Markov Chains and Mixing Times I: Existence and Uniqueness of the Stationary Distribution

Motivation

Finite Markov chains are the basic model for memoryless random dynamics: card shuffles, random walks on graphs and groups, Monte Carlo samplers, and queueing systems are all chains on a finite state space. The single most used fact about them is that an irreducible chain has exactly one stationary distribution — a probability vector $\pi$ with $\pi = \pi P$ — and that this $\pi$ is strictly positive and encodes the long-run behaviour of the chain through the return-time identity $\pi(x) = 1/\mathbb{E}_x(\tau_x^+)$. Every later result in the theory of mixing times (convergence theorems, coupling bounds, spectral methods, cutoff) is a statement about the distance of the chain from this $\pi$, so nothing in the subject can be formalized before this mission is.

This mission is the first in a series formalizing D. A. Levin, Y. Peres and E. L. Wilmer, Markov Chains and Mixing Times (AMS, 2009), covering Chapters 1–2: the basic vocabulary of finite chains (stochastic matrices, irreducibility, period, reversibility, time reversal, random walks on graphs and groups) and the classical examples of Chapter 2 (gambler's ruin, coupon collecting, the reflection principle for simple random walk on $\mathbb{Z}$). Later missions in the series build on the definitions published here.

Setting

A chain on a finite state space $\Omega$ is presented by its transition matrix, a matrix $P \in \mathbb{R}^{\Omega\times\Omega}$ with nonnegative entries whose rows sum to $1$. A distribution is a row vector $\mu$ with nonnegative entries summing to $1$; one step of the chain carries $\mu$ to $\mu P$, and the $t$-step transition probabilities are the entries of the matrix power $P^t$.

The chain is irreducible if for all states $x, y$ there is a $t$ with $P^t(x,y) > 0$. The period of a state $x$ is $\gcd\,\mathcal{T}(x)$ where $\mathcal{T}(x) = \{t \ge 1 : P^t(x,x) > 0\}$, and the chain is aperiodic if every state has period $1$. A distribution $\pi$ is stationary if $\pi P = \pi$, and $\pi$ and $P$ are in detailed balance (the chain is reversible) if $\pi(x)P(x,y) = \pi(y)P(y,x)$ for all $x,y$.

Trajectory events over a finite horizon are finite sums of path weights: a length-$t$ trajectory is a function $\omega : \{0,\dots,t\} \to \Omega$, with weight $\prod_{i<t} P(\omega_i, \omega_{i+1})$ conditional on its starting state. The tail probability $\mathbb{P}_x\{\tau_z^+ > t\}$ of the first hitting time $\tau_z^+ = \min\{t \ge 1 : X_t = z\}$ is the sum of the weights of the trajectories from $x$ that avoid $z$ at times $1,\dots,t$, and expectations of hitting times are recovered by the tail-sum formula $\mathbb{E}(Y) = \sum_{t\ge0} \mathbb{P}\{Y > t\}$, formalized as a tsum over $t$.

Formalization targets

Goal

$\text{$P$ stochastic and irreducible on a finite nonempty $\Omega$} \;\Longrightarrow\; \exists!\, \pi,\ \pi = \pi P .$

This is Corollary 1.17 of the book. It asserts only existence and uniqueness, leaving the finer structure of $\pi$ to the milestones; it is the weakest statement on which the rest of the series can stand, which is why it is the goal.

Milestones toward and around the goal

The milestone list follows the book's own route: well-definedness of the period (Lemma 1.6), positivity of some matrix power for irreducible aperiodic chains (Proposition 1.7), finiteness of expected hitting times (Lemma 1.13), existence of a positive stationary distribution together with

$$\pi(x)\,\mathbb{E}_x(\tau_x^+) = 1$$

(Proposition 1.14), constancy of harmonic functions (Lemma 1.16), stationarity from detailed balance (Proposition 1.19), the stationary and reversible measure $\pi(x) = \deg(x)/2|E|$ of simple random walk on a graph (Examples 1.12 and 1.20), the time reversal $\hat P$ and its path-reversal identity (Proposition 1.22), and the random walks on finite groups of Section 2.6 (Propositions 2.12–2.14). From Chapter 2 the list adds the gambler's ruin formulas $\mathbb{P}_k\{X_\tau = n\} = k/n$ and $\mathbb{E}_k(\tau) = k(n-k)$ (Proposition 2.1), the coupon collector expectation $n\sum_{k\le n} 1/k$ and tail bound $e^{-c}$ (Propositions 2.3 and 2.4), and the reflection principle and the bound $\mathbb{P}_k\{\tau_0 > r\} \le 12k/\sqrt{r}$ for simple random walk on $\mathbb{Z}$ (Lemma 2.18 and Theorem 2.17).

Significance

The result itself. Existence and uniqueness of $\pi$ is the pivot on which the entire quantitative theory turns: it defines the target of convergence, and the identity $\pi(x)\,\mathbb{E}_x(\tau_x^+) = 1$ ties the stationary measure to return times, which later missions use for hitting-time and cover-time results. Detailed balance is the practical tool by which stationary measures of graph and group walks are computed, and the Chapter 2 examples (gambler's ruin, coupon collecting, reflection) are the standard building blocks reused throughout the book — the coupon collector bound, for instance, is exactly the estimate behind the $n \log n + cn$ analysis of the top-to-random shuffle in a later mission of this series.

Formalizing it. Mathlib currently has no theory of finite Markov chains: no stochastic-matrix predicate, no stationary distribution, no periodicity, no hitting times. Everything proved in this mission is new formal mathematics, and the definition layer published here (mm_basic, mm_path, mm_classical) is the shared foundation that all twelve subsequent missions of the series import. All results are classical and have textbook proofs; none has a machine-checked proof.

Difficulty

The delicate point is the existence proof. The natural first idea — extract $\pi$ from an eigenvector of $P^{\mathsf T}$ for eigenvalue $1$, or invoke a fixed-point theorem — either does not give positivity and nonnegativity without further work, or uses compactness machinery (Brouwer) that is unavailable. The book's proof instead builds $\tilde\pi(y) = \mathbb{E}_z(\text{visits to } y \text{ before } \tau_z^+)$ and verifies $\tilde\pi P = \tilde\pi$ by reindexing trajectory sums; formalizing it requires managing infinite series of path sums (summability from the geometric tail bound of Lemma 1.13, exchanging tsum with finite sums, splitting a trajectory at its last step). The uniqueness half is linear algebra via constancy of harmonic functions (Lemma 1.16), which is elementary but requires a maximum-principle argument over a finite state space. The reflection principle and Theorem 2.17 are finite combinatorics on $\pm 1$ paths — the bijection is easy to describe and fiddly to implement.

