Markov chain Monte Carlo turns hard integration problems into long simulations: to estimate an expectation $E_\pi f$ one runs a Markov chain with stationary distribution $\pi$ and reports the sample average $\bar f_n$. The ergodic theorem guarantees $\bar f_n \to E_\pi f$, but honest error bars require more: a central limit theorem

$$\sqrt{n}(\bar f_n - E_\pi f) \to_d N(0, \sigma_f^2).$$

On general state spaces this is famously delicate - a merely ergodic chain with a square-integrable functional can fail the CLT, so the classical theory trades convergence rates (drift, minorization, geometric or polynomial total-variation rates) and mixing conditions ($\alpha$-, $\rho$-, $\varphi$-mixing) against moment conditions on $f$. This mission formalizes G. L. Jones's survey "On the Markov chain central limit theorem" (Probability Surveys, 2004): the drift-condition CLTs of Meyn-Tweedie and Jarner-Roberts, the classical mixing CLTs of Ibragimov-Linnik, Doukhan-Massart-Rio and Billingsley, the characterizations via uniform integrability and boundedness in probability, and their assembly into the summary theorem: six practically checkable regimes - from polynomial ergodicity with bounded functionals to uniform ergodicity with second moments - each of which guarantees the CLT for every initial distribution. The stationarity, total-variation and mixing infrastructure is general state space and reusable well beyond this mission.

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 whole series so far lived in discrete time on finite state spaces. Chapters 20–21 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) lift both restrictions. Running the jumps of a chain at the arrivals of a rate-one Poisson clock produces the continuous-time chain, whose transition semigroup — the heat kernel — converges to stationarity for every irreducible chain, with no aperiodicity hypothesis: continuous time washes out periodicity. On countable state spaces, existence of a stationary distribution is no longer automatic, and the trichotomy of transience, null recurrence, and positive recurrence replaces it; the convergence theorem survives exactly on the positive-recurrent class. The chapter's crown jewel — and this mission's goal — is Pólya's theorem: simple random walk on the lattice $\mathbb Z^d$ is recurrent in dimensions one and two and transient in dimension three and higher. "A drunk man will find his way home, but a drunk bird may get lost forever."

Setting

Continuous time (Ch. 20). For a finite chain $P$, the heat kernel at time $t\ge0$ is defined by Poissonization,

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

the law at time $t$ of a walk taking $P$-steps at Poisson arrival times ($=e^{t(P-I)}$ as a matrix exponential). With $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$, the continuous distance and mixing time are $d^{\mathrm{cont}}(t)=\max_x\|H_t(x,\cdot)-\pi\|_{TV}$ and $t^{\mathrm{cont}}_{\mathrm{mix}}(\varepsilon)=\inf\{t\ge0:d^{\mathrm{cont}}(t)\le\varepsilon\}$. The lazy version of a discrete chain is $\tfrac12(I+P)$; from Mission VII, the spectral gap $\gamma$ is $1-\lambda_2$ with $\lambda_2$ the largest eigenvalue $\ne1$. The product chain on a product of $n$ coordinate spaces picks a uniform coordinate and updates it by that coordinate's chain.

Countable state spaces (Ch. 21). A chain on a countable state space $V$ is a function $P$ with nonnegative entries and rows summing to one (as convergent series); $t$-step probabilities $P^t(x,y)$ are defined recursively, and trajectory probabilities are countable sums of path weights. The first return time to $x$ is $\tau_x^+=\min\{t\ge1:X_t=x\}$; a state is recurrent when $\mathbb P_x\{\tau_x^+<\infty\}=1$ (tails tend to zero), positive recurrent when moreover $\mathbb E_x(\tau_x^+)<\infty$ (tails summable), and null recurrent when recurrent but not positive recurrent. A stationary distribution is a nonnegative $\pi$ summing to one with $\pi P=\pi$ (as convergent series). Simple random walk on $\mathbb Z^d$ steps from $x$ to one of its $2d$ nearest neighbours uniformly at random.

Formalization targets

Goal

Pólya's theorem (§21.2, Examples 21.8–21.9), the capstone of Chapters 20–21:

1. for $d\le2$, simple random walk on $\mathbb Z^d$ is recurrent — it returns to its starting point with probability one;
2. for $d\ge3$, it is transient — with positive probability it never returns.

