The stopped covariance quantity of the bootstrap - The KLS theorem and its methods

Part of the fixed-cut archive, Chapter The fixed cut: the bootstrap; the reading order is on the full proofs page.

Overview. This dossier proves Theorem 33.2 (Theorem D28.1). For a near-worst isotropic log-concave law and a balanced cut, the stopped Cheeger excess is bounded by $Te_0$ plus $h_\mu$ times window, exit and covariance terms ((D28.9)). In that bound the covariance interface enters only through the cut-stopped quantity $\widehat\Xi_{T,\eta}$ of (D28.8). With $\eta=T^{1/3}$ it becomes $Te_0+2h_\mu(T^{4/3}+\widehat\Xi_{T,\eta})$ ((D28.10)). The result is only a comparison: no dimension-free bound on $\widehat\Xi_{T,\eta}$ is proved and KLS is not inferred.

At each deterministic time, Lemma 32.3 and Lemma 33.1 bound the expected stopped perimeter by $h_\mu/2+e_0$ ((D28.12)).
Lemma 33.2 and the tangent inequality (D28.13) bound the posterior Cheeger term from below on $\{t<\tau_\eta\}$ with covariance error $Z_t$ , stopped rather than unstopped ((D28.14)).
The near-worst ratio $\rho\ge1-\varepsilon$ ((D28.16)) and a two-case sign analysis combine steps 1–2 into the pointwise comparison (D28.20).
Doob’s weak $L^2$ inequality for the stopped mass martingale and the bracket bound (D28.3) control the exit probability $P_t$ by $t+\int_0^tZ_s\,\dd s$ ((D28.22)). Tonelli then integrates this in time ((D28.23)).
Integrating step 3 and inserting step 4 gives (D28.9). The choice $\eta=T^{1/3}$ with $\varepsilon\le T^{1/3}$ and $T<1/8$ gives the clean form with $C=2$ .

Setup and integrability. For a log-concave probability measure $\nu$ on $\mathbb R^n$ , let $h_\nu$ denote its Cheeger constant, and let $h_n^\star$ be the infimum of $h_\nu$ over isotropic log-concave laws in dimension $n$ . Fix an isotropic log-concave law $\mu$ and a measurable cut $E$ . Along stochastic localization write

p_t=\mu_t(E),\qquad q_t=1-p_t,\qquad s_t=p_tq_t, \qquad A_t=\operatorname{Cov}(\mu_t),

(D28.1)

and let

\bar e_t(E)=\mu_t^+(E)-h_{\mu_t}\min(p_t,q_t), \qquad X_t=(\lambda_{\max}(A_t)-1)_+.

(D28.2)

The process $p_t$ is a bounded continuous martingale. If $\delta_t=m_t^E-m_t^{E^c}$ and $r_t=s_t|\delta_t|^2$ , then the standard two-color identities give

\dd[p]_t=s_tr_t\,\dd t, \qquad s_tr_t\leq\frac14\lambda_{\max}(A_t) \leq\frac14(1+X_t).

(D28.3)

Indeed, $s_t\delta_t\delta_t^T\preceq A_t$ , its sole nonzero eigenvalue is $r_t$ , and $s_t\leq1/4$ .

For $0<\eta<1/2$ , set

\tau_\eta=\inf\{t:|p_t-1/2|>\eta\}.

(D28.4)

The process $(t,\omega)\mapsto X_t\mathbf1_{\{t<\tau_\eta\}}$ is nonnegative and jointly measurable. Moreover, taking traces in the covariance SDE, localizing its martingale part, and then using Fatou gives $\mathbb E\operatorname{Tr}A_t\leq\operatorname{Tr}A_0=n$ . Hence

0\leq\mathbb E[X_t\mathbf1_{\{t<\tau_\eta\}}]\leq n,

(D28.5)

so all deterministic-time expectations and finite-horizon integrals below are finite. If $\mu^+(E)=\infty$ , the claimed estimate is automatic in the extended sense. We therefore prove the only nontrivial case $\mu^+(E)<\infty$ , where Lemma 32.3 makes $\mu_t^+(E)$ an integrable nonnegative supermartingale.

Theorem D28.1 (Stopped covariance-interface refinement; Theorem 33.2)

Let $n\geq2$ , let $\varepsilon\in(0,1]$ , and let $\mu$ be isotropic and log-concave on $\mathbb R^n$ , with

h_\mu\leq(1+\varepsilon)h_n^\star.

(D28.6)

Let $E$ be any measurable set with $\mu(E)=1/2$ , put

e_0=e_0(E)=\bar e_0(E),

(D28.7)

and fix $\eta\in(0,1/2)$ . For $T>0$ , define the cut-stopped covariance interface

\widehat\Xi_{T,\eta}(\mu,E) :=\int_0^T\mathbb E\!\left[X_s\mathbf1_{\{s<\tau_\eta\}}\right]\dd s.

