Symmetric exclusion process under diffusive scaling

Assume the Axiom of Choice, meaning every family of nonempty sets has a choice function. Fix an integer $d\ge 1$ and an integer $N\ge 3$. Let ${\mathrm{\Lambda }}_N=(\mathbb{Z}\mathrm{/}N\mathbb{Z}{)}^d$, with each coordinate represented by $0,1,\dots ,N-1$, and let ${\mathcal{E}}_N$ be the set of unordered nearest-neighbor bonds $\{x,x+e_i\}$, where $1\le i\le d$ and addition is modulo $N$. The condition $N\ge 3$ makes this a simple graph with each bond listed once as $x$ ranges over ${\mathrm{\Lambda }}_N$. The site graph is connected because any two coordinate tuples are joined by finitely many steps of $+e_i$ or $-e_i$.

The configuration space is ${\mathrm{\Omega }}_N=\{\eta :{\mathrm{\Lambda }}_N\to \{0,1\}\}$, ordered lexicographically by its binary occupation vector in the fixed site order; this labels it as a finite set $\{1,\dots ,2^{N^d}\}$. The occupation ${\eta }_x$ is $1$ for an occupied site and $0$ for a vacant site. For $b=\{x,y\}\in {\mathcal{E}}_N$, let ${\eta }^b$ be the configuration obtained by swapping ${\eta }_x$ and ${\eta }_y$.

The coordinates $x\mathrm{/}N$ have dimensionless spacing $1\mathrm{/}N$ on the unit torus, and $t$ is dimensionless diffusive time: it is the microscopic unit-rate bond clock accelerated by the factor $N^2$. Thus the continuum heat equation below has normalized diffusion coefficient $D=1$.

Define a rate matrix $L_N$ on ${\mathrm{\Omega }}_N$ by giving each distinct state change $\eta \to {\eta }^b\mathrm{\ne }\eta$ rate $N^2$ and setting the diagonal entry to the negative sum of the off-diagonal row. If the two occupations on a bond agree, swapping them produces no state change and contributes no off-diagonal transition. On a column observable $f:{\mathrm{\Omega }}_N\to \mathbb{R}$, use the row-generator convention $(L_{N}f)(\eta )={\sum }_{\xi \mathrm{\ne }\eta }{L}_N(\eta ,\xi )(f(\xi )-f(\eta ))$; a row probability vector evolves by $p^{\mathrm{\prime }}=p{L}_N$. The symmetric simple exclusion process (SSEP) is the uniformized finite-state chain with rate matrix $L_N$ and any specified initial probability vector. The finite-generator existence result supplies its all-pairs transition matrices $e^{t{L}_N}$ and càdlàg paths. Every jump conserves ${\sum }_{x\in {\mathrm{\Lambda }}_N}{\eta }_x$.

For $k=0,1,\dots ,N^d$, the fixed-occupancy sector is ${\mathrm{\Omega }}_{N,k}=\{\eta \in {\mathrm{\Omega }}_N:{\sum }_x{\eta }_x=k\}$. When $0<k<N^d$, this sector has at least two states and its transition graph is connected by the fixed-occupancy connectivity lemma. Setting every sector energy to $0$, $\beta =1$, and each allowed transition rate to $N^2$ makes this sector a finite reversible stochastic rate model with uniform stationary weights. The sectors $k=0$ and $k=N^d$ are singleton constant chains. The full SSEP state graph is reducible across particle-number sectors; the generic finite rate-matrix construction, not the irreducible reversible-model definition, defines the full process.

Let ${\mathbb{T}}^d=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^d$ with normalized Haar measure $dx$. For $x\in {\mathrm{\Lambda }}_N$, write $x\mathrm{/}N\in {\mathbb{T}}^d$ for the point with coordinates represented in $[0,1)$. The empirical measure and its action on a smooth periodic test function $G$ are

${\pi }_t^N:=N^{-d}{\sum }_{x\in {\mathrm{\Lambda }}_N}{\eta }_x(t){\delta }_{x\mathrm{/}N},⟨{\pi }_t^N,G⟩:={\int }_{{\mathbb{T}}^d}Gd{\pi }_t^N=N^{-d}{\sum }_{x\in {\mathrm{\Lambda }}_N}{\eta }_x(t)G(x\mathrm{/}N)\mathrm{.}$

Here ${\delta }_y$ denotes unit point mass at $y$. Thus ${\pi }_t^N$ is a finite positive measure of total mass at most $1$, not necessarily a probability measure. For a smooth periodic $G$, define its discrete Laplacian on grid points by

${\mathrm{\Delta }}_{N}G(x\mathrm{/}N):=N^2{\sum }_{i=1}^d[G((x+e_i)\mathrm{/}N)+G((x-e_i)\mathrm{/}N)-2G(x\mathrm{/}N)]\mathrm{.}$

The Axiom of Choice

Finite reversible stochastic rate model

Finite-state generator chain

Uniformization constructs every finite-state rate matrix chain

Connectedness of fixed-occupancy exclusion sectors