hypotheses; the lamplighter matrix and its product stationary distribution are explicit definitions. Lemma 19.3 is stated for a single reversible chain at all times $t$.

Selected references

• D. A. Levin, Y. Peres, E. L. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, 2009. https://documents.epfl.ch/groups/i/ip/ipg/www/2013-2014/Random_Walks/markovmixing.pdf
• D. Aldous, P. Diaconis, Shuffling cards and stopping times, Amer. Math. Monthly 93 (1986). https://doi.org/10.1080/00029890.1986.11971821
• P. Diaconis, The cutoff phenomenon in finite Markov chains, Proc. Natl. Acad. Sci. USA 93 (1996). https://doi.org/10.1073/pnas.93.4.1659
• Y. Peres, D. Revelle, Mixing times for random walks on finite lamplighter groups, Electron. J. Probab. 9 (2004). https://doi.org/10.1214/EJP.v9-198

Motivation

Every mixing bound in Missions II–IX ultimately leaned on either coupling or the spectrum, and the spectral route demanded reversibility. Chapter 17 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) opens a third route: martingales — processes whose conditional expected increment vanishes — and the beautiful evolving-set process of Morris and Peres built from them. The payoff theorem of the chapter bounds the mixing time of any lazy irreducible chain by its bottleneck constant, with no reversibility hypothesis anywhere — an honest strengthening of the spectral route through the Cheeger inequality, whose proof is a martingale analysis of a random sequence of sets growing and shrinking under the chain's flow. The same chapter proves the optional stopping theorem in the exact discrete form the rest of the series consumes, and applies the machinery to sharp return-probability estimates for lazy walks.

Setting

Throughout, $P$ is a chain on a finite state space $V$ with stationary distribution $\pi$; $\|\mu-\nu\|_{TV}=\max_{A}|\mu(A)-\nu(A)|$, $d(t)=\max_x\|P^t(x,\cdot)-\pi\|_{TV}$, and $t_{\mathrm{mix}}(\varepsilon)=\min\{t:d(t)\le\varepsilon\}$ as in the earlier missions; $\pi_{\min}=\min_x\pi(x)$. The chain is lazy when $P(x,x)\ge\tfrac12$ at every state.

A martingale adapted to the chain is a family $M_t$ of functions of the trajectory up to time $t$ such that the conditional expectation of $M_{t+1}$, given the trajectory so far, equals $M_t$ — for a finite chain, a pointwise finite-sum identity. A stopping time $\tau$ is a $\{0,1\}$-valued stopping rule in the sense of Mission III: whether to stop at time $t$ depends only on the trajectory up to $t$.

From Mission IV: the edge measure is $Q(x,y)=\pi(x)P(x,y)$, with $Q(S,y)=\sum_{x\in S}Q(x,y)$, and the bottleneck constant $\Phi_\star$ is the minimum over sets $S$ with $0<\pi(S)\le\tfrac12$ of $\Phi(S)=Q(S,S^c)/\pi(S)$. The evolving-set process is the Markov chain on subsets of $V$ in which, from the current set $S$, one draws $u$ uniform on $(0,1]$ and passes to the superlevel set

$$S'=\Bigl\{y\;:\;\frac{Q(S,y)}{\pi(y)}\ge u\Bigr\}$$

— states currently receiving a large share of the flow out of $S$ are likely to join, states receiving little are likely to leave.

Formalization targets

Goal

Theorem 17.10 (Morris–Peres), the capstone of Chapter 17: for any lazy irreducible chain — reversibility not assumed —

$$t_{\mathrm{mix}}(\varepsilon)\;\le\;\frac{2}{\Phi_\star^{2}}\,\log\!\Bigl(\frac{1}{\varepsilon\,\pi_{\min}}\Bigr).$$

Milestones

• Corollary 17.7, the Optional Stopping Theorem — if $M$ is a martingale adapted to the chain, bounded uniformly by a constant, and $\tau$ is an almost surely finite stopping time, then $\mathbb E_x(M_\tau)=M_0(x)$: stopping a fair game at a fair time wins nothing.
• Lemma 17.12 — the evolving-set process, started from the singleton $\{x\}$, recovers the chain: $P^t(x,y)=\dfrac{\pi(y)}{\pi(x)}\,\mathbb P_{\{x\}}\{y\in S_t\}$.
• Lemma 17.13 — for the evolving-set process, the stationary mass $\pi(S_t)$ of the current set is a martingale.
• Theorem 17.17 — return probabilities of the lazy random walk on a graph of maximum degree $\Delta$: $\bigl|P^t(x,x)-\pi(x)\bigr|\le\sqrt2\,\Delta^{5/2}/\sqrt t$, an application of the evolving-set machinery.

