Glauber 过程的拌和时间（第一篇）
Chapter 1
1.1 The Glauber dynamics
In this paper we consider the following problem:
Given a simple undirected graph \(G=(V,E)\), a set of forbidden vertices \(X\subseteq V\), and a positive activity parameter \(\lambda>0\), compute the partition function
\[Z(G)=\sum_{\substack{I\in\mathcal{I}(G),\,I\cap X=\varnothing}}\lambda^{|I|}, \]where \(\mathcal{I}(G)\) denotes the family of independent sets of \(G\), and each independent set \(I\) is weighted by \(\lambda^{|I|}\).
Throughout this paper, we will assume \(V=[n]=\{1,\cdots,n\}\).
The following process is called the (single-site) Heat-Bath Glauber dynamics:
- Initially, set every vertex to white.
- For a given number of iterations \(T\), repeat the following for \(T\) steps: at each step sample a vertex \(i\in[n]\) uniformly at random and perform the single-site update:
- If \(i\in X\) or any neighbor of \(i\) is black, then force \(i\) to remain white.
- Otherwise, set \(i\) to black with probability \(\frac{\lambda}{\lambda+1}\) and to white with probability \(\frac{1}{\lambda+1}\).
- After the iterations, output the set \(I\) of all black vertices as an independent set.
Throughout the process, the set of black vertices is always an independent set of \(G\).
We will prove the following mixing-time result. If there exists an integer degree bound \(\Delta\geq 3\) and a constant slackness \(0<\alpha<1\) such that
then the Heat-Bath Glauber dynamics has mixing time
Therefore, it is an excellent Monte Carlo sampler.
1.2 \(\mathsf{P}\) is ergodic and reversible w.r.t. \(\mu\)
Represent a coloring by \(\sigma:[n]\to\{0,1\}\), where \(\sigma(i)=1\) means vertex \(i\) is black (belongs to the independent set), and \(\sigma(i)=0\) means vertex \(i\) is white.
Define the state space
That is, \(\Omega\) is the set of all colorings that satisfy "forbidden vertices are white" and "black vertices form an independent set". Such colorings are called valid. Throughout the remainder of the text, all colorings refer to valid colorings unless explicitly stated otherwise.
On \(\Omega\) define the Gibbs distribution (with activity \(\lambda\)):
where the partition function is
For each vertex \(i\in[n]\) define the single-site update matrix \(\mathsf{P}_i\) as the operation that resamples the color of \(i\) according to the conditional Gibbs distribution while keeping \(\sigma_{[n]\setminus\{i\}}\) fixed. More formally:
A simple calculation shows that if at some step we sample \(i\), then the algorithm's action coincides with performing the update using \(\mathsf{P}_i\).
The overall chain has transition matrix \(\mathsf{P}=\frac{1}{n}\sum_{i=1}^n\mathsf{P}_i\).
Clearly \(\mathsf{P}\) is a row-stochastic matrix. We prove \(\mathsf{P}\) satisfies the detailed-balance equation with respect to \(\mu\):
We verify case by case:
- If \(\sigma=\tau\), then both sides equal \(\mu(\sigma)\mathsf{P}(\sigma\to\sigma)\).
- If \(\sigma,\tau\) differ on one vertex, call it \(i\). In this case both sides are equal to \(w(\sigma,\tau)=\frac{1}{n}\cdot\frac{\mu(\sigma)\mu(\tau)}{\mu(\sigma)+\mu(\tau)}\).
- If \(\sigma,\tau\) differ on two or more vertices, then \(\mathsf{P}(\sigma\to\tau)=\mathsf{P}(\tau\to\sigma)=0\).
Hence \(\mathsf{P}\) is time-reversible with respect to \(\mu\).
Finally, it helps to visualize a graph \(\mathcal{G}\) on the \(N=|\Omega|\) distinct colorings:
- For each coloring \(\sigma\), put a self-loop at \(\sigma\) with weight \(\mu(\sigma)\mathsf{P}(\sigma\to\sigma)\).
- For each pair of colorings \(\sigma\) and \(\tau\) that differ on one vertex, put an undirected edge between \(\sigma\) and \(\tau\) with weight \(w(\sigma,\tau)\).
On this graph \(\mathcal{G}\), since \(\mu(\sigma)>0\) for all colorings \(\sigma\), all edge weights (including self-loops) are positive. Furthermore, the weights of all incident edges sum to \(\mu(\sigma)\).
We can see \(\mathsf{P}\) is irreducible and aperiodic by looking at this graph:
- \(\mathsf{P}\) is irreducible because every two colorings \(\sigma,\tau\) are connected in \(\mathcal{G}\), since they are both connected to the all-white coloring \(\iota\) (via the path that removes black vertices one by one).