Milestones

• Theorem 20.1 — for any irreducible finite chain, aperiodic or not, the heat kernel converges: $d^{\mathrm{cont}}(t)\to0$ as $t\to\infty$.
• Theorem 20.3 — the two-way comparison between lazy discrete and continuous mixing: eventual $\varepsilon$-mixing of the lazy chain at time $k$ gives $2\varepsilon$-mixing of the heat kernel at time $k$, and $\varepsilon$-mixing of the heat kernel at time $m$ gives $2\varepsilon$-mixing of the lazy chain at time $4m$.
• Theorem 20.6 — the spectral bound for reversible chains: $\bigl|H_t(x,y)-\pi(y)\bigr|\le\sqrt{\pi(y)/\pi(x)}\;e^{-\gamma t}$.
• Theorem 20.7 — mixing of continuous-time product chains: with all coordinate gaps $\ge\gamma$ and coordinate stationary masses bounded below, $t^{\mathrm{cont}}_{\mathrm{mix}}(\varepsilon)\le(2\gamma)^{-1}n\log n+\gamma^{-1}n\log(1/(c_0\varepsilon))$, with a matching $\tfrac{n}{2\gamma}\log n$ lower bound when the gaps are equal — the $n\log n$ product-chain phenomenon.
• Proposition 21.3 — the recurrence dichotomy on countable spaces: a state is recurrent exactly when its Green's function $\sum_tP^t(x,x)$ diverges, and for an irreducible chain one recurrent state makes all states recurrent.
• Theorem 21.12 — an irreducible countable chain is positive recurrent if and only if it has a stationary distribution.
• Lemma 21.13 (Kac) — for an irreducible chain with stationary distribution $\pi$ and any nonempty set $S$: $\sum_{x\in S}\pi(x)\,\mathbb E_x(\tau_S^+)=1$; in particular $\mathbb E_x(\tau_x^+)=1/\pi(x)$.
• Theorem 21.14 — the convergence theorem on countable spaces: an irreducible, aperiodic, positive recurrent chain has a unique stationary distribution $\pi$, and $\|P^t(x,\cdot)-\pi\|_{TV}\to0$ from every start.
• Theorem 21.17 — in the null recurrent case, $P^t(x,y)\to0$ for all pairs of states: no stationary profile is approached.

Significance

The results. Theorem 20.1 explains why laziness and aperiodicity pervade the discrete theory — periodicity is an artifact of the discrete clock. The product-chain theorem 20.7 is the cleanest instance of the $n\log n$ paradigm (independent coordinates mix in relaxation time $\times\log(\text{number of coordinates})$) and the template for the hypercube cutoff of Mission XI. Chapter 21's trichotomy is the backbone of applied Markov chain theory — queueing, branching, renewal — and Kac's lemma with the convergence theorem 21.14 is the standard equipment of any probability course. Pólya's theorem is one of the most celebrated results of twentieth-century probability, the birth of the random-walk-in-dimension-$d$ paradigm.

Formalizing them. Mathlib has no continuous-time Markov chains, no Poissonization, and no recurrence/transience theory (its PMF random walks stop far short). The countable-state layer built here — summable stationary equations, tail-sum return times, the recurrence dichotomy — is the missing infrastructure for formalized applied probability; Pólya's theorem is a famous target in its own right, and the $d\ge3$ half has never been formalized in any assistant to our knowledge.

Difficulty

The heat kernel is an infinite series of matrices: convergence (dominated by the Poisson weights), the semigroup property, and the interchange of the series with matrix products and limits must all be established by hand over tsum. Theorem 20.1 avoids aperiodicity by the number-theoretic fact that the Poisson distribution smears over residue classes — formally, the continuous chain is automatically aperiodic because $H_t(x,x)>0$ for $t>0$. The product-chain bounds need the $\ell^2$ machinery of Mission VII applied coordinatewise and a careful union bound; the lower bound is a Gaussian-free second-moment argument. On the countable side, everything is series bookkeeping in the absence of Fintype: the recurrence dichotomy is a generating-function (renewal) identity $G(x,x)=1/\mathbb P_x\{\tau_x^+=\infty\}$ handled through partial sums; Kac's lemma is a mass-transport double-count over trajectories; and Theorem 21.14 needs an aperiodicity-based coupling on a countable product space, the technical summit of the mission. Pólya's theorem itself combines a local central-limit-type estimate for the return probabilities ($P^{2t}(0,0)\asymp t^{-d/2}$, obtained by Stirling in $d=1,2$ and by a comparison argument in higher dimension) with the dichotomy of Proposition 21.3.

Formalization scope

Chapter 20 lives on finite state spaces: the heat kernel is a tsum over $k$ of Poisson weights times matrix powers (summability is provable, not assumed), continuous distance is a supremum over states, and the continuous mixing time is an sInf over nonnegative reals (junk $0$ if the set were empty — excluded under the theorems' hypotheses). Discrete-vs-continuous comparison (Theorem 20.3) is stated with eventual thresholds ($\exists K,\forall k\ge K$), matching the book's asymptotic phrasing. Chapter 21 lives on a Countable type: stochasticity and stationarity are HasSum statements, $t$-step powers are defined recursively with tsum convolutions, return-time tails are countable sums of path weights over finite horizons, and recurrence/positive recurrence are the tail-limit and tail-summability conditions above — measure theory never enters. Pólya's theorem is stated for the origin of $\mathbb Z^d$ with the walk defined by nearest-neighbour steps; the $d\le2$ and $d\ge3$ halves are separate conjuncts of one statement.

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
• G. Pólya, Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Math. Ann. 84 (1921). https://doi.org/10.1007/BF01458701
• W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., Wiley, 1968.
• D. Aldous, J. A. Fill, Reversible Markov Chains and Random Walks on Graphs, 2002. https://www.stat.berkeley.edu/~aldous/RWG/book.html

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

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

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