Significance

The results. The optional stopping theorem is the workhorse identity of discrete probability — the earlier missions' gambler's-ruin and hitting-time computations are all instances, and Missions XI–XII cite it again. The Morris–Peres theorem is the strongest known elementary relation between geometry and mixing: for reversible chains it recovers the Cheeger-based bound $t_{\mathrm{mix}}\lesssim\Phi_\star^{-2}\log(1/\pi_{\min})$ of Mission VII, but it needs no reversibility, and its proof technique — controlling the root $\sqrt{\pi(S_t)}$ as a supermartingale — introduced evolving sets as a tool that has since produced heat-kernel decay, isoperimetric mixing profiles, and bounds for non-reversible and time-inhomogeneous chains well beyond the book.

Formalizing them. Mathlib's martingale library lives in measure-theoretic generality; this mission's chain-adapted martingales are self-contained finite objects (families of functions on trajectory spaces), so the optional stopping theorem here is independent of, and complementary to, the measure-theoretic one. Evolving sets exist in no proof assistant; the process is a genuinely novel formalization target — a Markov chain whose states are Finsets, defined through interval lengths of a uniform variable.

Difficulty

The optional stopping theorem needs the dominated-convergence step (bounded martingale, a.s. finite time) rendered as an elementary tail estimate — the series $\sum_t\mathbb P\{\tau=t\}\,M_t$ must be shown summable and equal to $M_0$ by an exchange of finite sums with a limit. The evolving-set transition probabilities are interval lengths: the probability of passing from $S$ to $T$ is the length of the set of $u\in(0,1]$ whose superlevel set is exactly $T$, which the formalization encodes by explicit upper and lower thresholds (a min over $T$ and a max over $T^c$ of the clipped ratios $Q(S,y)/\pi(y)$); establishing that these lengths sum to one over $T$, and that the process has the two martingale properties, is delicate finite-order-statistics reasoning. The goal theorem then runs a supermartingale argument on $\sqrt{\pi(S_t)}$: laziness keeps the thresholds in $[\tfrac12,1]$, an expansion estimate converts the bottleneck constant into a per-step multiplicative decay of $\mathbb E\sqrt{\pi(S_t)\bigl(1-\pi(S_t)\bigr)}$, and Lemma 17.12 converts that decay into a total-variation bound. Theorem 17.17 composes the same machinery with a Cauchy–Schwarz step. None of this exists in any library; the auxiliary supermartingale lemmas are welcome as separate contributions.

Formalization scope

Martingales, stopping times, and stopped expectations are the trajectory-calculus objects of Missions I and III: finite sums over paths weighted by $\prod P(\omega_i,\omega_{i+1})$, with expectations over the stopping time as tsums in $t$ (non-summable families sum to $0$; the a.s.-finiteness hypothesis is the statement that the stopping mass sums to $1$). The evolving-set matrix is defined by the clipped-threshold formula described above — an explicit real matrix on Finset V — and the goal and lemmas assume $\pi$ positive and $P$ lazy exactly where the book does. Theorem 17.17 is stated for the lazy walk on a connected graph with positive degrees, with $\pi(x)=\deg(x)/2|E|$ written out. Real-valued bounds on the natural-valued $t_{\mathrm{mix}}$ are direct inequalities on the cast, with no hidden rounding.