- \(\mathsf{P}\) is aperiodic because every coloring \(\sigma\) has a self-loop in \(\mathcal{G}\).
Hence \(\mathsf{P}\) is irreducible and aperiodic on \(\Omega\), and therefore converges to \(\mu\) as its unique stationary distribution.
1.3 \(\mu\) is self-similar
In this paper we often examine conditional distributions of \(\mu\), so we make the following definitions:
-
A partial coloring \(\sigma:U\to\{0,1\}\) is like a coloring but defined on a subset \(U\subseteq[n]\).
(Again, all partial colorings are valid partial colorings unless explicitly stated otherwise.)
-
A conditional distribution \(\mu^\sigma:\Omega^\sigma\to[0,1]\) is the law of a full coloring sampled from \(\mu\) conditioned on agreeing with \(\sigma\) on its ground set \(U\).
-
A restricted distribution \(\mu_S:\Omega_S\to[0,1]\) is the law of the coloring on \(S\) when a full coloring is drawn from \(\mu\).
-
A restricted conditional distribution \(\mu^\sigma_S:\Omega^\sigma_S\to[0,1]\) is the law of the coloring on \(S\) when a full coloring is drawn from \(\mu^\sigma\). If \(\sigma\) has ground set \(U=[n]\setminus S\), we call this a boundary-conditioned distribution on \(S\).
In these definitions, the conditional state space \(\Omega^\sigma\) is the set of all full colorings that agree with the partial coloring \(\sigma\) on its ground set \(U\). The restricted (conditional) state spaces \(\Omega_S\) and \(\Omega^\sigma_S\) are the sets of partial colorings on \(S\) obtainable from some full coloring in \(\Omega\) and \(\Omega^\sigma\), respectively.
The (restricted) conditional distributions are well-defined because setting \([n]\setminus U\) to all white is always valid.
We state the self-similarity property of \(\mu\):
For any subset \(S\subseteq[n]\) and any coloring \(\sigma\) with ground set \([n]\setminus S\) (a boundary condition), let \(Y\subseteq S\) be the subset of \(S\) that is either forbidden by \(X\) or adjacent to at least one black vertex in \([n]\setminus S\) as determined by \(\sigma\).
Then the Gibbs distribution on the induced subgraph \(G[S]\) with forbidden set \(Y\) is exactly \(\mu^\sigma_S\).
In future discussions we sometimes need to prove that certain properties hold for all boundary-conditioned distributions. From this observation, we can save effort by proving the property only for the global Gibbs distribution \(\mu\) itself, since such a proof generalizes to all graphs and forbidden sets, including \(G[S]\) with forbidden set \(Y\).
This also justifies why the problem statement includes the forbidden set \(X\): it is precisely the structure needed to express both the original problem and every subproblem that arises under possible boundary conditions.
Chapter 2
2.1 Functional analysis
We will perform some functional analysis. Here are the definitions:
The inner product \(\langle\cdot,\cdot\rangle_\mu\) is defined by
Let \(\nu\) be any probability distribution. By right-multiplying the row-stochastic matrix \(\mathsf{P}\), \(\nu\mathsf{P}\) is the pushforward distribution defined by
(Equivalently, \((\nu\mathsf{P})(y)\) is the probability that \(I_{i+1}=y\) when the law of \(I_i\) is \(\nu\).)
Let \(f\) be any function. By left-multiplying by the row-stochastic matrix \(\mathsf{P}\), \(\mathsf{P}f\) is the one-step averaged function defined by
(Equivalently, \((\mathsf{P}f)(x)\) is the expected value of \(f(I_{i+1})\) when \(I_i=x\).)
The variance \(\text{Var}_\mu(\cdot)\) is defined by
The following computation
helps view \(\text{Var}_\mu(f)\) from a more local standpoint.
Finally, the Dirichlet energy \(\mathfrak{D}(\cdot,\cdot)\) is defined by
The motivation behind this definition will be revealed in future discussions.
The following computation
helps view \(\mathfrak{D}(f,f)\) from a more local standpoint.
2.2 Poincaré inequality
The Poincaré inequality is an important tool for proving upper-bounds on the mixing time.
For a nonnegative real number \(\gamma\geq 0\), we say the Poincaré inequality holds with parameter \(\gamma\) if and only if
A larger \(\gamma\) corresponds to a strictly stronger statement.