Formalization scope

States form a Fintype with decidable equality; chains are Matrix V V ℝ with the row-stochasticity predicate IsStochastic; distributions are functions V → ℝ with the predicate IsDist. Everything is distribution-side: no probability space or measure theory is used. The period is formalized as $\sup\{d : d \mid t \text{ for all } t \in \mathcal{T}(x)\}$, which equals $\gcd \mathcal{T}(x)$ when $\mathcal{T}(x) \neq \varnothing$ and takes the junk value $0$ otherwise. Expectations of hitting times are tsums of tail probabilities, with the usual junk value $0$ for non-summable families — the statements are arranged (e.g. multiplicatively, $\pi(x)\cdot\mathbb{E}_x(\tau_x^+) = 1$) so that junk values cannot make them vacuously true. Existence statements carry a Nonempty V hypothesis; irreducibility on the empty space is vacuous, and without nonemptiness the goal would be false, not trivial. The coupon collector and the walk on $\mathbb{Z}$ are presented directly by their driving randomness (uniform draws Fin t → Fin n, uniform sign strings Fin r → Bool), so those probabilities are elementary counting; in particular the reflection principle is stated as an equality of cardinalities of sets of sign strings — this is equivalent to the probabilistic statement because all $2^r$ strings are equally likely.

Contributions welcome beyond the milestone list: simp lemmas for the definition layer, the taboo-matrix representation of avoidance probabilities (useful for Lemma 1.13), and any interface lemmas connecting pathWeight sums to matrix powers — these will be reused by every later mission in the series.

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
• J. R. Norris, Markov Chains, Cambridge University Press, 1998. https://doi.org/10.1017/CBO9780511810633
• D. Aldous, J. A. Fill, Reversible Markov Chains and Random Walks on Graphs, 2002 (unfinished monograph). https://www.stat.berkeley.edu/~aldous/RWG/book.html

Motivation

The first mission of this series established that an irreducible finite Markov chain has a unique stationary distribution $\pi$. The present mission, covering Chapters 3–4 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009), answers the two questions that make that fact useful. First, the inverse problem of sampling: given a target distribution $\pi$ — uniform over proper colorings, a Gibbs measure, a posterior — how does one build a chain whose stationary distribution is $\pi$? The Metropolis and Glauber constructions of Chapter 3 are the universal answers, and they are the engine of Markov chain Monte Carlo across statistical physics, Bayesian statistics, and approximate counting. Second, the convergence question: in what sense, and how fast, does an irreducible aperiodic chain approach $\pi$? Chapter 4 introduces the total variation distance, proves the Convergence Theorem — geometric convergence to stationarity — and defines the mixing time, the parameter the entire remainder of the book estimates.

Setting

All chains live on a finite state space $V$ and are presented by row-stochastic matrices, with the definitions of Mission I. The total variation distance between distributions $\mu$ and $\nu$ is

$$\|\mu-\nu\|_{\mathrm{TV}} = \max_{A\subseteq V}\,|\mu(A)-\nu(A)|,$$

the maximal discrepancy over events. A coupling of $\mu$ and $\nu$ is a distribution on $V\times V$ whose marginals are $\mu$ and $\nu$. For a chain $P$ with stationary $\pi$ one sets

$$d(t)=\max_x \|P^t(x,\cdot)-\pi\|_{\mathrm{TV}},\qquad \bar d(t)=\max_{x,y}\|P^t(x,\cdot)-P^t(y,\cdot)\|_{\mathrm{TV}},$$

and the mixing time is $t_{\mathrm{mix}}(\varepsilon)=\min\{t : d(t)\le\varepsilon\}$, with $t_{\mathrm{mix}}=t_{\mathrm{mix}}(1/4)$.

The Metropolis chain for a target $\pi$ and a symmetric proposal chain $\Psi$ accepts a proposed move $x\to y$ with probability $1\wedge \pi(y)/\pi(x)$; a general (not necessarily symmetric) base chain is handled by the ratio $\bigl(\pi(y)\Psi(y,x)\bigr)/\bigl(\pi(x)\Psi(x,y)\bigr)\wedge 1$. The Glauber dynamics for a distribution $\pi$ on configurations $V^{\text{sites}}$ picks a uniform site and re-samples its value from $\pi$ conditioned on the rest.

Formalization targets

Goal

$\text{$P$ irreducible and aperiodic}\;\Longrightarrow\;\exists\,\alpha\in(0,1),\ C>0:\quad d(t)\le C\alpha^{t}.$

This is Theorem 4.9, the Convergence Theorem. It asserts only the geometric shape of convergence, leaving all quantitative rates to later missions, which is why it is the goal.

Milestones

The milestones are the chapter's working parts: stationarity and reversibility of the Metropolis chain for symmetric and general base chains (§3.2, Exercise 3.1), stationarity and reversibility of the Glauber dynamics (§3.3, Exercise 3.2); the three characterizations of total variation distance — the half-$\ell^1$ formula (Proposition 4.2 with Remark 4.3), the supremum over $[-1,1]$-bounded test functions (Proposition 4.5), and the coupling characterization with an optimal coupling attaining it (Proposition 4.7 with Remark 4.8); the comparison $d\le\bar d\le 2d$ (Lemma 4.11) and submultiplicativity $\bar d(s+t)\le\bar d(s)\bar d(t)$ (Lemma 4.12); the standard mixing-time consequences $d(\ell\, t_{\mathrm{mix}}(\varepsilon))\le(2\varepsilon)^\ell$ and $t_{\mathrm{mix}}(\varepsilon)\le\lceil\log_2\varepsilon^{-1}\rceil\, t_{\mathrm{mix}}$ (§4.5); and the equality of distance to stationarity for a group walk and its inverse walk (Lemma 4.13 and Corollary 4.14).

Significance

The results. The Convergence Theorem is the qualitative foundation on which quantitative mixing theory stands: it guarantees that $t_{\mathrm{mix}}(\varepsilon)$ is finite, so every bound in Missions III–XIII is a bound on a well-defined quantity. The TV characterizations are used constantly — the coupling characterization is the engine of Mission III, the half-$\ell^1$ formula of every explicit computation. The Metropolis and Glauber stationarity results justify the chains analyzed in Missions III (colorings, hardcore), VIII (path coupling) and IX (Ising). Submultiplicativity of $\bar d$ is what makes $t_{\mathrm{mix}}$ a meaningful single number.

Formalizing them. None of this exists in Mathlib: there is no total variation distance for finitely supported distributions, no coupling theory, no mixing time, no MCMC correctness statement. The definition layer published here (TV distance, $d$, $\bar d$, $t_{\mathrm{mix}}$, couplings, Metropolis, Glauber) is imported by every subsequent mission of the series.