Selected references

• D. A. Levin, Y. Peres, E. L. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, 2009. https://documents.epfl.ch/groups/i/ip/ipg/www/2013-2014/Random_Walks/markovmixing.pdf
• B. Morris, Y. Peres, Evolving sets, mixing and heat kernel bounds, Probab. Theory Related Fields 133 (2005). https://doi.org/10.1007/s00440-005-0434-7
• D. Williams, Probability with Martingales, Cambridge University Press, 1991. https://doi.org/10.1017/CBO9780511813658

Motivation

The Ising model is statistical mechanics' fruit fly: spins $\pm1$ on the vertices of a graph, neighbours preferring to agree, a single parameter — the inverse temperature $\beta$ — tuning the strength of that preference. Chapter 15 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) studies the Glauber dynamics for this model and exhibits, on concrete graphs, the phenomenon that made mixing times a subject: a dynamical phase transition. At high temperature (small $\beta$) the dynamics mixes in $O(n\log n)$ steps on any bounded-degree graph; on the complete graph the same dynamics passes, as $\beta$ crosses an explicit threshold, from $O(n\log n)$ mixing to exponentially slow mixing. The chapter also introduces two comparison tools of independent value — the sensitivity of the spectral gap to edge removal, and the block dynamics comparison — and the Kenyon–Mossel–Peres bound for trees, all formalized in this mission.

Setting

Spin configurations on a finite graph $G$ with vertex set of size $n$ and maximum degree $\Delta$ are functions $\sigma$ assigning $\pm1$ to each vertex (encoded over Booleans, $\mathrm{true}\mapsto+1$). The Ising model at inverse temperature $\beta>0$ is the Gibbs distribution

$$\pi(\sigma)\;=\;\frac{e^{\,\beta\sum_{\{v,w\}\in E}\sigma(v)\sigma(w)}}{Z(\beta)},$$

each edge counted once, $Z(\beta)$ the normalizing partition function. The Glauber dynamics for $\pi$ picks a uniform vertex and re-samples its spin from $\pi$ conditioned on all other spins — concretely, the new spin at $v$ is $+1$ with probability $\bigl(1+\tanh(\beta S)\bigr)/2$ where $S$ is the sum of the neighbouring spins.

The yardsticks are as in the earlier missions: $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$ is the total variation distance, $d(t)=\max_\sigma\|P^t(\sigma,\cdot)-\pi\|_{TV}$, and $t_{\mathrm{mix}}(\varepsilon)=\min\{t:d(t)\le\varepsilon\}$. From Mission VII: an eigenvalue of a chain is a real $\lambda$ with $Pf=\lambda f$ for some nonzero $f$; $\lambda_2$ is the largest eigenvalue $\ne1$, the spectral gap is $\gamma=1-\lambda_2$, $\lambda_\star$ the largest $|\lambda|$ over eigenvalues $\ne1$, and the relaxation time is $t_{\mathrm{rel}}=(1-\lambda_\star)^{-1}$.

Formalization targets

Goal

Theorem 15.1, the high-temperature fast-mixing theorem: if $\Delta\tanh\beta<1$ then the Glauber dynamics on any graph satisfies

$$t_{\mathrm{mix}}(\varepsilon)\;\le\;\Bigl\lceil\frac{n\,\bigl(\log n+\log(1/\varepsilon)\bigr)}{1-\Delta\tanh\beta}\Bigr\rceil,$$

together with its refinement for graphs with all degrees even, where the condition relaxes to $(\Delta/2)\tanh(2\beta)<1$.