In fact, the best \(\gamma\) that can appear in the Poincaré inequality is \(1-\lambda_2\), where \(\lambda_2\) is the second-largest eigenvalue of the transition matrix \(\mathsf{P}\). Equivalently,
Let \(N=|\Omega|\). Because \(\mathsf{P}\) is ergodic and reversible with respect to \(\mu\), Markov chain theory tells us that
Take an orthonormal eigenbasis \(\varphi_1,\varphi_2,\cdots,\varphi_N\) (with respect to \(\langle\cdot,\cdot\rangle_\mu\)) corresponding to the eigenvalues \(\lambda_1,\lambda_2,\cdots,\lambda_N\), with \(\varphi_1=\mathbf{1}\).
(Note that \(\varphi_1=\mathbf{1}\) because \(\mathsf{P}\mathbf{1}=\mathbf{1}\). For \(i\geq2\), \(\varphi_i\,\bot_{\mu}\,\varphi_1\), so \(\langle\varphi_i,\mathbf{1}\rangle_\mu=0\) and \(\mathbb{E}_\mu[\varphi_i]=0\).)
For any function \(f:\Omega\to\mathbb{R}\), decompose
Writing \(\text{Var}_\mu(f)\) and \(\mathfrak{D}(f,f)\) in terms of these coefficients yields
and
Thus the ratio \(\frac{\mathfrak{D}(f,f)}{\text{Var}_\mu(f)}\) is a weighted average of the numbers \(\{1-\lambda_i\}_{i=2,\cdots,N}\) with weights proportional to \(\alpha_i^2\). Therefore it is always at least \(1-\lambda_2\), and equality holds when \(\alpha_3=\cdots=\alpha_N=0\) but \(\alpha_2\neq0\).
2.3 PI implies rapid mixing
If the Poincaré inequality holds with parameter \(\gamma\), we prove
Note that this statement is valid only in the present context of the Heat-Bath Glauber dynamics.
For \(0\leq k\leq n\), the residual variance of \(k\)-pinnings is defined by
which denotes the average, over a uniformly random \(k\)-subset \(S\) and over \(\sigma\sim\mu_S\), of the variance of \(f\) on the complement \([n]\setminus S\). Note that \(V^{(0)}(f)=\text{Var}_\mu(f)\) and \(V^{(n)}(f)=0\).
It turns out that \(V^{(n-1)}(f)\) coincides with the Dirichlet energy \(\mathfrak{D}(f,f)\):
Here, the second equality is the nontrivial step: it follows from the fact that the variance of the two-point distribution \(\left\{\left(f_0,\frac{\mu_0}{\mu_0+\mu_1}\right),\left(f_1,\frac{\mu_1}{\mu_0+\mu_1}\right)\right\}\) equals \(\frac{\mu_0\mu_1}{(\mu_0+\mu_1)^2}(f_0-f_1)^2\).
The \(\chi^2\) divergence \(D_{\chi^2}(\cdot\,\|\,\cdot)\) is defined by
We wish to prove that the Poincaré inequality implies contraction of the \(\chi^2\) divergence:
Consequently,
In our chain \(I_0=\iota\), so the distribution of \(I_0\) is \(\delta_\iota\), and \(D_{\chi^2}(\delta_\iota\,\|\,\mu)=\frac{1}{\mu(\iota)}-1\). The distribution of \(I_t\) is \(\delta_\iota\mathsf{P}^t\).
By the Cauchy–Schwarz inequality,
Applying this to \(\delta_\iota\mathsf{P}^t\) gives
Requiring the left-hand side to be at most \(\varepsilon\) and solving for \(t\) yields the claimed mixing-time bound
For two distributions \(\nu,\mu\), let the density function be \(f=\frac{\nu}{\mu}\). (Then \(\mathbb{E}_\mu[f]=\langle f,\mathbf{1}\rangle_\mu=1\).) We compute:
which, together with \(V^{(n-1)}(f)=\mathfrak{D}(f,f)\) and the Poincaré inequality \(\mathfrak{D}(f,f)\geq\gamma\cdot V^{(0)}(f)\), implies \(D_{\chi^2}(\nu\mathsf{P}\,\|\,\mu)\leq(1-\gamma)D_{\chi^2}(\nu\,\|\,\mu)\).
Chapter 3
3.1 Spectral independence
Let \(f:\Omega\to\mathbb{R}\) be any function. Define \(\mu_1\) and \(f_1\) as the distribution on \(1\)-pinnings and function on \(1\)-pinnings, supported on the \(2n\) single-site colorings (NOT necessarily valid):
For a certain \(\eta\geq 0\), we say that \(\mu\) satisfies \(\eta\)-spectral independence if for every nonempty subset \(S\subseteq[n]\) and every \(\sigma\in\Omega_S\), the boundary-conditioned distribution \(\xi=\mu^\sigma_{[n]\setminus S}\) satisfies \(\eta\)-local spectral independence:
The quantity \(\text{Var}_{\xi_1}(f_1)\) will be referred to as the observed variance of \(1\)-pinnings.
