Second Order Polynomial Field
We start with the simplest non-trivial example for independent random walkers started from a product measure with homogeneous Poisson marginals. To illustrate our point let us start with a simple computation, which contains all the important ingredients of the more general Theorem 3.1 below. Consider the field
$$\begin{aligned} X_N^{(2)}(\eta ;\varphi ):=X_N(2\delta _0,\eta ,\varphi )=\sum _{x \in \mathbb {Z}^d}\varphi \left( \tfrac{x}{N}\right) \, D(2\delta _x, \eta ) \end{aligned}$$
(17)
The notation \(X_N^{(2)}\) suggests that this is in some sense the ”second order” polynomial field. In the orthogonal polynomial language, this is the field of the second order Charlier polynomial:
$$\begin{aligned} D(2\delta _x, \eta )=\eta _x(\eta _x-1)-2\rho (\eta _x-\rho )-\rho ^2 \end{aligned}$$
(18)
recall from earlier that
$$\begin{aligned} a ( 2 \delta _0) = \int (D(2\delta _x,\eta ))^2 d\nu _\rho (\eta ) \end{aligned}$$
then we have the following.
Proposition 3.1
The second order polynomial field \(X_N^{(2)}(\eta ;\varphi )\) is such that
-
1.
For \(t>0\) we have
$$\begin{aligned} \mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N^{(2)}(\eta (t);\varphi )\, X_N^{(2)}(\eta (0);\varphi )\right] =a ( 2 \delta _0) \,\sum _{x,y \in \mathbb {Z}^d}\varphi (\tfrac{x}{N})\varphi (\tfrac{y}{N}) (p_t(x,y))^2 \end{aligned}$$
(19)
-
2.
As a consequence, for \(t>0\) we have
$$\begin{aligned} \lim _{N\rightarrow \infty }\mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N^{(2)}(\eta (N^ 2t);\varphi ) X_N^{(2)}(\eta (0);\varphi )\right] = \frac{d\cdot a ( 2 \delta _0) }{(2\pi t)^d}\int _{\mathbb {R}^{2d}} e^{-\frac{d|x-y|^2}{t}} \varphi (x)\varphi (y) dx dy \end{aligned}$$
(20)
Proof
The first statement follows from self-duality and Lemma 2.1. For the second statement we use that \(\varphi \) has compact support, call this support S, and define
$$\begin{aligned} M := \max \{ d(x,y) : x, y \in S\} \end{aligned}$$
(21)
it follows from Theorem 6.2 that there exists \(c= c(M)\) such that
$$\begin{aligned} \sup _{x:|x|\le M N \sqrt{t}} p^{RW}_{N^2t} (x) \le \bar{p}_{N^2t}(x) \left( 1+ \frac{c}{ N\sqrt{t}}\right) \end{aligned}$$
with \( \bar{p}_{t}(\cdot )\) as defined in (71). Then from (28) it follows that
$$\begin{aligned}&\mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N^{(2)}(\eta (t);\varphi )X_N^{(2)}(\eta (0);\varphi )\right] \\&\qquad \qquad = a ( 2 \delta _0) \sum _{x,y \in S}\varphi (\tfrac{x}{N})\varphi (\tfrac{y}{N})\,\bar{p}_{N^2t}(x) \bar{p}_{N^2t}(y) \left( 1+ \frac{c}{ N\sqrt{t}}\right) ^2 \\&\qquad \qquad = a ( 2 \delta _0) \cdot \frac{d}{(2\pi t)^{d}}\cdot \frac{1}{N^{2d}}\sum _{x,y \in S}\varphi (\tfrac{x}{N})\varphi (\tfrac{y}{N}) e^{-\frac{d(z-y)^2}{tN^2}} \left( 1+ \frac{c}{ N\sqrt{t}}\right) ^2 \end{aligned}$$
and letting \(N \rightarrow \infty \) we obtain the r.h.s. of (20). \(\square \)
In the current context the Boltzmann–Gibbs principle for the fluctuation field of the function \( f = \eta _0 ( \eta _0 -1)\) is a consequence of Proposition 3.1. We make this statement more transparent with the following corollary
Corollary 3.1
The field \(X_N^{(2)}(\eta (N^ 2 t);\varphi ) \) is such that for all \(T>0\) and for all N big enough
$$\begin{aligned} \frac{1}{N^d}\int _0^T\int _0^T\mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N^{(2)}(\eta (N^ 2 t);\varphi ) X_N^{(2)}(\eta (N^2 s);\varphi )\right] \, ds \, dt\le C(T) N^{-\frac{2d}{ 2 + d}} \end{aligned}$$
(22)
More precisely, (20) gives a better estimate of the order of the covariance of the fluctuation field in the diffusive time-scale as \(N\rightarrow \infty \).