Difficulty

The tempting proof of Theorem 4.9 via spectral decomposition fails twice: it needs reversibility, which the theorem does not assume, and spectral machinery that arrives only in Mission VII. The book's proof is the Doeblin decomposition: by Proposition 1.7 some power satisfies $P^r(x,y)\ge\delta\pi(y)$, so $P^r=(1-\theta)\Pi+\theta Q$ with $\Pi$ the rank-one matrix of rows $\pi$, and induction gives $P^{rk}=(1-\theta^k)\Pi+\theta^kQ^k$. The formal work is matrix algebra with careful bookkeeping of the remainder chain $Q$, plus the monotonicity of $d$ needed to interpolate between multiples of $r$. For Proposition 4.7 the delicate half is constructing the optimal coupling: mass $\mu\wedge\nu$ on the diagonal and the normalized product of the positive parts off it, with the degenerate case $\mu=\nu$ handled separately. The Glauber stationarity statement must be phrased with care because configurations outside the support of $\pi$ have junk rows; the formalization asserts stochasticity only at supported configurations, and detailed balance globally.

Formalization scope

Total variation distance is defined as the supremum over events, $\bigsqcup_{A}\,|\mu(A)-\nu(A)|$ over Finset V, exactly as in (4.1); the half-$\ell^1$ formula is a milestone, not the definition. The mixing time is sInf of the set $\{t : d(t)\le\varepsilon\}$ in $\mathbb N$ (junk value $0$ if empty — impossible under the goal theorem). Couplings are distributions on the product with prescribed marginals; no probability-space machinery is used. The mixing-time inequalities are stated with the integer-rounding slack made explicit (e.g. $\lceil\log_2\varepsilon^{-1}\rceil$ via Nat.ceil of a real logarithm) so that no statement is true only "up to rounding". The Metropolis definitions use total real division, so the hypotheses require $\pi>0$ pointwise; this matches the book, which divides by $\pi(x)$ throughout.

Welcome contributions beyond the milestones: simp lemmas for tvDist, monotonicity of $d$ and $\bar d$ in $t$, and triangle-inequality infrastructure — all reused by Missions III–XIII.

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
• N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21 (1953). https://doi.org/10.1063/1.1699114
• W. Doeblin, Exposé de la théorie des chaînes simples constantes de Markov à un nombre fini d'états, Rev. Math. Union Interbalkan. 2 (1938).

Motivation

The Convergence Theorem of Mission II says an irreducible aperiodic chain mixes geometrically, but with constants coming from a crude Doeblin decomposition — useless for actual chains, whose state spaces are exponentially large. Chapters 5–6 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) develop the two classic probabilistic techniques that give useful upper bounds: coupling — run two copies of the chain jointly so that they meet quickly, and read a TV bound off the meeting time — and strong stationary times — random times at which the chain is exactly stationary, independent of the time. The flagship application is the top-to-random shuffle: repeatedly take the top card of a deck of $n$ cards and reinsert it at a uniform position; the deck is well mixed after $n\log n + cn$ shuffles, with error at most $e^{-c}$.

Setting

A Markovian coupling of a chain $P$ (with the stay-together convention (5.2)) is a chain $Q$ on pairs whose coordinate marginals are both $P$ and which moves diagonal states to diagonal states. The coupling time $\tau_{\mathrm{couple}}$ is the hitting time of the diagonal; its tails are expressed with the trajectory calculus of Mission I.

A randomized stopping time is presented by its stopping rule: for each time $t$ and trajectory prefix $\omega$, a number $s_t(\omega)\in[0,1]$, the conditional probability of stopping at $t$ given the trajectory so far and no earlier stop. A strong stationary time for the chain started at $x$ is an almost surely finite such $\tau$ with

$$\mathbb P_x\{\tau=t,\ X_\tau=y\}=\mathbb P_x\{\tau=t\}\,\pi(y),$$

i.e. $X_\tau\sim\pi$ independent of $\tau$. The separation distance is $s_x(t)=\max_y\,[1-P^t(x,y)/\pi(y)]$.

The mission also fixes the concrete chains it bounds: the lazy walk on the discrete torus $\mathbb Z_n^d$, the Metropolis chain on proper $q$-colorings of a graph, the Glauber dynamics of the hardcore model with fugacity $\lambda$, and the top-to-random shuffle on decks of $n$ cards (states are arrangements, i.e. permutations; position $0$ is the top).

Formalization targets

Goal

Let $d(t)= \max_x\|P^t (x,\cdot)−\pi\|_{TV}$,

$$d\bigl(\lceil n\log n+\alpha n\rceil\bigr)\;\le\;e^{-\alpha}\qquad(\alpha>0)$$

for the top-to-random shuffle on $n\ge2$ cards — display (6.16) of the book, the chapter's flagship bound, proved by combining the strong stationary time $\tau_{\mathrm{top}}$ with the coupon collector tail of Mission I.

Milestones

Theorem 5.2 and Corollary 5.3 (the coupling bound: $\|P^t(x,\cdot)-P^t(y,\cdot)\|_{\mathrm{TV}}\le\mathbb P_{x,y}\{\tau_{\mathrm{couple}}>t\}$, hence $d(t)\le\max_{x,y}\mathbb P_{x,y}\{\tau_{\mathrm{couple}}>t\}$); Theorem 5.5 ($t_{\mathrm{mix}}(\varepsilon)\le c(d)\,n^2\log_2\varepsilon^{-1}$ for the lazy torus walk); Theorem 5.7 (Metropolis colorings, $q>3\Delta$: mixing in $O(n\log n)$); Theorem 5.8 (hardcore Glauber, $\lambda<(\Delta-1)^{-1}$: mixing in $O(n\log n)$); Proposition 6.1 with Example 6.7 ($\tau_{\mathrm{top}}$ is a strong stationary time); Lemma 6.11 ($s_x(t)\le\mathbb P_x\{\tau>t\}$); Lemma 6.13 ($\|P^t(x,\cdot)-\pi\|_{\mathrm{TV}}\le s_x(t)$); Proposition 6.10 ($d(t)\le\max_x\mathbb P_x\{\tau>t\}$).

Significance

The results. The coupling bound is the single most used upper-bound technique in the subject: Missions VIII (path coupling), IX (Ising) and XI (cutoff examples) all instantiate it. Strong stationary times and separation distance return in Mission XI (separation cutoff) and underlie perfect sampling in Mission XIII. The three concrete bounds (torus, colorings, hardcore) are the standard first applications and give the first polynomial mixing results of the series; the colorings and hardcore chains are the objects of intense ongoing research on sampling thresholds.