By our earlier statement of the self-similarity of \(\mu\), it suffices to prove \(\eta\)-local spectral independence only for the global Gibbs distribution; it then automatically generalizes to all boundary-conditioned distributions.
By the law of total variance,
Therefore
Note that these inequalities can be chained together: for any \(0\leq m\leq n-1\),
3.2 The influence matrix
The (folded) influence matrix \(\Psi_\mu\in\mathbb{R}^{n\times n}\) of the global Gibbs distribution \(\mu\) is defined by
This definition assumes \(X=\varnothing\).
If instead \(X\neq\varnothing\), then \(\Psi_\mu\) should be defined as the \(n'\times n'\) matrix on the remaining \(n'=n-|X|\) vertices.
We prove that \(\lambda_{\max}(\Psi_\mu)\leq1+\eta\) implies that \(\mu\) satisfies \(\eta\)-local spectral independence.
It suffices to prove \(\frac{\text{Var}_{\mu_1}(f_1)}{\text{Var}_\mu(f)}\leq\frac{1+\eta}{n}\) in the case \(X=\varnothing\).
If \(X\neq\varnothing\), consider the graph \(G\) defined on \([n]\setminus X\) instead of \([n]\). In that scenario \(\text{Var}_\mu(f)\) is unchanged, while \(\text{Var}_{\mu_1}(f_1)\) is multiplied by \(\frac{n'}{n}\), and the inequality \(\frac{\text{Var}_{\mu_1}(f_1)}{\text{Var}_\mu(f)}\leq\frac{1+\eta}{n'}\) in the \([n]\setminus X\) scenario translates back to the desired inequality \(\frac{\text{Var}_{\mu_1}(f_1)}{\text{Var}_\mu(f)}\leq\frac{1+\eta}{n}\) in the \([n]\) scenario.
Now assume \(X=\varnothing\). In this case every vertex has nonzero probability of being colored \(0\) and of being colored \(1\).
Our goal is to show \(\frac{\text{Var}_{\mu_1}(f_1)}{\text{Var}_\mu(f)}\leq\frac{1+\eta}{n}\). Note that shifting \(f\) by an additive constant (which also shifts \(f_1\) by the same constant) does not change \(\text{Var}_{\mu_1}(f_1)\) or \(\text{Var}_\mu(f)\); hence we may assume \(\mathbb{E}_{\mu_1}[f_1]=\mathbb{E}_\mu[f]=0\), in which case \(\text{Var}_{\mu_1}(f_1)=\mathbb{E}_{\mu_1}[f_1^2]\). We compute
We wish to express the right-hand side in the form \(\langle f,\mathsf{P}f\rangle_\mu\). By definition,
so define
With this choice, the previous identity holds.
Note that this \(\mathsf{P}\) is a row-stochastic matrix satisfying
Therefore \(\mathsf{P}\) is time-reversible with respect to \(\mu\).
Applying the same argument as used earlier for such matrices, we obtain:
where \(\lambda_2(\mathsf{P})\) denotes the second-largest eigenvalue of the constructed matrix \(\mathsf{P}\).
Note that the notation \(\mathsf{P}\) is used differently only in this Section 3.2. Everywhere else in this paper, both before and after this section, \(\mathsf{P}\) denotes the \(N\times N\) transition matrix of the Heat-Bath Glauber dynamics. In this section alone, however, \(\mathsf{P}\) refers to the \(N\times N\) matrix we have constructed so that \(\mathbb{E}_{\mu_1}[f_1^2]=\langle f,\mathsf{P}f\rangle_\mu\).
The rank of \(\mathsf{P}\) does not exceed \(2n\). Define
Then \(\mathsf{P}=UV^T\). Define \(\mathsf{Q}=V^TU\). We can compute
It is a linear-algebra fact that the spectrum (multiset of eigenvalues) of \(\mathsf{P}\) (the \(N\) eigenvalues) equals the spectrum of \(\mathsf{Q}\) (the \(2n\) eigenvalues) together with \(N-2n\) zeros. Therefore \(\lambda_2(\mathsf{P})=\lambda_2(\mathsf{Q})\).
Note that this \(\mathsf{Q}\) is a row-stochastic matrix satisfying
Therefore \(\mathsf{Q}\) is time-reversible with respect to \(\mu_1\).