Proof
Given the fact that the RHS of (20) has an indetermination at \(t=0\). Hence we derive the following estimate for the integrand in (22)
$$\begin{aligned}&\frac{1}{N^d}\mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N^{(2)}(\eta (N^ 2 t);\varphi ) X_N^{(2)}(\eta (N^2 s);\varphi )\right] \\&\qquad \qquad =\,K_\rho \, \frac{1}{N^d} \sum _{x \in \mathbb {Z}^d}\varphi (\tfrac{x}{N}) p_{N^2(t-s)}(x,y) \sum _{y \in \mathbb {Z}^d} \varphi (\tfrac{y}{N}) p_{N^2(t-s)}(x,y) \\&\qquad \qquad \le \,K_{\rho } p_{N^2(t-s)}(0,0) \Vert \varphi \Vert _1 \mathbb {E}_x \varphi (\tfrac{X_t}{N}) \\&\qquad \qquad \le \,K_{\rho } p_{N^2(t-s)}(0,0) \Vert \varphi \Vert _1 \Vert \varphi \Vert _{\infty } \end{aligned}$$
at this point we could have concluded (22) by naively estimating \( p_{N^2(t-s)}(0,0) \) by one. Nevertheless our aim is to provide a more quantitative statement. Hence, we distinguished the cases \( |t-s| \ge \epsilon _N\) and \( |t-s| < \epsilon _N\) where \(\epsilon _N\) is to be optimized. By the LCLT
$$\begin{aligned} p_{N^2(t-s)}(0,0) \le \frac{d}{(2\pi N^2(t-s))^{d/2}} \end{aligned}$$
(23)
then
$$\begin{aligned} p_{N^2(t-s)}(0,0) \le {\left\{ \begin{array}{ll} \frac{d}{N^d \epsilon _N^{d/2}}, &{} \text {if}\ |t-s| \ge \epsilon _N \\ 1 &{} \text {if}\ |t-s| < \epsilon _N \end{array}\right. } \end{aligned}$$
(24)
Hence the integral is bounded by
$$\begin{aligned}&\int _0^T\int _0^T \frac{1}{N^d}\mathbb {E}_{\nu _\rho }\left[ X_N^{(2)}(\eta (N^ 2 t);\varphi ) X_N^{(2)}(\eta (N^2 s);\varphi )\right] \, ds \, dt \nonumber \\&\qquad \qquad \qquad \le K_{\rho } \Vert \varphi \Vert _1 \Vert \varphi \Vert _{\infty } \frac{T^2}{2} \left[ \frac{d}{N^d \epsilon _N^{d/2}} + d \epsilon _N \right] \end{aligned}$$
(25)
Assume \(\epsilon _N \) is of the form \( N^{-\alpha }\), optimality then comes from solving for \(\alpha \)
$$\begin{aligned} N^{-\alpha } = N^{-d} N^{d/2 \alpha } \end{aligned}$$
after elementary computations we find \(\alpha = \frac{2d}{d+2}\). Which in fact not only shows that the Boltzmann–Gibbs principle holds, but also provides us with a better estimate of the order of convergence. \(\square \)
Back to the second order polynomial fluctuation fields, and for the sake of transparency, we make explicit the dependency on the “coordinate points” \(x_1, x_2 \) and redefine the fields in terms of the orthogonal duality polynomials as follows:
$$\begin{aligned} X_N^{(2)}(x_1, x_2,\eta ;\varphi ):=\sum _{x \in \mathbb {Z}^d}\varphi \Big (\tfrac{x}{N}\Big ) D(\delta _{x_1+x}+\delta _{x_2+x} , \eta ) \end{aligned}$$
(26)
Notice then, that in Proposition 3.1 we treated for \(x_1=x_2=0\). It is necessary then to verify that Proposition 3.1 is not only result of this particular choice we made, consider then for \(x_1\not =x_2\) the field
$$\begin{aligned} X_N^{(2),\not =}(x_1, x_2,\eta ,\varphi )= \sum _{x \in \mathbb {Z}^d} \varphi (\tfrac{x}{N})(\eta _{x+x_1}-\rho )(\eta _{x+x_2}-\rho ) \end{aligned}$$
(27)
where the upper index \(\not =\) refers to the fact that \(x_1\not = x_2\). We then have the following analogous of Proposition 3.1.