Formalizing them. Nothing here exists in Mathlib. The novel infrastructure is the stopping-rule formalization of randomized stopping times over trajectory prefixes — measure-theory-free, but expressive enough for the strong stationarity identity — and the Markovian-coupling predicate on pair chains. Both are reused later in the series (Matthews method, cutoff, CFTP).

Difficulty

The coupling bound itself is short once Proposition 4.7 (Mission II) is available; the work is in the applications. For the torus, the coordinatewise coupling requires assembling $d$ one-dimensional couplings and bounding the coupling time by a sum of one-dimensional meeting times — the formal bookkeeping of "couple coordinate by coordinate" is the real cost, and the constant $c(d)$ absorbs it. For Theorem 5.7 and 5.8 the argument is a grand coupling over all colorings/configurations simultaneously; the formal statements quantify only over the resulting bound, but a solver must build the coupling. For Proposition 6.1, the crux is the induction "given $k$ cards under the original bottom card, all $k!$ orders are equally likely" — an exchangeability argument that must be carried through the stopping-rule encoding. Lemma 6.11 is where the definition of strong stationarity does its work; the naive attempt to prove Proposition 6.10 directly from the coupling characterization fails, which is exactly why separation distance is introduced.

Formalization scope

Couplings of chains are transition matrices on $V\times V$ with marginal conditions stated row by row; the stay-together convention is part of the predicate, matching (5.2). Stopping rules take the trajectory prefix (which includes the starting state), so times "depending on the starting position" are covered; strong stationarity packages the stopping rule bounds, almost-sure finiteness ($\sum_t\mathbb P_x\{\tau=t\}=1$), and the product identity. The colorings chain lives on the subtype of proper colorings; the hardcore chain is the Glauber dynamics of Mission II restricted to the subtype of hardcore configurations (transitions never leave it). Mixing-time upper bounds carry an explicit $+1$ for integer rounding where the book's real-valued display would otherwise be false for the integer-valued $t_{\mathrm{mix}}$. The torus statement fixes $\varepsilon\le 1/2$; for $\varepsilon$ near $1$ the display is false as stated in the book.

Welcome contributions: interface lemmas between setAvoidTailProb of the pair chain and the two coordinates; the taboo-matrix form of coupling-time tails; exchangeability infrastructure for the deck chains (reused in Mission V).

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, J. A. Fill, Strong stationary times via a new form of duality, Ann. Probab. 18 (1990). https://doi.org/10.1214/aop/1176990628

Motivation

Missions II–III produce upper bounds on mixing times. Whether such a bound is sharp is a different question: an $O(n^2)$ bound on a chain that actually mixes in $n\log n$ steps hides the truth. Chapter 7 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) develops the standard toolkit of lower bounds: the counting and diameter bounds (a chain cannot spread faster than its transition graph allows), the bottleneck ratio (a chain cannot mix faster than it crosses its worst cut), and distinguishing statistics (a chain is far from stationarity as long as some statistic separates $P^t(x,\cdot)$ from $\pi$ by several standard deviations). The bottleneck bound — closely related to the conductance of the chain and, via Mission VII, to the Cheeger inequality — is the single most important obstruction result in the subject: it is how slow mixing (torpid mixing of Ising at low temperature, in Mission IX) is proved.

Setting

For a chain $P$ with stationary distribution $\pi$, the edge measure is $Q(x,y)=\pi(x)P(x,y)$, the bottleneck ratio of a set $S$ of states is

$$\Phi(S)=\frac{Q(S,S^{c})}{\pi(S)},\qquad Q(S,S^c)=\sum_{x\in S,\,y\notin S}\pi(x)P(x,y),$$

and the bottleneck ratio of the chain is $\Phi_\star=\min\{\Phi(S):\pi(S)\le\tfrac12,\ S\neq\varnothing\}$. The maximal one-step out-degree is $\Delta=\max_x|\{y:P(x,y)>0\}|$; the diameter is measured in the graph joining $x\ne y$ when $P(x,y)+P(y,x)>0$. For a statistic $f:V\to\mathbb R$ and distribution $\mu$, $\mathbb E_\mu(f)$ and $\operatorname{Var}_\mu(f)$ are the finite-sum expectation and variance, and $\mu f^{-1}$ denotes the pushforward of $\mu$ under $f$.

Formalization targets

Goal

$$t_{\mathrm{mix}}\;\ge\;\frac{1}{4\Phi_\star}.$$

This is Theorem 7.3, the bottleneck-ratio lower bound, the chapter's central theorem.

Milestones

The counting bound $t_{\mathrm{mix}}(\varepsilon)\ge\log\bigl(|V|(1-\varepsilon)\bigr)/\log\Delta$ for chains with uniform stationary distribution (§7.1.1, display (7.2)); the diameter bound: any two states satisfy $\mathrm{dist}(x_0,y_0)\le 2\,t_{\mathrm{mix}}(\varepsilon)$ for $\varepsilon<1/2$ (§7.1.2, display (7.3)); Proposition 7.8 (a statistic separating means by $r$ standard deviations forces $\|\mu-\nu\|_{\mathrm{TV}}\ge1-4/(4+r^2)$); Lemma 7.9 (projection under a statistic does not increase TV distance); Proposition 7.13 (the lazy hypercube walk satisfies $d(\tfrac12 n\log n-\alpha n)\ge 1-8e^{1-2\alpha}$); and Proposition 7.14 (the top-to-random shuffle needs $n\log n-O(n)$ shuffles, matching the upper bound of Mission III).

Significance

The results. Together with Mission III this pins the top-to-random shuffle at $n\log n\pm O(n)$ — the first sharp mixing result of the series, and the prototype of the cutoff phenomenon formalized in Mission XI. Proposition 7.13 similarly matches the hypercube upper bound and feeds the cutoff analysis. The bottleneck bound is used in Mission IX to prove exponentially slow mixing of the mean-field Ising model at low temperature, and its two-sided refinement is the Cheeger inequality of Mission VII.

Formalizing them. Mathlib has no notion of conductance/bottleneck ratio of a chain, no distinguishing-statistic method, and no mixing-time lower bound of any kind. The pushforward and variance infrastructure over finitely supported distributions is elementary but new, and reusable wherever second-moment methods appear (Wilson's method in Mission VII).