Finally, we only need to prove \(\lambda_2(\mathsf{Q})=\frac{1}{n}\lambda_{\max}(\Psi_\mu)\). A direct computation reveals that
where \(\mathsf{R}\big((i,s)\to(j,t)\big)=\mu_1\big((j,t)\big)=\frac{1}{n}\>\!\text{Pr}_{\sigma\sim\mu}\big[\sigma_j=t\big]\), which is used to remove the leading eigenvalue \(\lambda_1(\mathsf{Q})=1\).
It is a linear-algebra fact that the spectrum of \(\mathsf{Q}-\mathsf{R}\) (the \(2n\) eigenvalues) is equal to the spectrum of \(\mathsf{Q}\) with one copy of the eigenvalue \(1\) removed and a single \(0\) inserted. This is because \(\mathsf{R}\mathbf{1}=\mathbf{1}\) and for any \(f\,\bot_{\mu_1}\,\mathbf{1}\) (i.e. \(\langle f,\mathbf{1}\rangle_{\mu_1}=0\)) we have \(\mathsf{R}f=\mathbf{0}\).
Hence the entries of \(n\big(\mathsf{Q}-\mathsf{R}\big)\) are
Noting that \(\text{Pr}_{\sigma\sim\mu}\big[\sigma_j=0\big|\sigma_i=s\big]+\text{Pr}_{\sigma\sim\mu}\big[\sigma_j=1\big|\sigma_i=s\big]=\text{Pr}_{\sigma\sim\mu}\big[\sigma_j=0\big]+\text{Pr}_{\sigma\sim\mu}\big[\sigma_j=1\big]=1\), the two block columns are negatives of each other. Thus
Therefore the spectrum of \(n\big(\mathsf{Q}-\mathsf{R}\big)\) (the \(2n\) eigenvalues) equals the spectrum of \(\Psi_\mu\) (the \(n\) eigenvalues) together with \(n\) zeros.
In summary, the maximum possible value of \(\frac{\text{Var}_{\mu_1}(f_1)}{\text{Var}_\mu(f)}\) is \(\lambda_2(\mathsf{P})\); moreover, \(\lambda_2(\mathsf{P})=\lambda_2(\mathsf{Q})=\frac{1}{n}\lambda_{\max}(\Psi_\mu)\).
3.3 \(\eta\) is upper-bounded by \(8(\lambda_c+1)\alpha^{-1}-1\)
3.3.1 Sensitivity interpretation
Section 3.3 is devoted to proving
for any simple undirected graph \(H\) with forbidden set \(X=\varnothing\), assuming \(H\) has maximum degree at most \(\Delta\) and the activity parameter satisfies the same relation as in our problem statement:
where the integer degree bound \(\Delta\geq 3\) and the slackness \(0<\alpha<1\) are as stated at the beginning of this paper. The vertex \(\tau\in H\) is an arbitrary vertex of \(H\), called the root.
It is a linear-algebra fact that no eigenvalue of a matrix can exceed the sum of absolute values of the entries in any fixed row. Therefore \(\boldsymbol{(*)}\) implies \(\lambda_{\max}(\Psi_H)\leq1+\eta\) for \(\eta=8(\lambda_c+1)\alpha^{-1}-1\), hence \(\mu\) satisfies \(\eta\)-local spectral independence for this \(\eta\).
(Our notation for the influence matrix changes from \(\Psi_\mu\) to \(\Psi_H\) to emphasize the underlying graph \(H\).)
Imagine the activity parameter of each vertex may be distinct. Then the activity array \(\boldsymbol{\lambda}\) is an input. It turns out that
To see this, note the following. When \(\lambda_u\) is multiplied by \(1+\varepsilon\), the weight of every independent set containing \(u\) is scaled by \(1+\varepsilon\). Because these sets contribute a \(p\)-fraction of the total weight under \(\mu_{H,\boldsymbol{\lambda}}\), the partition function \(Z_H(\boldsymbol{\lambda})\) is therefore scaled by \(1+p\varepsilon\).
One then shows
where
denotes the ratio of the probability that \(\tau\) is contained in \(I\) to the probability that \(\tau\) is not contained in \(I\).
Indeed,
3.3.2 Potential method on SAW-tree
We build the self-avoiding-walk tree \(T=T_{\textsf{SAW}}(H,\tau)\) of \(H\) rooted at \(\tau\). It is straightforward that
This is because, by the construction of the self-avoiding-walk tree, each node (a clone) of \(T\) maps back to a unique donor in the original graph \(H\), and each clone copies the donor's activity parameter. Hence a perturbation of \(\lambda_u\) in \(H\) corresponds to simultaneously perturbing the activity of every clone in \(T\) whose donor is \(u\), so the sensitivity at \(u\) in \(H\) is at most the sum of sensitivities over all such clones in \(T\).