Proposition 3.2
The second order polynomial fluctuation field \(X_N^{(2),\not =}(x_1, x_2,\eta ;\varphi )\) is such that
-
1.
For \(t>0\) we have
$$\begin{aligned}&\mathbb {E}_{\nu _{\bar{\rho }}}(X_N^{(2),\not =}(x_1, x_2,\eta (t);\varphi )X_N^{(2),\not =}(x_1, x_2,\eta (0);\varphi )) \nonumber \\&\qquad \qquad = a( \delta _{x_1} + \delta _{x_2})\sum _{x,y \in \mathbb {Z}^d}\varphi \left( \tfrac{x}{N}\right) \varphi \left( \tfrac{y}{N}\right) p_t(x+x_1,x+x_2;y+x_1,y+x_2) \nonumber \\&\qquad \qquad \qquad +\,a( \delta _{x_1} + \delta _{x_2})\sum _{x,y \in \mathbb {Z}^d}\varphi \left( \tfrac{x}{N}\right) \varphi \left( \tfrac{y}{N}\right) p_t(x+x_1,x+x_2;y+x_2,y+x_1)\nonumber \\ \end{aligned}$$
(28)
-
2.
As a consequence, for \(t>0\) we have
$$\begin{aligned}&\lim _{N\rightarrow \infty } \mathbb {E}_{\nu _{\bar{\rho }}}\left( X_N^{(2),\not =}(x_1, x_2,\eta (N^ 2t);\varphi ) X_N^{(2),\not =}(x_1, x_2,\eta (0);\varphi )\right) \nonumber \\&\qquad \qquad =\frac{2 a( \delta _{x_1} + \delta _{x_2}) d}{(2\pi t)^d}\int _{\mathbb {R}^{2d}} e^{-\frac{d|x-y|^2}{t}} \varphi (x)\varphi (y) dx dy. \end{aligned}$$
(29)
Proof
The argument for the first statement is similar to the one in the proof of Proposition 3.1, the difference is that now
$$\begin{aligned} D(\delta _{x+x_1} +\delta _{x+x_2},\eta )= (\eta _{x+x_1}-\rho )(\eta _{x+x_2}-\rho ) \end{aligned}$$
is the product of two first order Charlier polynomials, which by the assumption of factorized polynomials allows us to proceed in the same way than before. Furthermore, in this case we have
$$\begin{aligned}&p_t(\delta _{x+x_1}+ \delta _{x+x_2},\delta _{y+x_1}+ \delta _{y+x_2} ) \nonumber \\&\qquad \qquad = p_t(x+x_1,x+x_2;y+x_1,y+x_2) + p_t(x+x_1,x+x_2;y+x_2,y+x_1)\nonumber \\ \end{aligned}$$
(30)
which is the source of the second term in (28). In the second statement is necessary to verify that \(x_1\) and \(x_2\) do not play a role in the leading order
$$\begin{aligned}&\mathbb {E}_{\nu _{\bar{\rho }}}(X_N^{(2),\not =}(x_1, x_2,\eta (N^ 2t);\varphi ) X_N^{(2),\not =}(x_1, x_2,\eta (0);\varphi )) \nonumber \\&\qquad \qquad =a( \delta _{x_1} + \delta _{x_2})\sum _{x,y \in \mathbb {Z}^d}\varphi (\tfrac{x}{N})\varphi (\tfrac{y}{N}) p_{N^2t}(x+x_1,x+x_2;y+x_1,y+x_2) \nonumber \\&\qquad \qquad \qquad +\,a( \delta _{x_1} + \delta _{x_2})\sum _{x,y \in \mathbb {Z}^d}\varphi (\tfrac{x}{N})\varphi (\tfrac{y}{N}) p_{N^2t}(x+x_1,x+x_2;y+x_2,y+x_1)\nonumber \\ \end{aligned}$$
(31)
The first term in the RHS of (31) can be treated in the same way than before. For the second term, we just have to notice
$$\begin{aligned} |x+x_1-y-x_2|^2 +|x+x_2-y-x_1|^2 = 2|x-y|^2 +2|x_1-x_2|^2 \end{aligned}$$
and proceed in the same way. \(\square \)
Now we show how to generalize this result and discuss the case of higher order fields.