Difficulty

The bottleneck theorem's proof is short but exact: it hinges on the identity $\pi(S)\|\mu_S P-\mu_S\|_{\mathrm{TV}}=Q(S,S^c)$ for $\pi$ conditioned on $S$, followed by a telescoping estimate of $\|\mu_S P^t-\mu_S\|_{\mathrm{TV}}$; the formal cost is manipulating conditioned measures and one-sided TV sums (Remark 4.3 from Mission II). For Proposition 7.8, the second-moment argument runs through Chebyshev on both distributions plus optimization of a threshold — the constants $4/(4+r^2)$ are exact, not asymptotic, so the formal inequalities must be done carefully. Proposition 7.13 requires the binomial mean/variance computation for Hamming weight under both $\pi$ and $P^t(\mathbf 1,\cdot)$, including the negative-correlation bound for unrefreshed coordinates. The naive route to a lower bound — "the chain has not left a small set, so it is far from $\pi$" — is precisely the counting bound and is too weak for the sharp results; the statistics method is what closes the gap.

Formalization scope

$\Phi_\star$ is an infimum over the subtype of nonempty sets with $\pi(S)\le\tfrac12$; on a one-point space this subtype is empty and the infimum takes a junk value, making the goal trivially true there (the bound carries content only for $|V|\ge2$, as in the book). The counting bound divides by $\log\Delta$ with total division ($\Delta\le1$ gives a trivially true statement). The diameter bound is stated for arbitrary pairs of states through SimpleGraph.dist of the transition graph, which subsumes the book's diameter formulation. Propositions 7.13 and 7.14 are stated for all integer times $t\le\frac12 n\log n-\alpha n$ (resp. $t\le n\log n-\alpha n$), using monotonicity of $d$ instead of evaluating at a real-valued time — this avoids floor artifacts while keeping the book's content. Proposition 7.14 quantifies "$\alpha$ large, then $n$ large" exactly as the book's iterated limit.

Welcome contributions: monotonicity of $d(t)$ in $t$; conditioned-measure lemmas; variance API for distExp/distVar.

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
• M. Jerrum, A. Sinclair, Approximating the permanent, SIAM J. Comput. 18 (1989). https://doi.org/10.1137/0218077
• D. Aldous, P. Diaconis, Shuffling cards and stopping times, Amer. Math. Monthly 93 (1986). https://doi.org/10.1080/00029890.1986.11971821

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

A reversible Markov chain is an electrical network: states are nodes, and the conductance $c(x,y)=\pi(x)P(x,y)$ turns hitting probabilities into voltages and hitting times into resistances. This dictionary, going back to Kakutani and popularized by Doyle and Snell, converts probabilistic estimates into the physical laws of circuits — series/parallel reduction, energy minimization, monotonicity under edge removal. Chapters 9–11 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) develop the dictionary and its two crown results: the commute time identity of Chandra–Raghavan–Ruzzo–Smolensky–Tiwari, $\mathbb E_a(\tau_b)+\mathbb E_b(\tau_a)=c_G\,R(a\leftrightarrow b)$, and the Matthews method bounding cover times by hitting times with harmonic-number precision.

Setting

A network is a symmetric nonnegative conductance function $c$ on pairs of vertices; the associated walk moves with probabilities

$$P(x,y)=c(x,y)/c(x)$$

where $c(x)=\sum_y c(x,y)$, and $c_G=\sum_x c(x)$. A function $h$ is harmonic at $x$ if $h(x)=\sum_y P(x,y)h(y)$. The voltage with boundary values $1$ at $a$ and $0$ at $z$ is $W(x)=\mathbb P_x\{\tau_a<\tau_z\}$, the harmonic extension of its boundary data; the current flowing out of $a$ has strength $\|I\|=\sum_y c(a,y)\,[W(a)-W(y)]$, and the effective resistance is $R(a\leftrightarrow z)=\|I\|^{-1}$. A flow from $a$ to $z$ is an antisymmetric edge function obeying the node law off $\{a,z\}$; its energy is $\mathcal E(\theta)=\sum_e\theta(e)^2/c(e)$. Hitting times $\tau_S=\min\{t\ge0:X_t\in S\}$, their expectations, the Green's function $G_{\tau_z}(a,x)$, the maximal hitting time $t_{\mathrm{hit}}$, and the cover time $t_{\mathrm{cov}}$ (expected time to visit every state, maximized over starts) all use the trajectory calculus of Mission I.

Formalization targets

Goal

$$\mathbb E_a(\tau_b)+\mathbb E_b(\tau_a)\;=\;c_G\,R(a\leftrightarrow b).$$

This is Proposition 10.6, the commute time identity — the exact bridge between the probabilistic and electrical sides, and the engine of the transience/recurrence theory of Mission XII.

Milestones

Reversibility and stationarity of the network walk with $\pi(x)=c(x)/c_G$ (§9.1); Proposition 9.1 (existence and uniqueness of harmonic extensions with given boundary values, $h(x)=\mathbb E_x f(X_{\tau_B})$); Lemma 9.6 (the Green's function identity $G_{\tau_z}(a,a)=c(a)R(a\leftrightarrow z)$); Theorem 9.10 (Thomson's principle: $R(a\leftrightarrow z)$ is the minimal energy of a unit flow, attained); Theorem 9.12 (Rayleigh monotonicity: lowering conductances raises effective resistance); Lemma 10.1 (the random target lemma: $\sum_y\mathbb E_a(\tau_y)\pi(y)$ does not depend on $a$); Corollary 10.8 (the resistance triangle inequality); Theorem 11.2 (Matthews: $t_{\mathrm{cov}}\le t_{\mathrm{hit}}\,(1+\tfrac12+\dots+\tfrac1n)$); Proposition 11.4 (the matching Matthews lower bound over subsets).

Significance

The results. The commute time identity computes hitting times from circuit reductions — this is how hitting times on trees, tori and glued graphs are actually evaluated — and, through Thomson and Rayleigh, makes them monotone under graph operations, something invisible probabilistically. The Matthews bounds pin cover times up to a $\log n$ factor in complete generality; they are the tool behind cover-time results for lamplighter groups in Mission XI's sequel. Green's function identities feed Mission XII's recurrence theory, where $R(a\leftrightarrow\infty)$ decides transience.

Formalizing them. Mathlib has graph Laplacians but no electrical network theory: no effective resistance, no flows, no energy, no Thomson/Rayleigh, no hitting or cover times. This mission publishes that layer over the trajectory calculus of Mission I. It is the most reusable single block of the series outside Missions I–II: effective resistance on finite networks is of independent interest to combinatorics (spanning trees, spectral sparsification) well beyond mixing times.

