BKL: cumulant energy estimates - The KLS theorem and its methods

Part of the Bizeul–Klartag–Lehec proof, Chapter Bizeul–Klartag–Lehec: cumulants and suspension; the reading order is on the full proofs page.

Overview. This dossier reconstructs Section 5 of Bizeul et al., 2026, version 1. An Itô computation in the inverse covariance metric produces a positive next-order energy. The moving metric costs only a quadratic factor in the cumulant order. The infinite-time conclusion is taken along a sequence on which the terminal expectation tends to zero.

Author. plan_framework researcher, unknown, 2026-10-06.

Dependencies. Definition 8.1 and Lemma 8.3, including its finite-time integrability and third-order matrix bound. No dimension-free estimate for higher cumulants or KLS bound is used.

Theorem 133.1 (Energy inequality)

For the localization process of Lemma 8.3, a deterministic unit vector $u$ , and $m\ge3$ , let $\mathcal E_m(t)=|\kappa_m(t)(u)|_t^2$ . There is a universal constant $C\ge1$ such that

\operatorname{drift}\mathcal E_m(t)\ge \tfrac12\mathcal E_{m+1}(t)-Cm^2\mathcal E_m(t) -2\langle\kappa_m(t)(u),L_m(t)(u)\rangle_t.

(133.1)

In particular, if

I_m:=\int_0^\infty\mathbb E\mathcal E_m(t)\,dt<\infty, \qquad J_m:=\int_0^\infty\mathbb E|L_m(t)(u)|_t^2\,dt<\infty,

(133.2)

then

\mathcal E_m(0)+\tfrac12\int_0^\infty\mathbb E\mathcal E_{m+1}(t)\,dt \le Cm^2 I_m+2\sqrt{I_mJ_m}.

(133.3)

The choice $C=17$ is sufficient.

Proof 133.1

Put $r=m-1$ , $T_t=\kappa_m(t)(u)$ and $W_{i,t}=\kappa_{m+1}(t)(u,\cdot,\ldots,\cdot,A_t^{-1/2}e_i)$ . The dynamics give

dA=\sum_iH_i\,dB_i-A\,dt,\qquad dT=\sum_iW_i\,dB_i-(mT+L_m(u))\,dt.

(133.4)

For positive definite $A$ define $F(A,T)=\langle T,(A^{-1})^{\otimes r}T\rangle$ . For any fixed symmetric positive definite $P$ ,

F(A,T)=F(PAP,P^{\otimes r}T).

(133.5)

This identity follows from $P(PAP)^{-1}P=A^{-1}$ and also holds for the directional derivatives of $F$ , with the corresponding congruence applied to directions. At the particular time where the generator is evaluated choose $P=A^{-1/2}$ . This is a change of variables in a derivative at a fixed point, not a stochastic change of coordinates; it adds no drift. Use tildes for the transformed tensors, and let $G_i=PH_iP$ . The first derivative is evaluated in the direction $(-I,-m\widetilde T-\widetilde L)$ and the quadratic directions are $(G_i,\widetilde W_i)$ .

For symmetric $H$ , write $H^{(s)}$ for its action on slot $s$ of an order- $r$ tensor, and put $S_H=\sum_{s=1}^r H^{(s)}$ . Different slots commute. Multiplying the expansions of $(I+\varepsilon H)^{-1}$ in the $r$ slots gives

((I+\varepsilon H)^{-1})^{\otimes r} =I-\varepsilon S_H+\frac{\varepsilon^2}{2} \left(S_H^2+\sum_{s=1}^r(H^{(s)})^2\right)+O(\varepsilon^3).

(133.6)

It follows by multiplying on each side by $\widetilde T+\varepsilon W$ that the coefficient of $\varepsilon$ in $F(I+\varepsilon H,\widetilde T+\varepsilon W)$ is

2\langle\widetilde T,W\rangle-\langle\widetilde T,S_H\widetilde T\rangle,

(133.7)

and its coefficient of $\varepsilon^2$ is

|W|^2-2\langle W,S_H\widetilde T\rangle +\tfrac12|S_H\widetilde T|^2 +\tfrac12\sum_{s=1}^r|H^{(s)}\widetilde T|^2.

(133.8)

These are the first derivative and half the second derivative respectively. The deterministic direction has $S_{-I}=-rI$ and therefore contributes $-(m+1)|\widetilde T|^2-2\langle\widetilde T,\widetilde L\rangle$ . Consequently the full drift equals

\begin{split} &-(m+1)|\widetilde T|^2-2\langle\widetilde T,\widetilde L\rangle +\sum_i|\widetilde W_i|^2 -2\sum_i\langle\widetilde W_i,S_{G_i}\widetilde T\rangle\\ &\hspace{12mm}+\tfrac12\sum_i|S_{G_i}\widetilde T|^2 +\tfrac12\sum_{i,s}|G_i^{(s)}\widetilde T|^2. \end{split}

(133.9)

Discard the last two nonnegative sums. Young’s inequality bounds the absolute value of the cross term by $\frac12\sum_i|\widetilde W_i|^2+2\sum_i|S_{G_i}\widetilde T|^2$ . The third-order matrix estimate is $\sum_iG_i^2\le8I$ . Thus

\sum_i|S_{G_i}\widetilde T|^2 \le r\sum_{s=1}^r\sum_i|G_i^{(s)}\widetilde T|^2 \le8r^2|\widetilde T|^2.

(133.10)

Also $|\widetilde T|^2=\mathcal E_m$ , $\sum_i|\widetilde W_i|^2=\mathcal E_{m+1}$ and $\langle\widetilde T,\widetilde L\rangle= \langle\kappa_m(u),L_m(u)\rangle_t$ . The drift lower bound follows because $m+1+16(m-1)^2\le17m^2$ for $m\ge3$ .

By the finite-time integrability established in Lemma 8.3, the expectation of the Itô stochastic integral on $[0,T]$ is zero. Equivalently, one may first stop in the parameter space and then remove the stop using its deterministic finite-order posterior energy bounds. Integrating the drift gives

\begin{split} \mathcal E_m(0)+\tfrac12\int_0^T\mathbb E\mathcal E_{m+1}(t)\,dt \le{}&\mathbb E\mathcal E_m(T)+Cm^2\int_0^T\mathbb E\mathcal E_m(t)\,dt\\ &+2\int_0^T\mathbb E\langle\kappa_m(t)(u),L_m(t)(u)\rangle_t\,dt. \end{split}

(133.11)

The absolute value of the last integral is at most $\sqrt{I_mJ_m}$ by Cauchy–Schwarz on the product of time and probability spaces. Since the nonnegative function $t\mapsto\mathbb E\mathcal E_m(t)$ is integrable, there exists an increasing deterministic sequence $T_j\to\infty$ for which $\mathbb E\mathcal E_m(T_j)\to0$ . In fact one can select $T_j$ in successive unit intervals whose energy integral tends to zero. Monotone convergence on the two nonnegative energy integrals along this sequence proves the claimed inequality, and proves finiteness of the next-order integral at the same time. Integrability alone is not being used to assert terminal decay at every time. For arbitrary fixed $u$ , all terms are homogeneous of degree two in $u$ , so the same conclusions follow by scaling; the case $u=0$ has all terms zero.

Fences respected. There is no assigned bounded_by fence. The conclusion requires both explicitly stated integrability hypotheses. They are discharged by the coupled induction, not by an assumption of the desired higher-order estimate.

References¶

Bizeul, P., Klartag, B., & Lehec, J. (2026). Presenting a Proof of the Kannan–Lovasz–Simonovits Conjecture. https://arxiv.org/abs/2610.05474v1