(D28.8)

Then

\begin{split} \int_0^T\mathbb E\!\left[ \bar e_t(E)\mathbf1_{\{t<\tau_\eta\}}\right]\dd t \leq{}&T e_0+h_\mu\left[ \left(\frac\varepsilon2+\eta\right)T +\frac{T^2}{16\eta^2}\right.\\ &\left.\hspace{31mm} +\left(\frac{T}{8\eta^2}+\frac14\right) \widehat\Xi_{T,\eta}(\mu,E)\right]. \end{split}

(D28.9)

Consequently, if $0<T<1/8$ , $\eta=T^{1/3}$ , and $\varepsilon\leq T^{1/3}$ , then

\int_0^T\mathbb E\!\left[ \bar e_t(E)\mathbf1_{\{t<\tau_\eta\}}\right]\dd t \leq T e_0+C h_\mu\left( T^{4/3}+\widehat\Xi_{T,\eta}(\mu,E)\right)

(D28.10)

with a universal constant; for example, $C=2$ is valid.

Proof D28.1

For a fixed deterministic time $t>0$ , abbreviate

\mathcal A_t=\{t<\tau_\eta\},\qquad P_t=\mathbb P(\tau_\eta\leq t),\qquad Z_t=\mathbb E[X_t\mathbf1_{\mathcal A_t}], \qquad a=\frac12-\eta.

(D28.11)

The perimeter supermartingale is used only at the deterministic time $t$ :

\begin{split} \mathbb E[\mu_t^+(E)\mathbf1_{\mathcal A_t}] &\leq\mathbb E\mu_t^+(E)\leq\mu^+(E)\\ &=\frac{h_\mu}{2}+e_0. \end{split}

(D28.12)

The last equality follows from balance and Lemma 33.1. In particular, no optional-stopping assertion for the perimeter process is used.

On $\mathcal A_t$ one has $\min(p_t,q_t)\geq a$ . The posterior covariance remains nondegenerate at finite times because localization changes the initial law by a strictly positive density on the same full-dimensional support. Lemma 33.2 and

\lambda^{-1/2}\geq1-\frac12(\lambda-1)_+, \qquad \lambda>0,

(D28.13)

therefore yield the stopped lower bound

\begin{split} \mathbb E\!\left[ h_{\mu_t}\min(p_t,q_t)\mathbf1_{\mathcal A_t}\right] &\geq h_n^\star a\, \mathbb E\!\left[ \lambda_{\max}(A_t)^{-1/2}\mathbf1_{\mathcal A_t}\right]\\ &\geq h_n^\star a\left(1-P_t-\frac{Z_t}{2}\right). \end{split}

(D28.14)

Crucially, the covariance error here is $Z_t$ , not the larger unstopped expectation $\mathbb E X_t$ .

Set

\rho=\frac{h_n^\star}{h_\mu}, \qquad B_t=1-P_t-\frac{Z_t}{2}.

(D28.15)

By isotropy and near-worstness,

\frac1{1+\varepsilon}\leq\rho\leq1, \qquad\text{so}\qquad \rho\geq1-\varepsilon.

(D28.16)

Because the lower bound in (D28.14) can be negative, the sign of $B_t$ must be retained.

Suppose first that $B_t\geq0$ . Combining (D28.12)– (D28.16), and multiplying by the lower bound for $\rho$ only in this nonnegative case, gives

\begin{split} \mathbb E[\bar e_t(E)\mathbf1_{\mathcal A_t}]-e_0 &\leq h_\mu\left\{\frac12-(1-\varepsilon)aB_t\right\}\\ &\leq h_\mu\left( \eta+\frac{P_t}{2}+\frac{Z_t}{4}+\frac\varepsilon2\right). \end{split}

(D28.17)

Indeed,

\frac12-a(1-P_t)=\eta+aP_t\leq\eta+\frac{P_t}{2}, \qquad \frac{aZ_t}{2}\leq\frac{Z_t}{4}, \qquad \varepsilon aB_t\leq\frac\varepsilon2,

(D28.18)

where $0\leq B_t\leq1$ in this case.

If instead $B_t<0$ , discard the nonnegative posterior Cheeger term in (D28.12). Since the case assumption is precisely $P_t/2+Z_t/4>1/2$ , one obtains

\mathbb E[\bar e_t(E)\mathbf1_{\mathcal A_t}]-e_0 \leq\frac{h_\mu}{2} <h_\mu\left( \eta+\frac{P_t}{2}+\frac{Z_t}{4}+\frac\varepsilon2\right).