Difficulty

The identity chain behind the goal runs: Green's function of the stopped walk $\to$ escape probability $\mathbb P_a\{\tau_z<\tau_a^+\}=\bigl(c(a)R(a\leftrightarrow z)\bigr)^{-1}$ (via harmonic uniqueness) $\to$ the Aldous–Fill occupation identity $G_\tau(a,x)=\mathbb E_a(\tau)\pi(x)$ for stopping times with $X_\tau=a$ — each step is a manipulation of infinite series of trajectory sums whose exchange steps (splitting a path at its first visit, last-exit decompositions) need summability from Mission I's Lemma 1.13. Thomson's principle is a finite-dimensional convex minimization: existence of the minimizer needs a compactness or completing-the-square argument, and the identification of the minimizer with the current flow needs the cycle law; the naive "differentiate the energy" route must be made exact. Matthews' method is a clean but genuinely clever argument — a uniformly random ordering of targets and the harmonic-number telescoping; the formal cost is the exchangeability of the randomized order against the chain, handled combinatorially.

Formalization scope

Networks are functions $c:V\times V\to\mathbb R$ with a symmetry-and-nonnegativity predicate; loops are permitted; connectivity enters as irreducibility of the induced walk. The voltage is defined probabilistically as $\mathbb P_x\{\tau_a<\tau_z\}$ (the book's harmonic characterization is Proposition 9.1); $R(a\leftrightarrow z)$ is the reciprocal of the explicit current strength, with total division junk when $a,z$ are disconnected — statements carry irreducibility so this does not arise. Energy counts each undirected edge once, formalized as half the ordered double sum, and $0^2/0=0$ handles absent edges. Cover times are tail sums of the explicit event "some state unvisited". The Matthews lower bound is stated with an arbitrary lower bound $m$ for the pairwise hitting times of the subset $A$ — equivalent to the book's min over pairs and easier to instantiate.

Welcome contributions: series/parallel reduction laws, the cycle and node law API for flows, escape-probability lemmas — all reused in Mission XII's infinite-network arguments.

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. G. Doyle, J. L. Snell, Random Walks and Electric Networks, MAA, 1984. https://arxiv.org/abs/math/0001057
• A. K. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky, P. Tiwari, The electrical resistance of a graph captures its commute and cover times, STOC 1989. https://doi.org/10.1145/73007.73062
• P. Matthews, Covering problems for Markov chains, Ann. Probab. 16 (1988). https://doi.org/10.1214/aop/1176991894

Motivation

How fast does a Markov chain forget its starting point? For a reversible chain the complete answer is coded in the eigenvalues of its transition matrix: the largest eigenvalue is always $1$, and the size of the gap between $1$ and the rest of the spectrum is the chain's fundamental time constant — a large gap means fast mixing, a small gap means slow mixing. Chapters 12–13 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) develop this spectral theory and culminate in the discrete Cheeger inequality of Jerrum–Sinclair and Lawler–Sokal, which says that the spectral gap $\gamma$ (an analytic quantity, defined below) and the bottleneck constant $\Phi_\star$ (a geometric quantity from Mission IV, also recalled below) control each other:

$$\frac{\Phi_\star^2}{2}\;\le\;\gamma\;\le\;2\,\Phi_\star.$$

A chain mixes rapidly exactly when its state space has no bottleneck. This inequality is the backbone of the Markov-chain approach to approximate counting and of spectral graph theory at large; alongside it the chapters provide Wilson's method — the sharpest general technique for mixing-time lower bounds — and the comparison machinery that transfers spectral estimates between chains.

Setting

All chains live on a finite state space $V$. A chain with transition matrix $P$ and stationary distribution $\pi$ is reversible when the detailed balance equations $\pi(x)P(x,y)=\pi(y)P(y,x)$ hold; reversibility is the standing assumption of both chapters. The yardsticks of the series are the total variation distance $\|\mu-\nu\|_{TV}=\max_{A\subseteq V}|\mu(A)-\nu(A)|$, the worst-case distance to stationarity $d(t)=\max_x\|P^t(x,\cdot)-\pi\|_{TV}$, and the mixing time $t_{\mathrm{mix}}(\varepsilon)=\min\{t: d(t)\le\varepsilon\}$, with $t_{\mathrm{mix}}=t_{\mathrm{mix}}(1/4)$; we write $\pi_{\min}=\min_x\pi(x)$.

The spectral vocabulary: an eigenfunction of $P$ is a nonzero $f:V\to\mathbb R$ with $Pf=\lambda f$, where $(Pf)(x)=\sum_yP(x,y)f(y)$; the number $\lambda$ is then an eigenvalue. Out of the spectrum one forms

• $\lambda_2$, the largest eigenvalue different from $1$, and the spectral gap $\gamma=1-\lambda_2$;
• $\lambda_\star$, the largest absolute value of an eigenvalue different from $1$, the absolute gap $\gamma_\star=1-\lambda_\star$, and the relaxation time $t_{\mathrm{rel}}=1/\gamma_\star$.

Functions on $V$ carry the weighted inner product $\langle f,g\rangle_\pi=\sum_xf(x)g(x)\pi(x)$ — the geometry, denoted $\ell^2(\pi)$, in which a reversible $P$ is self-adjoint — and the Dirichlet form

$$\mathcal E(f)=\tfrac12\sum_{x,y}\bigl[f(x)-f(y)\bigr]^2\,\pi(x)P(x,y),$$

the average squared variation of $f$ along the chain's transitions. Finally, from Mission IV: the bottleneck ratio of a set $S$ of states is $\Phi(S)=\sum_{x\in S,\,y\notin S}\pi(x)P(x,y)\,/\,\pi(S)$, the conditional probability at stationarity of escaping $S$ in one step, and the bottleneck constant is $\Phi_\star=\min\{\Phi(S):\varnothing\ne S,\ \pi(S)\le\tfrac12\}$.

Formalization targets

Goal

Theorem 13.14, the discrete Cheeger inequality: for a reversible irreducible chain,