Higher Order Fields
Let \(k\in \mathbb {N}\) and denote by \(\mathbf {x}\in \mathbb {Z}^{kd}\) the coordinates vector \(\mathbf {x}:=(x_1,\ldots ,x_k)\), with \(x_i\in {\mathbb {Z}}^d\), \(i=1,\ldots , k\). We denote by \(\xi (\mathbf {x})\) the configuration associated to \(\mathbf {x}\), i.e. \(\xi _x(\mathbf {x})=\sum _{i=1}^k\mathbf {1}_{x=x_i}\). We define \(||\mathbf {x}||:=||\xi (\mathbf {x})||=k\). Here \(x_i\) is the position of the i-th particle, where particles are labeled in such a way that the dynamics is symmetric. For a more extensive explanation of the labeled dynamics we refer the reader to [5]. We denote by \(\hat{\tau }_z\), \(z\in \mathbb {Z}^d\) the shift operator acting on the coordinate representation:
$$\begin{aligned} \hat{\tau }_z \mathbf {x}= (z+x_1, \ldots , z+x_k), \qquad \text {and then} \qquad \tau _z\xi = \xi (\hat{\tau }_z \mathbf {x}) \end{aligned}$$
(32)
Because of the translation invariance of the dynamics we have that
$$\begin{aligned} p_t(\xi (\hat{\tau }_y \mathbf {x}), \xi (\hat{\tau }_z \mathbf {x}))=p_t(\xi (\mathbf {x}), \xi (\hat{\tau }_{z-y} \mathbf {x})) \end{aligned}$$
(33)
With an abuse of notation, we keep denoting by \(p_t(\mathbf {x}, \mathbf {y})\) the transition probability of the labeled particles in the coordinate representation.
Remark 3.1
The relation between the transition probabilities in the coordinate and in the configuration representations is given by
$$\begin{aligned} p_t(\xi (\mathbf {x}),\xi (\mathbf {y}))=\sum _{\mathbf {x}': \xi (\mathbf {x}')=\xi (\mathbf {y})} p_t(\mathbf {x}, \mathbf {x}') \end{aligned}$$
(34)
Notice that it is presicely from relation (34) that a factor of 2 appears in Proposition 3.2 and not in Proposition 3.1. We can expect that in this general setting the difference among cases will become more cumbersome. To avoid any further notational difficulties we introduce the following:
Let \(\mathscr {P}_k\) be the set of permutations of \(\{1,\ldots , k\}\), for \(\sigma ,\sigma ' \in \mathscr {P}_k \) we define the following equivalence relation:
$$\begin{aligned} \sigma \sim \sigma ' \quad \text {mod} \quad \mathbf {x} \qquad \text {iff} \quad x_{\sigma (i)}=x_{\sigma '(i)} \quad \forall i \in \{1,\ldots ,k\} \end{aligned}$$
(35)
and define \(\mathscr {P}_k(\mathbf {x}):=\mathscr {P}_k/\sim _{\mathbf {x}}\). Then we have
$$\begin{aligned} |\mathscr {P}_k(\mathbf {x})|=\frac{k!}{\prod _{i\in \mathbb {Z}^d}\xi _i(\mathbf {x})!} \end{aligned}$$
(36)
For each \(\sigma \in \mathscr {P}_k(\mathbf {x})\) we define the new coordinate vector \(\mathbf {x}^{(\sigma )}\) such that
$$\begin{aligned} \mathbf {x}^{(\sigma )}_i= x_{\sigma (i)} \end{aligned}$$
(37)
thus we can write