Milestones

• Lemma 15.2 (the tanh lemma) — the elementary inequalities about $x\mapsto\tanh(\beta(x+1))-\tanh(\beta(x-1))$ (symmetry, monotonicity, the bounds by $2\tanh\beta$ and, at odd integers, by $\tanh 2\beta$) that drive the one-site coupling contraction.
• Theorem 15.3 — the dynamical phase transition on the complete graph at $\beta=\alpha/n$: (i) for $\alpha<1$, $t_{\mathrm{mix}}(\varepsilon)\le n(\log n+\log(1/\varepsilon))/(1-\alpha)$; (ii) for $\alpha>1$, there are $r(\alpha),C>0$ with $t_{\mathrm{mix}}\ge C\,e^{r n}$ — split here into a fast half and a slow half.
• Theorem 15.4 — on the $n$-cycle, at every $\beta>0$, mixing is $n\log n$ up to explicit constants: $(1+o(1))\,\frac{n\log n}{2c_O(\beta)}\le t_{\mathrm{mix}}(\varepsilon)\le(1+o(1))\,\frac{n\log n}{c_O(\beta)}$ with $c_O(\beta)=1-\tanh(2\beta)$.
• Theorem 15.6 (Kenyon–Mossel–Peres) — on the rooted $b$-ary tree of depth $k$ with $n_k$ vertices, the relaxation time is polynomial at every temperature: $t_{\mathrm{rel}}\le n_k^{\,c_T(\beta,b)}$ with $c_T(\beta,b)=2\beta(3b+1)/\log b+1$.
• Proposition 15.7 — removing $r$ edges changes the spectral gap of the Glauber dynamics by at most a factor $e^{2\beta(\Delta+2r)}$.
• Theorem 15.9 — the block-dynamics comparison: if blocks $V_1,\dots,V_b$ cover the vertex set, each of size at most $M$, each vertex in at most $M_\star$ blocks, then the spectral gap $\gamma_B$ of the block dynamics and the gap $\gamma$ of the single-site dynamics satisfy $\gamma_B\le M^2M_\star\,(4e^{2\beta\Delta})^{M+1}\,\gamma$.

Significance

The results. Theorem 15.1 is the standard fast-mixing criterion for Glauber dynamics, and the model application of the path-coupling technique of Mission VIII. Theorem 15.3 exhibits, in the cleanest possible setting, the correspondence between the static phase transition of the mean-field Ising model and the dynamical transition of its Glauber dynamics — the phenomenon at the heart of Markov-chain approaches to statistical physics. The tree bound of Kenyon–Mossel–Peres and the block-dynamics comparison are the standard tools for spatially structured spin systems; block dynamics in particular is the engine of recursive gap bounds on trees and lattices.

Formalizing them. Nothing about the Ising model, Gibbs distributions, or Glauber dynamics exists in Mathlib. The Gibbs-distribution layer (weights, partition functions, conditional single-site laws with their $\tanh$ closed forms) is foundational for any future formalization of statistical mechanics; the phase-transition theorem 15.3(ii) would be, to our knowledge, the first formalized instance of exponentially slow mixing driven by an energy barrier.

Difficulty

The high-temperature theorem is path coupling (Mission VIII) plus the tanh lemma: the one-site coupling of two adjacent configurations contracts the Hamming metric at rate $1-\bigl(1-\Delta\tanh\beta\bigr)/n$, and every analytic input is in Lemma 15.2 — which is why that elementary lemma is a milestone of its own. The slow-mixing half of Theorem 15.3 runs through the bottleneck bound of Mission IV: the magnetization performs a one-dimensional walk in a double-well free-energy landscape, and the bottleneck at zero magnetization has exponentially small stationary mass — a large-deviations estimate carried out with binomial coefficients. The cycle's lower bound needs Wilson's method (Mission VII) with an explicit eigenfunction. The tree and block theorems are exercises in the comparison technology of Mission VII (Dirichlet forms, canonical paths through block updates); their constants are crude but the inductive structure is delicate. Everything sits on the subtlety that the state space $\{\pm1\}^V$ has size $2^n$, so all "polynomial" bounds are polynomial in $n$, not in the size of the state space.

Formalization scope

The Gibbs distribution is defined by explicit finite sums (weight over partition function, total division); no positivity side conditions are needed since the weights are exponentials. The Glauber dynamics is the generic single-site heat bath of Mission II applied to the Ising distribution, so the $\tanh$ closed form is a provable lemma, not a definition. Asymptotic statements ($o(1)$, "for sufficiently large $n$") are rendered with explicit $\exists N,\forall n\ge N$ quantifiers and a free precision parameter $\delta$; the phase-transition constants $r(\alpha),C$ are existentially quantified. The tree is encoded as words of length $\le k$ over an alphabet of size $b$; the block dynamics re-samples a uniformly chosen block from the conditional Gibbs distribution, with the convention that a conditioning of zero mass yields a zero row (total division), which the theorems' hypotheses exclude on the support. The even-degree refinement of the goal is stated as a second conjunct with its own hypothesis.