$$\frac{\Phi_\star^2}{2}\;\le\;\gamma\;\le\;2\,\Phi_\star.$$

Milestones

• Lemma 12.1 — every eigenvalue satisfies $|\lambda|\le1$; for an irreducible chain the eigenfunctions of the eigenvalue $1$ are the constant functions; an irreducible aperiodic chain does not have $-1$ as an eigenvalue.
• Lemma 12.2 — the spectral representation: a reversible chain admits eigenfunctions $f_1,\dots,f_{|V|}$, orthonormal with respect to $\langle\cdot,\cdot\rangle_\pi$, with eigenvalues $\lambda_j$, such that $P^t(x,y)/\pi(y)=\sum_jf_j(x)f_j(y)\lambda_j^t$ for all $t,x,y$.
• Theorem 12.3 — mixing is at most relaxation times a log factor: $t_{\mathrm{mix}}(\varepsilon)\le\log\bigl(1/(\varepsilon\,\pi_{\min})\bigr)\,t_{\mathrm{rel}}+1$.
• Theorem 12.4 — mixing is at least the relaxation time: $t_{\mathrm{mix}}(\varepsilon)\ge(t_{\mathrm{rel}}-1)\log\bigl(1/(2\varepsilon)\bigr)$.
• §12.3.1 — the model computation: simple random walk on the $n$-cycle has the numbers $\cos(2\pi j/n)$, $j=0,\dots,n-1$, among its eigenvalues.
• Theorem 13.1 — if every pair of states admits a coupling of the two one-step distributions that contracts some metric $\rho$ on $V$ by a factor $\theta$ in expectation, then $\lambda_\star\le\theta$.
• Theorem 13.5, Wilson's method — an eigenfunction $\Phi$ with eigenvalue $\lambda\in(\tfrac12,1)$ whose one-step increments have second moment at most $R$ yields the explicit lower bound $t_{\mathrm{mix}}(\varepsilon)\ge\bigl[2\log(1/\lambda)\bigr]^{-1}\bigl[\log\bigl((1-\lambda)\Phi(x)^2/(2R)\bigr)+\log\bigl((1-\varepsilon)/\varepsilon\bigr)\bigr]$.
• Lemmas 13.11–13.12 — the variational characterization: $\gamma$ is the minimum of $\mathcal E(f)$ over functions with mean zero ($\sum_xf(x)\pi(x)=0$) and unit norm ($\langle f,f\rangle_\pi=1$), and the minimum is attained.
• Lemma 13.22 — the comparison method: if a second reversible chain $\tilde P$ on the same space, with stationary distribution $\tilde\pi$, Dirichlet form $\tilde{\mathcal E}$, and gap $\tilde\gamma$, satisfies $\tilde{\mathcal E}(f)\le B\,\mathcal E(f)$ for every $f$, then $\tilde\gamma\le\bigl[\max_x\pi(x)/\tilde\pi(x)\bigr]B\,\gamma$.

Significance

The results. Theorems 12.3–12.4 sandwich the mixing time between $t_{\mathrm{rel}}$ and $t_{\mathrm{rel}}\log(1/\pi_{\min})$ — the fundamental equivalence of spectral and mixing estimates for reversible chains, prerequisite for the cutoff criterion of Mission XI. The Cheeger inequality converts isoperimetry into spectral bounds; it is the mathematical core of the Jerrum–Sinclair program of polynomial-time approximate counting, and its graph version underlies expander theory. Wilson's method produced the sharp lower bounds for adjacent transpositions and hypercube-type chains; the comparison lemma is the engine behind the shuffle bounds cited in Mission V (§16.1).

Formalizing them. Mathlib has the spectral theorem for symmetric matrices but nothing connecting spectra to Markov chains: no spectral gap, no relaxation time, no Dirichlet forms, no Cheeger inequality in any form. A formalized discrete Cheeger inequality would be a landmark reusable well outside this series (spectral graph theory, expanders); the eigenvalue and Rayleigh-quotient layer built here is what Missions IX (tree relaxation, block dynamics) and XI (cutoff criterion) consume.

Difficulty

Everything routes through one change of basis: conjugating $P$ by the diagonal matrix with entries $\sqrt{\pi(x)}$ produces a matrix that is symmetric precisely because the chain is reversible, so Mathlib's spectral theorem applies — but transporting the resulting eigenbasis back to $\ell^2(\pi)$, keeping track of orthonormality with respect to the weighted inner product, is a genuine formal-linear-algebra project; nothing about it is deep, all of it is fussy. The definitions of $\lambda_2$ and $\lambda_\star$ as suprema over the set of non-unit eigenvalues (finite, and nonempty once $|V|\ge2$) must be reconciled with the eigenbasis enumeration before any variational argument runs. The upper half of Cheeger is direct from the variational characterization — test it on the indicator function of a bottleneck set $S$, recentred to have mean zero; the lower half is the hard half: the standard proof takes an optimal $f$, decomposes it over its level sets $\{f>c\}$, and applies Cauchy–Schwarz twice, and formalizing that level-set sweep is the main effort of the mission. Wilson's method needs a supermartingale-style iteration of the eigenfunction estimate; its constants are exact, so the inequalities cannot be rounded.

Formalization scope

Eigenvalues are defined by real eigenvectors ($\exists f\ne0,\ Pf=\lambda f$); for reversible chains this captures the whole spectrum, and all statements assume reversibility wherever the book does. $\lambda_2$ and $\lambda_\star$ are suprema of explicit sets of reals; on a one-point space these sets are empty and the supremum takes a junk value, so the affected statements carry the explicit hypothesis $|V|\ge2$, matching the book's implicit assumption of a non-degenerate chain. $t_{\mathrm{rel}}=(1-\lambda_\star)^{-1}$ with total inverse. Theorem 12.3 carries an explicit $+1$ absorbing the rounding of a real-valued bound to an integer time. The cycle eigenvalue statement exhibits eigenvalues (existence of eigenfunctions); completeness of that list is not asserted. The variational characterization asserts both the minimization identity and its attainment, so it can be used in either direction.

Welcome contributions: the symmetrization API (conjugation by $\mathrm{diag}(\sqrt{\pi})$), Rayleigh-quotient lemmas, level-set (layer-cake) infrastructure — all reused by Missions IX and XI.

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
• M. Jerrum, A. Sinclair, Approximating the permanent, SIAM J. Comput. 18 (1989). https://doi.org/10.1137/0218077
• G. F. Lawler, A. D. Sokal, Bounds on the L² spectrum for Markov chains and Markov processes, Trans. Amer. Math. Soc. 309 (1988). https://doi.org/10.1090/S0002-9947-1988-0930082-9
• D. B. Wilson, Mixing times of lozenge tiling and card shuffling Markov chains, Ann. Appl. Probab. 14 (2004). https://doi.org/10.1214/aoap/1042765669

Motivation

The coupling method of Mission III asks for a coupling of two copies of a chain from every pair of starting states — often painful to construct globally. Chapter 14 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) replaces that global demand by a local one. The path coupling technique of Bubley and Dyer says: put a connected graph structure on the state space, and couple one step of the chain only across edges; if each edge contracts in expectation, contraction propagates automatically along paths to arbitrary pairs of distributions. The bookkeeping runs through the transportation metric (Kantorovich distance) between distributions, whose theory — attainment by an optimal coupling, the triangle inequality — is developed on the way. The chapter's payoff is the sharpest elementary bound for sampling proper colorings (Theorem 14.8: the Glauber dynamics mixes in $O(n\log n)$ steps once $q>2\Delta$), and, through the sampling-to-counting reduction of Jerrum–Valiant–Vazirani, a polynomial-time approximation algorithm for counting colorings — the paradigm of the Markov chain Monte Carlo method as an algorithmic tool.