$$\begin{aligned} p_t(\xi (\mathbf {x}), \xi (\hat{\tau }_{z} \mathbf {x}))=\sum _{\mathbf {x}': \xi (\mathbf {x}')=\xi (\hat{\tau }_z \mathbf {x})} p_t(\mathbf {x}, \mathbf {x}')= \sum _{\sigma \in \mathscr {P}_k({\mathbf {x}})}p_t(\mathbf {x}, \hat{\tau }_z \mathbf {x}^{(\sigma )}) \end{aligned}$$
(38)
With a slight abuse of notation we denote by
$$\begin{aligned} X_N(\mathbf {x},\eta ,\varphi ):= \sum _{z\in \mathbb {Z}^d} \varphi \left( \frac{z}{N}\right) D(\hat{\tau }_z \mathbf {x},\eta ), \end{aligned}$$
(39)
define the k-th order fluctuation field associated to the k-particles configuration \(\mathbf {x}\). Then we have
Theorem 3.1
Let \(k:=||\mathbf {x}||\), then the k-th order fluctuation field \(X_N(\mathbf {x},\eta ,\varphi )\) is such that
-
1.
For all \(t>0\)
$$\begin{aligned}&\mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N(\mathbf {x},\eta (t),\varphi )X_N(\mathbf {x},\eta (0),\varphi )\right] \nonumber \\&\quad = a(\xi (\mathbf {x}))\sum _{\sigma \in \mathscr {P}_k({\mathbf {x}})} \sum _{y,z\in \mathbb {Z}^d} \varphi \left( \frac{y}{N}\right) \varphi \left( \frac{z}{N}\right) p_t( \mathbf {x}, \hat{\tau }_{z-y} \mathbf {x}^{(\sigma )}) \end{aligned}$$
(40)
-
2.
As a consequence, for \(t>0\)
$$\begin{aligned}&\lim _{N\rightarrow \infty } N^{d(k-2)}\mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N(\mathbf {x},\eta (N^2t),\varphi )X_N(\mathbf {x},\eta (0),\varphi )\right] \nonumber \\&\quad = |\mathscr {P}_k(\mathbf {x})| a(\xi (\mathbf {x})) \frac{ d^{k/2}}{(2\pi t)^{dk/2}}\int _{\mathbb {R}^{2d}} e^{-kd|z-y|^2/2t} \varphi (z)\varphi (y) dz dy \end{aligned}$$
(41)
Proof
The first statement of the theorem is a direct application of Lemma 2.1 and the fact that the function \(a(\cdot )\) is translation invariant, i.e. \(a(\xi (\hat{\tau }_z \mathbf {x}))=a(\xi (\mathbf {x}))\), for all \(z \in \mathbb {Z}^d\).
$$\begin{aligned}&\mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N(\mathbf {x},\eta (t),\varphi )X_N(\mathbf {x},\eta (0),\varphi )\right] \nonumber \\&\qquad \qquad = a(\xi (\mathbf {x}))\sum _{y,z\in \mathbb {Z}^d} \varphi \left( \frac{y}{N}\right) \varphi \left( \frac{z}{N}\right) p_t(\xi (\hat{\tau }_y \mathbf {x}), \xi (\hat{\tau }_z \mathbf {x})) \end{aligned}$$
(42)
Then, from (33) and (42) it follows that
$$\begin{aligned}&\mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N(\mathbf {x},\eta (t),\varphi )X_N(\mathbf {x},\eta (0),\varphi )\right] \nonumber \\&\qquad \qquad = a(\xi (\mathbf {x}))\sum _{\sigma \in \mathscr {P}_k({\mathbf {x}})} \sum _{y,z\in \mathbb {Z}^d} \varphi \left( \frac{y}{N}\right) \varphi \left( \frac{z}{N}\right) p_t( \mathbf {x}, \hat{\tau }_{z-y} \mathbf {x}^{(\sigma )}) \end{aligned}$$
(43)
For the second stament observe that from translation invariance we have
$$\begin{aligned} p^{\text {IRW}}_{N^2t}( \mathbf {x}, \hat{\tau }_{z-y} \mathbf {x})=\left( p_{N^2t}^{RW}(z-y)\right) ^k \end{aligned}$$