Selected references

• D. A. Levin, Y. Peres, E. L. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, 2009. https://documents.epfl.ch/groups/i/ip/ipg/www/2013-2014/Random_Walks/markovmixing.pdf
• C. Kenyon, E. Mossel, Y. Peres, Glauber dynamics on trees and hyperbolic graphs, FOCS 2001. https://doi.org/10.1109/SFCS.2001.959934
• D. A. Levin, M. J. Luczak, Y. Peres, Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability, Probab. Theory Related Fields 146 (2010). https://doi.org/10.1007/s00440-008-0189-z
• F. Martinelli, Lectures on Glauber dynamics for discrete spin models, Lectures on Probability Theory and Statistics (Saint-Flour XXVII), Springer, 1999. https://doi.org/10.1007/978-3-540-48115-7_2

Motivation

How many shuffles does a deck of cards need? The question created the modern theory of mixing times: Diaconis and Shahshahani's analysis of random transpositions (1981) and the Bayer–Diaconis "seven shuffles suffice" analysis of the riffle shuffle (1992) are its founding results. Chapters 8 and 16 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) treat the four standard shuffles: random transpositions (pick two cards, swap them), the riffle shuffle (cut and interleave, the Gilbert–Shannon–Reeds model), random adjacent transpositions, and the $L$-reversal chain of segment reversals — the last motivated by genome rearrangement, where a chromosome evolves by reversing segments and the mixing time measures evolutionary distance ("from shuffling cards to shuffling genes").

Setting

All four chains are random walks on a symmetric group; decks are encoded as in Mission III, an arrangement being a permutation with position $0$ on top. Random transpositions have increment distribution $\mu(\mathrm{id})=1/n$, $\mu(\text{transposition})=2/n^2$. The lazy adjacent-transposition walk puts mass $1/2$ on the identity and $1/[2(n-1)]$ on each $(i\ i{+}1)$. One inverse riffle shuffle assigns each card an independent uniform bit and moves the cards labeled $0$ to the top, preserving relative order; the (forward) riffle shuffle is its time reversal, given by transposing the inverse-shuffle matrix (the stationary distribution being uniform). The $L$-reversal chain on circular arrangements indexed by $\mathbb Z_n$ picks a position $i$ and a length $k<L$ uniformly and reverses the segment $[i,i+k]$.

Formalization targets

Goal

$$t_{\mathrm{mix}}\;\le\;(2+o(1))\,n\log n$$

for random transpositions on $n$ cards (Corollary 8.10), formalized as: for every $\delta>0$ there is $N$ with $t_{\mathrm{mix}}\le(2+\delta)n\log n$ for all $n\ge N$.

Milestones

Proposition 8.11 (random transpositions lower bound $t_{\mathrm{mix}}(\varepsilon)\ge\frac{n-1}{2}\log\bigl(\frac{(1-\varepsilon)n}{6}\bigr)$, via fixed points); Proposition 8.13 (riffle shuffle: $t_{\mathrm{mix}}\le 2\log_2(4n/3)+1$); Proposition 8.14 (riffle shuffle: $t_{\mathrm{mix}}(\varepsilon)\ge(1-\delta)\log_2 n$ for large $n$, by the counting bound of Mission IV); the random adjacent transpositions upper bound $t_{\mathrm{mix}}(\varepsilon)\le 2n^3\log_2 n$ for large $n$ (§16.1.2, display (16.4)) and lower bound $t_{\mathrm{mix}}\ge n^2(n-1)/16$ (§16.1.3, by following a single card); and Proposition 16.2 (the $L$-reversal chain with $L<n/2$ satisfies $d\bigl((1-\varepsilon)\frac n2\log n\bigr)\to1$, via conserved adjacencies).

Significance

The results. Random transpositions at $\frac12 n \log n$ (the constant $2$ here is not sharp; the sharp constant is part of the celebrated Diaconis–Shahshahani cutoff) and riffle at $\frac32\log_2 n$ are the emblematic mixing results outside of spin systems; the riffle bound is the mathematical content of "seven shuffles suffice" for $n=52$. The adjacent-transposition walk at order $n^3\log n$ is the basic example where geometry (diameter $\binom n2$) forces polynomial mixing. The $L$-reversal lower bound is the quantitative starting point of the biological application.