Setting

All chains live on a finite state space $V$ with a transition matrix $P$; $P^t(x,\cdot)$ is the time-$t$ distribution from $x$, $\|\mu-\nu\|_{TV}=\max_{A\subseteq V}|\mu(A)-\nu(A)|$ the total variation distance, $d(t)=\max_x\|P^t(x,\cdot)-\pi\|_{TV}$ the worst-case distance to the stationary distribution $\pi$, and $t_{\mathrm{mix}}(\varepsilon)=\min\{t:d(t)\le\varepsilon\}$ the mixing time. A coupling of distributions $\mu,\nu$ is a distribution $q$ on $V\times V$ with marginals $\mu$ and $\nu$.

Given a metric-like cost $\rho$ on pairs of states, the transportation metric between two distributions is the cheapest expected cost of moving one onto the other:

$$\rho_K(\mu,\nu)=\min\Bigl\{\sum_{x,y}\rho(x,y)\,q(x,y)\;:\;q\ \text{a coupling of}\ \mu,\nu\Bigr\}.$$

Given a connected graph structure $G$ on the state space with edge lengths $\ell\ge1$, the path metric $\rho(x,y)$ is the least total length of a $G$-path from $x$ to $y$.

For the colorings application: a $q$-coloring of the vertices of a graph is proper when adjacent vertices receive distinct colors, and the Glauber dynamics on proper colorings picks a uniform vertex and re-samples its color uniformly among the colors legal there; its stationary distribution is uniform on the proper colorings. Throughout, $n$ is the number of vertices and $\Delta$ the maximum degree of the graph being colored.

Formalization targets

Goal

Theorem 14.8, the capstone of Chapter 14: for the Glauber dynamics on proper $q$-colorings, if $q>2\Delta$ then

$$t_{\mathrm{mix}}(\varepsilon)\;\le\;\Bigl\lceil\frac{q-\Delta}{q-2\Delta}\;n\,\bigl(\log n-\log\varepsilon\bigr)\Bigr\rceil.$$

Milestones

• Lemma 14.3 and Remark 14.2 — the transportation distance is attained by an optimal coupling, and satisfies the triangle inequality (so it is a genuine metric on distributions).
• Theorem 14.6, path coupling (Bubley–Dyer) — if for every edge $\{x,y\}$ of a connected graph structure there is a coupling of the one-step distributions $P(x,\cdot),P(y,\cdot)$ contracting the path metric by $e^{-\alpha}$ in expectation, then one step of the chain contracts the transportation metric of arbitrary distribution pairs by $e^{-\alpha}$.
• Corollary 14.7 — under the same hypotheses, $d(t)\le e^{-\alpha t}\,\mathrm{diam}(V)$ and $t_{\mathrm{mix}}(\varepsilon)\le\lceil(\log\mathrm{diam}(V)-\log\varepsilon)/\alpha\rceil$, where $\mathrm{diam}(V)$ is the largest path-metric distance between two states.
• Theorem 14.12, approximate counting — for $q>2\Delta$ there is a randomized estimator, computed from an explicit polynomial number of independent uniform random seeds, which with probability at least $1-\eta$ estimates the number of proper $q$-colorings within a $(1\pm\varepsilon)$ factor: rapid sampling yields rapid approximate counting.

Significance

The results. Path coupling converted the coupling method from an art into a calculus: one bounds a single-edge contraction constant, and the machinery does the rest. It is the standard tool for Glauber dynamics on colorings, independent sets, and other constraint-satisfaction models, and the $q>2\Delta$ colorings bound is its flagship application. Theorem 14.12 is the discrete embodiment of the Jerrum–Valiant–Vazirani equivalence between approximate counting and sampling — the conceptual foundation of the entire MCMC approach to $\#\mathrm P$-hard counting problems.

Formalizing them. Mathlib has no transportation/Kantorovich metric in the finite setting, no path coupling, and nothing on approximate counting. The transportation-metric layer (optimal couplings, triangle inequality) is reusable far beyond this mission — it is the finite Wasserstein distance. The path-coupling theorem feeds directly into Mission IX (Ising) and is quoted throughout modern mixing literature.

Difficulty

The transportation metric asks for minimization over the (compact) polytope of couplings: attainment is a finite-dimensional compactness argument, and the triangle inequality requires gluing two optimal couplings along their common marginal — the classic construction that must be carried out with explicit finite sums here. Path coupling itself is an induction along geodesics of the path metric, with the subtlety that the composite coupling produced along a path need not be optimal, only admissible; the bookkeeping of the contraction constant through the induction is exactly the kind of argument Lean keeps honest. Theorem 14.8 instantiates the machinery: the single-edge coupling for colorings needs a careful case analysis of the proposed recolorings at the two endpoints (matching legal colors bijectively), and the contraction constant $(q-2\Delta)/(q-\Delta)$ emerges from counting disagreeing proposals. Theorem 14.12 layers a probabilistic-amplification argument (medians of means over independent runs) on top of the mixing bound; its combinatorial core — expressing $|\Omega|^{-1}$ as a telescoping product of marginal probabilities — is elementary but notation-heavy, and the formal statement quantifies over explicit seed spaces, so the whole estimator is a finite object.

Formalization scope

The transportation metric is an sInf over coupling costs (the coupling polytope is nonempty for genuine distributions, and attainment is part of the milestone, so the junk value never propagates); the path metric is an sInf over walk lengths in a connected graph. The Glauber dynamics on colorings is the restriction to proper colorings of the single-site heat-bath chain of Mission II, matching §3.3 of the book; its state space is the subtype of proper colorings, nonempty whenever $q>2\Delta$ (a fact the hypotheses of the goal supply). Mixing-time upper bounds are stated with the book's explicit ceilings, so no rounding slack is hidden. In Theorem 14.12 the estimator is presented concretely as a function of finitely many uniform seeds, and "with probability $\ge1-\eta$" is a counting inequality over the seed space — no measure theory enters. Contributions of intermediate lemmas (optimal-coupling gluing, geodesic decompositions, colorings edge-coupling) are welcome and will be reused by Mission IX.

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
• R. Bubley, M. Dyer, Path coupling: a technique for proving rapid mixing in Markov chains, FOCS 1997. https://doi.org/10.1109/SFCS.1997.646111
• M. Jerrum, A very simple algorithm for estimating the number of k-colorings of a low-degree graph, Random Structures Algorithms 7 (1995). https://doi.org/10.1002/rsa.3240070205
• M. Jerrum, L. Valiant, V. Vazirani, Random generation of combinatorial structures from a uniform distribution, Theoret. Comput. Sci. 43 (1986). https://doi.org/10.1016/0304-3975(86)90174-X

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