(44)
Define \(B_{M,N}:=\{x\in \mathbb {Z}^d: |x|\le N M\}\), then, since \(\varphi \) has a finite support we have that there exists \(M\ge 0\) such that, for
$$\begin{aligned}&\sum _{y,z\in \mathbb {Z}^d} \varphi \left( \frac{y}{N}\right) \varphi \left( \frac{z}{N}\right) p^{\text {IRW}}_{N^2t}( \mathbf {x}, \hat{\tau }_{z-y} \mathbf {x})\\&\qquad \qquad =\sum _{y,z\in B_{M,N}} \varphi \left( \frac{y}{N}\right) \varphi \left( \frac{z}{N}\right) \left( p_{N^2t}^{RW}(z-y)\right) ^k\\&\qquad \qquad =\left( \frac{\sqrt{d}}{(2\pi t)^{d/2}}\right) ^k\left( 1+\frac{c}{N\sqrt{t}}\right) ^k\frac{1}{N^{kd}}\sum _{y,z\in B_{M,N}} \varphi \left( \frac{y}{N}\right) \varphi \left( \frac{z}{N}\right) \, e^{-\frac{kd|\frac{z}{N}-\frac{y}{N}|^2}{2t}} \end{aligned}$$
for a suitable \(c=c(M)\), the last inequality coming from Theorem 6.2. We have
$$\begin{aligned}&\lim _{N\rightarrow \infty } \frac{1}{N^{2d}}\sum _{y,z\in \mathbb {Z}^d} \varphi \left( \frac{y}{N}\right) \varphi \left( \frac{z}{N}\right) e^{-\frac{kd|\frac{z}{N}-\frac{y}{N}|^2}{2t}}= \int _{\mathbb {R}^{2d}} \varphi \left( y\right) \varphi \left( z\right) \, e^{-\frac{kd|z -y|^2}{2t}} dxdz. \end{aligned}$$
\(\square \)
Quantitative Boltzmann–Gibbs Principle
On the same spirit than Corollary 3.1 we can now state a refined quantitative version of the Boltzmann–Gibbs principle for higher order fields.
Theorem 3.2
The field \(X_N^{(k)}(\eta (N^ 2 t);\varphi ) \) is such that for all \(T>0\) there exists C(T) such that for all N big enough
$$\begin{aligned} \frac{1}{N^d}\int _0^T\int _0^T\mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N(\mathbf {x},\eta (N^2 t),\varphi )X_N(\mathbf {x},\eta (N^2 s),\varphi ) \right] \, ds \, dt \le C(T) N^{-\frac{2(k-1) d}{ 2 +(k-1) d}} \end{aligned}$$
(45)
Proof
Analogously to the case of two particles ( see the proof of Corollary 3.1), and using observation (44) we first obtain the following estimate
$$\begin{aligned}&\frac{1}{N^d} \mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N(\mathbf {x},\eta (N^2 t),\varphi )X_N(\mathbf {x},\eta (N^2 s),\varphi ) \right] \nonumber \\&\qquad \qquad \le \left( p_{N^2(t-s)}^{RW}(0)\right) ^{k-1} |\mathscr {P}_k(\mathbf {x})| a(\xi (\mathbf {x})) \Vert \varphi \Vert _1 \Vert \varphi \Vert _{\infty } \end{aligned}$$
(46)
again, by the LCLT
$$\begin{aligned} \left( p_{N^2(t-s)}^{RW}(0)\right) ^{k-1} \le {\left\{ \begin{array}{ll} \frac{d}{N^{(k-1)d} \epsilon _N^{(k-1)d/2}}, &{} \text {if}\ |t-s| \ge \epsilon _N \\ 1, &{} \text {otherwise} \end{array}\right. } \end{aligned}$$
(47)
allowing us to bound the integral