Formalizing them. Random walks on symmetric groups and their mixing are entirely absent from Mathlib; so are the combinatorial devices these proofs run on — the Broder stopping time, rising sequences and the Eulerian-number analysis of riffle shuffles, single-card projections, conserved adjacencies. The riffle shuffle formalization via bit strings and the stable-sort permutation Tuple.sort gives a clean combinatorial model reusable for the cutoff analysis of Mission XI.

Difficulty

Corollary 8.10's route is the Broder strong stationary time: mark cards under a card-and-position scheme and prove that, conditioned on the marked set and its positions, the marked cards are uniformly ordered — an exchangeability induction on top of Mission III's stopping-rule framework; then the coupon-collector-style tail with an extra $\log n$ factor. Proposition 8.11 requires the fixed-point statistic: the expected number of untouched cards after $t$ transpositions and a second-moment bound, fed into Proposition 7.8. The riffle upper bound runs through inverse shuffles: after $t$ inverse shuffles the deck is a uniform stable sort of $t$-bit labels, and mixing reduces to the birthday problem for $2^t$ labels; formalizing "distinct labels imply uniform order" is the crux. The $L$-reversal lower bound needs a second-moment argument for the number of conserved adjacencies with the exact variance bookkeeping of the book. Throughout, the walk-on-group conventions (left vs right increments, forward vs inverse shuffle) are the classic source of silent errors; the formal statements pin them.

Formalization scope

All chains are explicit matrices on Equiv.Perm (Fin n) (or Equiv.Perm (ZMod n) for the circular $L$-reversal chain); increments act on the left, matching Mission I's group-walk convention. The riffle shuffle is defined as the transpose-reversal of the explicit inverse-riffle matrix — the two have equal distance to uniformity by Lemma 4.13 (Mission II), which a solver may use rather than re-derive. Asymptotic statements ($o(1)$, "for sufficiently large $n$") are spelled out with explicit $\forall\delta\,\exists N$ quantifiers; upper bounds carry a $+1$ where integer rounding requires it. The $L$-reversal family takes the length function $L(n)$ as a hypothesis-carrying parameter with $1\le L(n)<n/2$, and its lower bound is a genuine limit statement (Tendsto, distance to stationarity tending to $1$).

Welcome contributions: exchangeability and projection lemmas for deck chains; Eulerian/rising-sequence counting; the single-card chain projection (reused for Mission XI's cutoff examples).

Selected references

• D. A. Levin, Y. Peres, E. L. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, 2009. https://documents.epfl.ch/groups/i/ip/ipg/www/2013-2014/Random_Walks/markovmixing.pdf
• P. Diaconis, M. Shahshahani, Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete 57 (1981). https://doi.org/10.1007/BF00535487
• D. Bayer, P. Diaconis, Trailing the dovetail shuffle to its lair, Ann. Appl. Probab. 2 (1992). https://doi.org/10.1214/aoap/1177005705
• R. Durrett, Shuffling chromosomes, J. Theoret. Probab. 16 (2003). https://doi.org/10.1023/A:1024940217006

Motivation

The log-Sobolev constant $\alpha$ of a finite Markov chain controls how fast the chain forgets its starting point in relative entropy and in $\ell^2$, and it gives sharper mixing-time bounds than the spectral gap $\lambda$ when the stationary distribution has very small atoms. Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996)) made this constant a standard tool for finite chains. A practical obstacle runs through their paper: $\alpha$ is defined by a nonlinear variational problem, and exact values are known for very few chains. Lower bounds on $\alpha$ for a complicated chain are usually obtained by comparison with a simple chain whose constant is known exactly.

The Appendix of that paper computes such a reference value. For the chain that forgets its state in one step, $K(x,y)=\pi(y)$, it determines $\alpha$ in closed form for every positive $\pi$, and with it the log-Sobolev constant of every chain on two points and of the random walk on the complete graph.

Timeline. Rothaus's work on log-Sobolev inequalities (references [38–40] of the paper) studied the constant through its minimizers, the approach the Appendix follows. Diaconis and Saloff-Coste (1996) give the formula $\alpha=(1-2\pi_*)/\log(1/\pi_*-1)$, and note that Higuchi and Yoshida independently found the asymmetric two-point value.