Indeed, the \(n\) donors in \(H\) each have an associated set of clones in \(T\), and these clone-sets form a partition of all the nodes of the tree. Thus the contribution of any donor \(u\in H\) to the left-hand sum is at most the total contribution of its clones to the right-hand sum, and summing over all \(n\) donors yields the stated inequality.
Therefore it suffices to upper-bound the right-hand side by \(8(\lambda_c+1)\alpha^{-1}\).
We compute the result on \(T\) by a subtree DP. For each node \(u\) let the DP value be \(R_u\), with recursion
Except for the root \(\tau\), every node has \(0\leq d\leq\Delta-1\), while the root satisfies \(0\leq d\leq\Delta\).
Define the potential function
Set \(K=\varphi(R)\), so \(K_u=G_{d_u,\lambda_u}\big(K_{u_1},\cdots,K_{u_{d_u}}\big)\) where
For each node \(u\) of \(T\) create two subnodes \(u^{(1)}\) and \(u^{(2)}\). We include directed edges with the following weights:
- An edge from \(u^{(1)}\) to \(u^{(2)}\) with weight \(w\big(u^{(1)}\to u^{(2)}\big)=\dfrac{\partial K_u}{\partial\ln R_u}\).
- An edge from \(\tau^{(2)}\) to \(\tau^{(1)}\) with weight \(w\big(\tau^{(2)}\to\tau^{(1)}\big)=\dfrac{\partial\ln R_\tau}{\partial K_\tau}\).
- For each child \(v_i\) of \(u\), an edge from \(v_i^{(2)}\) to \(u^{(2)}\) with weight \(w\big(v_i^{(2)}\to u^{(2)}\big)=\partial_{K_{v_i}}G_{d_u,\lambda_u}\big(K_{v_1},\cdots,K_{v_{d_u}}\big)\).
Since \(\dfrac{\partial\ln R_u}{\partial\ln\lambda_u}=1\), we have
By the chain rule, \(\Psi_T(\tau\to u)\) equals the sum of products of edge weights along the unique nonempty path from \(u^{(1)}\) to \(\tau^{(1)}\).
Fix \(u^{(2)}\). We now examine the sum of the edge weights on \(v_i^{(2)} \to u^{(2)}\) over all \(d\) children \(v_i\) of \(u\). For a node \(u\) with DP value \(R\) and children \(v_1,\cdots,v_d\) with DP values \(R_1,\cdots,R_d\),
Fix \(R\). We show that the right-hand side of \((*)\) is maximized when \(R_1=\cdots=R_d\).
Set \(u_i=\ln(R_i+1)\geq 0\). Then \(\ln R=\ln\lambda-(u_1+\cdots+u_d)\) is fixed. Rewriting \((*)\) gives
Let \(h(u)=\sqrt{1-e^{-u}}\) for \(u\geq0\). Then
and \(h'\) is strictly decreasing, so \(h\) is concave on \([0,+\infty)\). Hence, under the constraint \(u_1+\cdots+u_d\) fixed, the sum \(\sum_{i=1}^d h(u_i)\) is maximized when \(u_1=\cdots=u_d\), which proves the claim.
3.3.3 Completion of the bound
Attach degree-completion leaves with \(R=0\) to each non-root node with \(d\leq \Delta-2\) until every non-root node has \(d=\Delta-1\), and adjust the root in the same way so that it has \(d=\Delta\). From now on set \(d:=\Delta-1\).
Then for any non-root \(u^{(2)}\), the sum of the edge weights \(w\big(v_i^{(2)}\to u^{(2)}\big)\) over all \(d\) children \(v_i\) of \(u\) does not exceed the supremum over \(x\geq 0\) of
For the root \(\tau^{(2)}\) simply replace \(d\) by \(d+1\) in this statement.
We call these two quantities \(Q_{\lambda,d}\) and \(Q_{\lambda,d+1}\), respectively.
We prove that \(Q_{\lambda,d+1}\leq\frac{d+1}{d}\cdot Q_{\lambda,d}\). Let \(x^*\) attain \(Q_{\lambda,d+1}\) in the \((d+1)\)-formula. Plugging \(x=x^*\) into the \(d\)-formula gives the same factor \(\sqrt{\frac{x}{x+1}}\) but a larger \(y\), hence a larger factor \(\sqrt{\frac{y}{y+1}}\). Therefore \(Q_{\lambda,d}\geq\frac{d}{d+1}\cdot Q_{\lambda,d+1}\), which rearranges to the claimed inequality.
Now we prove the crucial bound \(Q_{\lambda,d}\leq 1-\frac{\alpha}{2(\lambda_c+1)}\).