$$\begin{aligned}&\int _0^T\int _0^T \frac{1}{N^d}\mathbb {E}_{\nu _{\bar{\rho }}}\left[ X_N^{(2)}(\eta (N^ 2 t);\varphi ) X_N^{(2)}(\eta (N^2 s);\varphi )\right] \, ds \, dt \nonumber \\&\qquad \qquad \qquad \le |\mathscr {P}_k(\mathbf {x})| a(\xi (\mathbf {x})) \Vert \varphi \Vert _1 \Vert \varphi \Vert _{\infty } \frac{T^2}{2} \left[ \frac{d}{N^{(k-1)d} \epsilon _N^{(k-1)d/2}} + d \epsilon _N \right] \end{aligned}$$
(48)
the same anzats, \(\epsilon _N = N^{-\alpha }\), results on the optimal value
$$\begin{aligned} \alpha = \frac{2(k-1) d}{ 2 +(k-1) d}. \end{aligned}$$
(49)
\(\square \)
Fluctuation Fields of Projections on \({\mathscr {H}}_N\)
We can further generalize part (2) of Theorem 3.1 to a wider class of functions f. In this section we make such a generalization for a particular subset of \(L^2(\nu _\rho )\). For \(f\in L^2(\nu _\rho )\) we can use the fact that the union of the spaces \({\mathscr {H}}_n\) is dense in \(L^2(\nu _\rho )\) to express f as follows
$$\begin{aligned} f (\eta ) = \sum _{\begin{array}{c} n \ge 0 \\ \xi \in \Omega _f: \Vert \xi \Vert = n \end{array}} C_{n,\xi } D(\xi ,\eta ) \end{aligned}$$
(50)
for the rest of this section we restrict ourselves to the set of functions \(f\in L^2(\nu _\rho )\) satisfying the following condition
$$\begin{aligned} \sum _{ \xi , \xi ^\prime \in \Omega _f: \Vert \xi \Vert = \Vert \xi ^\prime \Vert } | C_{n,\xi } C_{n,\xi ^\prime } | a(\xi ^\prime ) < \infty \end{aligned}$$
(51)
In particular all linear combinations of orthogonal duality polynomials satisfy (51).
Theorem 3.3
Let f be a function such that the condition (51) is satisfied, and as before let \(f_{k-1}\) denote the projection of f on \({\mathscr {H}}_{k-1}\), then the field
$$\begin{aligned} \mathsf {X_N}(f-f_{k-1},\eta ;\varphi )=\sum _{x\in \mathbb {Z}^d} (\tau _x f (\eta )- \tau _x f_{k-1}(\eta )) \varphi \left( \frac{x}{N}\right) \end{aligned}$$
satisfies
$$\begin{aligned} \mathbb {E}_{\nu _{\bar{\rho }}}\left[ \mathsf {X_N}(f-f_{k-1},\eta ;\varphi )\mathsf {X_N}(f-f_{k-1},\eta (N^2 t);\varphi )\right] = O(N^{-d(k-2)}) \end{aligned}$$
Proof
After some simplifications due to orthogonality the field reads
$$\begin{aligned} \mathsf {X_N}(f-f_{k-1},\eta ;\varphi )= \sum _{x \in \mathbb {Z}^d} \varphi \left( \frac{x}{N}\right) \sum _{\begin{array}{c} n \ge k \\ \xi \in \Omega _f: \Vert \xi \Vert = n \end{array}} C_{n,\xi } \tau _x D( \xi ,\eta ) \end{aligned}$$
We then compute
$$\begin{aligned}&\mathbb {E}_{\nu _{\bar{\rho }}}\left[ \mathsf {X_N}(f-f_{k-1},\eta ;\varphi )\mathsf {X_N}(f-f_{k-1},\eta (N^2 t);\varphi )\right] \nonumber \\&\quad = \sum _{x, y} \varphi \left( \frac{x}{N}\right) \varphi \left( \frac{y}{N}\right) \sum _{\begin{array}{c} n \ge k \\ \xi \in \Omega _f: \Vert \xi \Vert = n \end{array}} \sum _{\begin{array}{c} l \ge k \\ \xi ^\prime \in \Omega _f: \Vert \xi ^\prime \Vert = l \end{array}} C_{n,\xi } C_{l,\xi ^\prime }\nonumber \\&\qquad \int \tau _x D(\xi ,\eta ) E_\eta \left[ \tau _y D(\xi ^\prime ,\eta (N^2 t)) \right] d \nu _{\bar{\rho }} (\eta ) \nonumber \\&\quad = \sum _{x, y} \varphi \left( \frac{x}{N}\right) \varphi \left( \frac{y}{N}\right) \sum _{\begin{array}{c} n \ge k \\ \xi \in \Omega _f: \Vert \xi \Vert = n \end{array}} \sum _{\begin{array}{c} l \ge k \\ \xi ^\prime \in \Omega _f: \Vert \xi ^\prime \Vert = l \end{array}} C_{n,\xi } C_{l,\xi ^\prime } \nonumber \\&\qquad \int \tau _x D(\xi ,\eta ) E_\eta \left[ \tau _y D(\xi ^\prime ,\eta (N^2 t)) \right] d \nu _{\bar{\rho }} (\eta ) \nonumber \\&\quad = \sum _{x, y} \varphi \left( \frac{x}{N}\right) \varphi \left( \frac{y}{N}\right) \sum _{\begin{array}{c} n \ge k \\ \xi \in \Omega _f: \Vert \xi \Vert = n \\ \xi ^\prime \in \Omega _f: \Vert \xi ^\prime \Vert = n \end{array}} C_{n,\xi } C_{n,\xi ^\prime } a(\xi ^\prime ) p_{N^2 t}(\tau _y \xi ^\prime , \tau _x \xi ) \end{aligned}$$
(52)
from the LCLT we can also obtain that
$$\begin{aligned} p_{N^2 t}(\tau _y \xi , \tau _x \xi ^\prime ) = \mathscr {O}(N^{-d \Vert \xi \Vert }) \end{aligned}$$
this, allows us to bound our expression of interest
$$\begin{aligned}&N^{d(k-2)} \mathbb {E}_{\nu _{\bar{\rho }}}\left[ \mathsf {X_N}(f-f_{k-1},\eta ;\varphi )\mathsf {X_N}(f-f_{k-1},\eta (N^2 t);\varphi )\right] \nonumber \\&\qquad \qquad \le N^{d(k-2)} \sum _{x, y} \varphi \left( \frac{x}{N}\right) \varphi \left( \frac{y}{N}\right) \sum _{\begin{array}{c} n \ge k \\ \xi \in \Omega _f: \Vert \xi \Vert = n \\ \xi ^\prime \in \Omega _f: \Vert \xi ^\prime \Vert = n \end{array}} \frac{M}{N^{dn}} | C_{n,\xi } C_{n,\xi ^\prime }| a(\xi ^\prime ) \nonumber \\&\qquad \qquad = \left( \frac{1}{N^{2d}}\sum _{x, y} \varphi \left( \frac{x}{N}\right) \varphi \left( \frac{y}{N}\right) \right) \sum _{\begin{array}{c} n \ge k \\ \xi \in \Omega _f: \Vert \xi \Vert = n \\ \xi ^\prime \in \Omega _f: \Vert \xi ^\prime \Vert = n \end{array}} \frac{M}{N^{d(n-k)}} | C_{n,\xi } C_{n,\xi ^\prime }| a(\xi ^\prime ) \end{aligned}$$
(53)
At this point we need to show that the last summation does not play a role in the leading order. But this comes from the fact that f satisfies condition (51). \(\square \)
Analogously to Theorem 3.2 we provide a quantitative version of the Boltzmann–Gibbs principle for the current setting.
Theorem 3.4
The field \(\mathsf {X_N}(f-f_{k-1},\eta ;\varphi )\) is such that for all \(T>0\) there exists C(T) such that for all N big enough
$$\begin{aligned}&\frac{1}{N^d}\int _0^T\int _0^T\mathbb {E}_{\nu _{\bar{\rho }}}\left[ \mathsf {X_N}(f-f_{k-1},\eta (N^2 t);\varphi ) \mathsf {X_N}(f-f_{k-1},\eta (N^2 s);\varphi ) \right] \, ds \, dt\nonumber \\&\qquad \qquad \qquad \le C(T) N^{-\frac{2(k-1) d}{ 2 +(k-1) d}}. \end{aligned}$$
(54)