Setting

Let $\mathcal X$ be a finite set and $K$ a Markov kernel on it: $K(x,y)\ge0$ and $\sum_y K(x,y)=1$. Let $\pi$ be an invariant probability, $\sum_x\pi(x)K(x,y)=\pi(y)$, with $\pi(x)>0$ for every $x$. Functions on $\mathcal X$ are real. Write $Kf(x)=\sum_yK(x,y)f(y)$, $\langle f,g\rangle=\sum_x f(x)g(x)\pi(x)$, $\|f\|_2^2=\langle f,f\rangle$, $E_\pi f=\sum_x f(x)\pi(x)$ and $\mathrm{Var}_\pi(f)=\|f-E_\pi f\|_2^2$.

The Dirichlet form is $\mathcal E(f,g)=\langle (I-K)f,g\rangle$. The entropy functional is

$$\mathcal L(f)=\sum_{x\in\mathcal X}|f(x)|^2\log\frac{|f(x)|^2}{\|f\|_2^2}\,\pi(x),$$

which is nonnegative and vanishes exactly when $|f|$ is constant. The spectral gap and the log-Sobolev constant are

$$\lambda=\inf\Big\{\frac{\mathcal E(f,f)}{\mathrm{Var}_\pi(f)}:\mathrm{Var}_\pi(f)\ne0\Big\},\qquad \alpha=\inf\Big\{\frac{\mathcal E(f,f)}{\mathcal L(f)}:\mathcal L(f)\ne0\Big\}.$$

Equivalently, $1/\alpha$ is the best constant $C$ in $\mathcal L(f)\le C\,\mathcal E(f,f)$.

The chain of the mission is $K(x,y)=\pi(y)$: at each step it draws a fresh point from $\pi$, ignoring its current state. Its invariant measure is $\pi$, and $\pi_*=\min_{\mathcal X}\pi$ is its smallest atom. In Lean the objects are dirichlet, entL, gap, logSobolev, indepChain and piMin in the namespace LogSobolevMC.TwoPoint.

Formalization targets

Goal: Theorem A.1

For a positive probability $\pi$ on $\mathcal X$ with $\pi_*<1/2$, the chain $K(x,y)=\pi(y)$ has

$$\alpha=\frac{1-2\pi_*}{\log(1/\pi_*-1)},$$

and in particular, for $\pi\equiv1/|\mathcal X|$ with $|\mathcal X|\ge3$, $\alpha=(1-2/|\mathcal X|)/\log(|\mathcal X|-1)$. The goal is an equality, so it contains both a log-Sobolev inequality with this constant and the existence of functions that make it sharp.

Milestones

1. Lemma 3.1: $2\alpha\le\lambda$ for every chain.
2. $\mathcal E(|f|,|f|)\le\mathcal E(f,f)$ (proof of Theorem A.1).
3. Either $\alpha=\lambda/2$, or the infimum is attained at a nonconstant nonnegative function (proof of Theorem A.1).
4. (A.3): a positive minimizer $u$ of a reversible chain satisfies $2u\log u-2u\log\|u\|_2-\alpha^{-1}(I-K)u=0$.
5. (A.1): the explicit pair $a=\log[(1-\theta)/\theta]/(1-2\theta)$, $s=(1-2\theta)/(2\theta(1-\theta))$ solves the two-equation system of the proof of Theorem A.2.
6. Theorem A.2: the two-point chain with both rows $(\theta,1-\theta)$, $0<\theta\le1/2$, has $\alpha(\theta)=(1-2\theta)/\log[(1-\theta)/\theta]$, with the limit value $1/2$ at $\theta=1/2$.

Consequences

Corollary A.3 (every two-point chain), Corollary A.4 ($\alpha(K)\ge(1-2\pi_*)\lambda(K)/\log[1/\pi_*-1]$ for every finite chain) and Corollary A.5 (the complete graph, $\alpha=(|\mathcal X|-2)/((|\mathcal X|-1)\log(|\mathcal X|-1))$, and $\alpha=1/(2\log2)$ on $\mathbb Z_3$).