Maximizing \(d\sqrt{\frac{x}{x+1}}\sqrt{\frac{y}{y+1}}\) over \(x\geq 0\) is equivalent to maximizing \(\frac{xy}{(x+1)(y+1)}\). Define \(z=\frac{1}{x+1}\in(0,1]\), so \(x=\frac{1}{z}-1\) and \(y=\lambda z^d\). Differentiating \(\frac{xy}{(x+1)(y+1)}\) with respect to \(x\) shows that its critical points correspond to the zeros of
Indeed, the derivative of \(\frac{xy}{(x+1)(y+1)}\) with respect to \(x\) has the same sign as
and, after expressing the result in terms of \(z=\frac{1}{x+1}\), this quantity has the opposite sign of
Therefore the critical points arise exactly at the solutions of \(h(z)=0\).
Moreover, \(h\) is strictly increasing on \([0,1]\), with \(h(0)=-d\) and \(h(1)=\lambda+1\), so \(h\) has a unique root \(z^*\in(0,1)\).
For \(\lambda=(1-\alpha)\lambda_c\), let \(x^*=\frac{1}{z^*}-1\) and \(y^*=\lambda(z^*)^d\), so
Let \(x_0,y_0,z_0\) denote the corresponding values when \(\lambda=\lambda_c\). Then \(z_0=\frac{d-1}{d}\) and \(x_0=y_0=\frac{1}{d-1}\), and \(Q_{\lambda_c,d}=1\).
Since \(\lambda(z^*)^{d+1}+(d+1)z^*-d=0\) and \(\lambda_c(z_0)^{d+1}+(d+1)z_0-d=0\), subtracting the two equations gives
By the mean value theorem there exists \(z_1\in[z_0,z^*]\) with
Hence
Next observe that
because
Therefore
This proves the bound for each non-root node's total child-edge weight.
Let \(w_{\max}\) and \(w_{\min}\) denote the maximum and minimum values, respectively, of the edge weight from a node's \((1)\) subnode to its own \((2)\) subnode. Since \(\frac{\partial K}{\partial\ln R}=R\varphi'(R)=\sqrt{\frac{R}{R+1}}\), we obtain
The final inequality is justified by the following claim: for any \(d\geq 2\) and \(0<\lambda\leq\lambda_c=\frac{d^d}{(d-1)^{d+1}}\), letting \(A=\lambda\) and \(B=\frac{\lambda}{(\lambda+1)^d}\), one has
Squaring both sides yields the equivalent inequality
where \(16_d=\frac{16d^2}{(d+1)^2}\) and \(15_d=16_d-1\). Note that
so \(L\) is convex on \([0,+\infty)\). Thus the maximum of \(L\) on \([0,\lambda_c]\) occurs at an endpoint. Since \(L(0)=1\leq 16_d\), it suffices to check \(L(\lambda_c)\leq 16_d\).
A Python program is used to verify the inequality \(L(\lambda_c)\leq 16_d\) by brute force for \(d\in\{2,\cdots,199\}\). For \(d\geq 200\), we instead apply an analytic argument. Observe that
and it is a routine calculus exercise to check that the right-hand side is strictly decreasing in \(d\). Its value at \(d=200\) has already fallen below \(15.5\). On the other hand, for \(d\geq 200\) we clearly have \(16_d \geq 15.8\). This completes the argument.
Chapter 4
4.1 Shattering lemma
In Section 4.1 we prove the shattering lemma, a graph-theoretic lemma whose sole purpose in this paper is to support the calculations in Section 4.3.
Shattering lemma: Fix any undirected graph \(G\). For any parameter \(0\leq\theta\leq1\) that is a multiple of \(\frac{1}{n}\), choose uniformly at random a subset \(U\) of size \(\theta n\) to keep and delete the other vertices. For any vertex \(u\), let \(C_u\) denote the largest connected component (among the kept vertices) that contains \(u\). Then for any \(\ell\geq 1\),
Proof of shattering lemma: Consider how many different vertex sets \(S\) contain \(u\), have size \(\ell\leq\theta n\), and are connected. For each such set, the probability that all vertices of \(S\) are kept equals
Next count how many different rooted trees \(T\) (rooted at \(u\)) of size \(\ell\) can be formed in \(G\). Two rooted trees \(T_1,T_2\) are considered different iff either the number of children or the chosen child-set of \(u\) differs, or some corresponding rooted child-subtrees differ.
By this definition, the number of rooted trees (rooted at \(u\)) of size \(\ell\) in \(G\) is at most the number of rooted trees (rooted at \(u\)) of size \(\ell\) in the infinite \(\Delta\)-regular tree \(\mathbb{T}_\Delta\).