(D28.19)

Thus both cases prove the stopped pointwise comparison

\mathbb E[\bar e_t(E)\mathbf1_{\{t<\tau_\eta\}}] \leq e_0+h_\mu\left( \frac\varepsilon2+\eta+\frac{P_t}{2}+\frac{Z_t}{4}\right).

(D28.20)

It remains to estimate $P_t$ without discarding the stopping indicator on $X_s$ . The process

M_u=p_{u\wedge\tau_\eta}-\frac12

(D28.21)

is a bounded continuous square-integrable martingale. On $\{\tau_\eta\leq t\}$ , continuity of $p$ implies $\sup_{u\leq t}|M_u|\geq\eta$ . The weak $L^2$ maximal inequality, martingale isometry at the bounded stopping time, and (D28.3) give

\begin{split} P_t &\leq\eta^{-2}\mathbb E M_t^2 =\eta^{-2}\mathbb E[p]_{t\wedge\tau_\eta}\\ &\leq\frac1{4\eta^2}\int_0^t \mathbb E\!\left[ \mathbf1_{\{s<\tau_\eta\}}(1+X_s)\right]\dd s\\ &\leq\frac1{4\eta^2}\left(t+\int_0^t Z_s\,\dd s\right). \end{split}

(D28.22)

The bracket integral naturally carries the indicator of the stopped interval. Whether its value at $s=\tau_\eta$ is included is immaterial for Lebesgue integration. Boundedness of $M$ and the integrability recorded above justify the isometry; no unbounded optional-stopping theorem is used.

All remaining integrands are nonnegative, so Tonelli’s theorem and (D28.22) imply

\begin{split} \int_0^T P_t\,\dd t &\leq\frac1{4\eta^2}\left[ \frac{T^2}{2}+ \int_0^T\int_0^t Z_s\,\dd s\,\dd t\right]\\ &=\frac1{4\eta^2}\left[ \frac{T^2}{2}+ \int_0^T(T-s)Z_s\,\dd s\right]\\ &\leq\frac1{4\eta^2}\left[ \frac{T^2}{2}+T\widehat\Xi_{T,\eta}(\mu,E)\right]. \end{split}

(D28.23)

Integrating (D28.20), using $\int_0^T Z_t\,\dd t=\widehat\Xi_{T,\eta}(\mu,E)$ , and inserting (D28.23) gives exactly (D28.9): the $P_t/2$ term contributes $T^2/(16\eta^2)$ and $T\widehat\Xi_{T,\eta}/(8\eta^2)$ , while the $Z_t/4$ term contributes $\widehat\Xi_{T,\eta}/4$ .

Finally, when $0<T<1/8$ , the choice $\eta=T^{1/3}$ satisfies $0<\eta<1/2$ . If also $\varepsilon\leq T^{1/3}$ , then

\left(\frac\varepsilon2+\eta\right)T \leq\frac32T^{4/3}, \qquad \frac{T^2}{16\eta^2}=\frac1{16}T^{4/3},

(D28.24)

and

\frac{T}{8\eta^2}+\frac14 =\frac{T^{1/3}}8+\frac14<\frac5{16}.

(D28.25)

Thus the whole bracket in (D28.9) is at most $2(T^{4/3}+\widehat\Xi_{T,\eta})$ , which proves (D28.10) with $C=2$ .

Hypotheses and dependency audit. The proof uses exactly: balance of $E$ ; log-concavity and isotropy of $\mu$ ; the near-worst comparison $h_\mu\leq(1+\varepsilon)h_n^\star$ ; the range $\varepsilon\in(0,1]$ ; the deterministic-time conclusion of Lemma 32.3; Lemma 33.1; Lemma 33.2; and the standard localization identities displayed in (D28.3). Finite initial perimeter is needed only for the nontrivial finite-valued case; infinite perimeter makes the assertion automatic. The clean specialization additionally uses exactly $0<T<1/8$ , $\eta=T^{1/3}$ , and $\varepsilon\leq T^{1/3}$ . No smoothness of the density or cut, no near-minimality of the cut, and no unstated covariance estimate are used.

Obstructions respected. The fence rem:profile-circularity is respected: the proof never treats the random mass-constrained isoperimetric profile as a supermartingale and never inserts a lower bound for that moving profile. Its only posterior isoperimetric input is the certified whitening comparison with the external worst-case constant $h_n^\star$ , coupled to the explicit near-worst hypothesis at time zero.

The fence rem:relative-ceiling is also respected. The theorem is only a comparison whose right-hand side contains the cut-dependent stopped interface $\widehat\Xi_{T,\eta}(\mu,E)$ . It proves no dimension-free or relative-scale upper bound for that interface (the elementary $nT$ integrability bound above is dimension dependent), and it does not infer KLS. In particular, stopping the interface does not by itself discharge the remaining covariance-occupation problem.