Significance

The result. Theorem A.1 is one of the few exact log-Sobolev constants known for a whole family of chains. Since this chain has $\lambda=1$, the formula shows that $\alpha$ can be smaller than $\lambda/2$ by a factor of order $\log(1/\pi_*)$, which answers the question on p. 715 of how far $2\alpha$ and $\lambda$ can be apart. Corollary A.4 turns this into a general tool: any spectral gap bound gives a log-Sobolev bound, losing only a factor $\log(1/\pi_*-1)/(1-2\pi_*)$. Corollary A.3 settles every chain on two points, and Corollary A.5 gives the complete graph, a frequent reference chain for comparison arguments.

Formalizing it. These results are proved in the paper. To our knowledge none of them, nor the log-Sobolev constant of a finite Markov chain itself, has a machine-checked formalization. The mission produces the definitions of $\mathcal E$, $\mathcal L$, $\lambda$ and $\alpha$ for finite chains, and exact values that later comparison arguments can cite.

Difficulty

The upper bound $\alpha\le(1-2\pi_*)/\log(1/\pi_*-1)$ needs only a good test function. The lower bound is the hard part: it is a statement about every function on $\mathcal X$, and $\mathcal L$ is neither quadratic nor convex. The obvious route, comparing with the spectral gap, loses exactly the logarithmic factor the theorem is about. Reducing to two-valued functions requires the infimum to be attained and analysed through its minimizers. The two-point problem that remains is a one-parameter inequality $l(s)\le a\,e(s)$ on an interval. The authors describe its solution as a calculus exercise that depended on numerical guessing, and it has removable singularities at $\theta=1/2$.

Formalization scope

$\mathcal X$ is a Fintype with decidable equality. $K$ is a Matrix V V ℝ and $\pi$ a function V → ℝ. Stochasticity, invariance, distributions and reversibility are the published MarkovMixing predicates (IsStochastic, IsStationary, IsDist, DetailedBalance) from mm_basic, and $\mathrm{Var}_\pi$ is MarkovMixing.distVar from mm_lower. Positivity of $\pi$ is a hypothesis throughout, as in §2.1. The infima $\lambda$ and $\alpha$ are Lean sInfs over exactly the sets of (2.4) and (3.1). $\mathcal E$ is the bilinear form $\langle (I-K)f,g\rangle$. $\lambda$ is the variational gap, not the eigenvalue-based MarkovMixing.spectralGap, which agrees with it only for reversible chains.

Disclosed additions, all at removable singularities of the printed formulas: $\pi_*<1/2$ in Theorem A.1 (and $|\mathcal X|\ge3$ in its uniform case); the explicit limit value $1/2$ at $\theta=1/2$ in Theorem A.2; strict $\pi(0)<\pi(1)$ in Corollary A.3; and $|\mathcal X|\ge3$ in Corollary A.5. In (A.3), reversibility, positivity of $u$ and $\alpha>0$ are added. Irreducibility is not assumed anywhere.

Trivializing formalizations are ruled out. A closed form evaluated at $0/0$ would be Lean's value $0$, so each such case is excluded or replaced by its limit. Real.log of a nonpositive number and an sInf over an empty or unbounded set do not occur under the stated hypotheses. Dropping the positivity or invariance of $\pi$, or replacing $\lambda$ by the eigenvalue gap, would change the statements. Stating the goal for two points only, or assuming the minimizer is two-valued, would state a step of the proof.

Contributions welcome: the general facts $\mathcal L\ge0$, $\mathcal E(f,f)=\tfrac12\sum_{x,y}(f(x)-f(y))^2K(x,y)\pi(x)$ and the existence of minimizers are reusable for every log-Sobolev development on finite spaces. The two-point calculus of Theorem A.2 is self-contained real analysis.

Selected references

• P. Diaconis and L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab. 6(3):695–750, 1996. https://doi.org/10.1214/aoap/1034968224
• D. A. Levin, Y. Peres and E. L. Wilmer, Markov Chains and Mixing Times, AMS, 2009 (source of the MarkovMixing definitions used here). https://doi.org/10.1090/mbk/058