Moreover, every connected vertex set \(S\) containing \(u\) admits at least one rooted spanning tree, and the mapping from such a connected set \(S\) to a rooted tree is injective. Hence the number of connected vertex sets \(S\) is bounded by the number of rooted trees in \(\mathbb{T}_\Delta\).
We compute the latter by a subtree DP/generating-function argument. Writing \(F(x)=x\big(1+F(x)\big)^\Delta\) and applying Lagrange inversion gives
Note that
Multiplying the two bounds yields
The final inequality is justified only for \(\ell \geq 2\). However, when \(\ell = 1\), the lemma holds trivially. This completes the proof of the shattering lemma.
4.2 Variance factorization
In Section 4.2 we prove variance factorization from perfect independence, a variance inequality whose purpose in this paper is also to support the calculations in Section 4.3.
Variance factorization from perfect independence: For an \(n\)-variate function \(F:X_1\times\cdots\times X_n\to\mathbb{R}\), if it satisfies perfect independence, i.e.
then the variance of \(F\) factorizes over the \(n\) components, namely
Proof of variance factorization from perfect independence: By the law of total variance,
Fix the values of \(x_1,\cdots,x_{i-1}\). It suffices to prove that
where \(F(x_i,\cdots,x_n)=F(x_1,\cdots,x_{i-1},x_i,\cdots,x_n)\) is redefined to omit the first \((i-1)\) fixed parameters.
This follows from
The only nontrivial inequality is the application of the Cauchy–Schwarz inequality, \(\mathbb{E}[G]^2 \leq \mathbb{E}[G^2]\), where \(G = F(x_i,\dots,x_n) - F(x'_i,\dots,x_n)\), and the expectation is taken over \(x_{i+1},\dots,x_n\).
4.3 Variance tensorization
Fix any function \(f:\Omega\to\mathbb{R}\). The discussions in Section 4.3 do not use any property of \(f\).
Choose a parameter \(0<\theta<0.1\) that is a multiple of \(\frac{1}{n}\). We will lower-bound \(V^{(n-1)}(f)\) in terms of \(V^{(0)}(f)\) by lower-bounding \(V^{((1-\theta)n)}(f)\) in terms of \(V^{(0)}(f)\) and then lower-bounding \(V^{(n-1)}(f)\) in terms of \(V^{((1-\theta)n)}(f)\).
In Section 3.1 we established
We use the lower bound \(1-x\geq e^{-1.1x}\) for \(0\leq x\leq\frac{1}{6}\). For \(n\geq n_0=\frac{6(1+\eta)}{\theta}\), we have
The final inequality is justified because \(\ln(n+1)\leq H_n\leq 1+\ln n\) for all \(n\geq1\).
Now return to the definition of \(V^{((1-\theta)n)}(f)\):
Consider the connected components \(C\) of \(G\big[[n]\setminus S\big]\). The colorings on distinct components are independent. By variance factorization from perfect independence,
A Poincaré inequality can be applied to each component \(C\) under the conditional distribution \(\mu^{\sigma\sqcup\tau}\):
There is a universal upper bound \(C_\textsf{PI}(|C|)\) on the Poincaré constant depending only on \(|C|\).
Merging \(\tau\) and \(\varsigma\) into a single \(\tau\), and substituting back and simplifying, we obtain
We state without justification (for now) that \(C_\textsf{PI}(|C|)=2|C|^2\eth^{2|C|}\) is valid, where \(\eth=\frac{1+\lambda}{\min\{1,\lambda\}}\).
Consider two (overlapping) regions according to \(\lambda\):
The first region: The \(0<\lambda\leq\frac{1}{\Delta}\) region lies entirely within the Dobrushin regime (defined by \(\lambda<\frac{1}{\Delta-1}\)). In this region, we apply the Dobrushin mixing-time bound (proven using a path-coupling argument, omitted here) to get
The second region: The \(\frac{1}{\Delta+1}\leq\lambda<\lambda_c\) region crosses the border of the Dobrushin regime and covers the rest of the uniqueness regime. In this region, we have \(\eth=\frac{1+\lambda}{\min\{1,\lambda\}}\leq\Delta+2\).
Let \(\theta=\frac{1}{1.1e\Delta\eth^2}\) and putting everything together gives a universal mixing-time bound of
However, this is still not the \(O\big(n\log n+n\log\varepsilon^{-1}\big)\) we claimed at the beginning of this paper.
The framework in the first half of this paper is variance tensorization. In the second half of this paper, we use a more refined version of this approach, called entropy tensorization, to obtain the desired result.
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