Theorem: Compact-test thermal limit

Under Assumption ass:peeling, fix $L>0$. There are constants $t_L,C_L>0$ such that, for $t\ge t_L$,

is represented on $[-L,L{]}^2$ by a continuous kernel $E_t(x,y)$ satisfying

Consequently, for $f,g\in {C^{\infty }}_c((-L,L))$,

This proves a local distributional thermal limit for one translated current covariance.

Proof. Affine translation and positive affine rescaling leave the singular ratio in (eq:Cp) unchanged. Define

Equation (eq:peeling) gives

For sufficiently large $t$, both $q_t^{\prime }$ and $q_{\infty }^{\prime }$ are bounded below by a positive constant on this fixed interval.

Set

Then $q(x)-q(y)=(x-y)D_q(x,y)$, and the singular pullback is

Write $m=(x+y)/2$ and $s=x-y$. The function $A_q$, expressed in $(m,s)$, is even in $s$, satisfies $A_q(m,0)=1$, and has ${\partial }_s{A}_q(m,0)=0$. Taylor's formula gives

The map $q\mapsto A_q$ is locally Lipschitz from $C^3$ to $C^2$ when $q^{\prime }$ has a common positive lower bound. Indeed, its derivatives through order two are finite sums of products of derivatives of $q$ through order three and inverse powers of $D_q$; the denominators are uniformly bounded away from zero. Thus (eq:qconvergence) implies

The distributional identity

now shows that the difference of the two singular pullbacks is the continuous kernel

The $q_{\infty }$ pullback is $C_{\kappa }$ by Lemma lem:pullback. This proves the first error term.

The smooth remainder contributes

On the fixed square, $p^{\prime }(t+x)=O_L(e^{-\kappa t})$, and $R$ is bounded near the limiting affine point. This contribution is $O_L(e^{-2\kappa t})$, proving (eq:localerror). Integrating the continuous kernel against $\overline{f}\otimes g$ proves (eq:smearederror).