Functional integral representations for self-avoiding walk

By “bosonic representations” we mean representations for random walk models in terms of ordinary Gaussian integrals. For our purposes, these integrals are in terms of a two-component field $(u_{x},v_{x})_{x\in\{1,\ldots,M\}}$, which is most conveniently represented by the complex pair $(\phi_{x},\bar{\phi}_{x})$, where

The differentials $d\phi_{x}$, $d\bar{\phi}_{x}$ are given by

and their product $d\bar{\phi}_{x}d\phi_{x}$ is given by

where we adopt the convention that differentials are multiplied together with the anticommutative wedge product; in particular $du_{x}du_{x}$ and $dv_{x}dv_{x}$ vanish and do not appear in the above product. This anticommutative product will play a central role when we come to fermions in Section 4, but until then plays no role beyond the formula (2.3). We are using the letter “$x$” as index for the field in anticipation of the fact that in our representations the field will be indexed by the space in which our random walks take steps.

We now briefly review some elementary properties of Gaussian measures. Let $C$ be an $M\times M$ complex matrix. We assume that $C$ has positive Hermitian part, i.e., $\sum_{x,y=1}^{M}\phi_{x}(C_{x,y}+\bar{C}_{y,x})\bar{\phi}_{y}>0$ for all nonzero $\phi\in\mathbb{C}^{M}$. Let $A=C^{-1}$. We write $d\mu_{C}$ for the Gaussian measure on ${\mathbb{R}}^{2M}$ with covariance $C$, namely

where $\phi A\bar{\phi}=\sum_{x,y=1}^{M}\phi_{x}A_{x,y}\bar{\phi}_{y}$, and where $Z_{C}$ is the normalization constant

We will need the value of $Z_{C}$ given in the following lemma.

A basic tool is the integration by parts formula given in the following lemma. The derivative appearing in its statement is defined by

With $\partial/\partial\bar{\phi}_{x}$ defined to be its conjugate, this leads to the equations

The equations

are simple consequences of Lemma 2.2. The last equality is a special case of Wick’s theorem, which provides a formula for the calculation of arbitrary moments of the Gaussian measure. We will only need the following special case of Wick’s theorem, in which a particular Gaussian expectation is evaluated as the permanent of a submatrix of $C$.

Our setting throughout the paper is a fixed finite set $\Lambda=\{1,2,\ldots,M\}$ of cardinality $M\geq 1$. Given points $a,b\in\Lambda$, a walk $\omega$ from $a$ to $b$ is a sequence of points $x_{0}=a,x_{1},x_{2},\ldots,x_{n}=b$, for some $n\geq 0$. We write $|\omega|$ for the length $n$ of $\omega$. Sometimes it is useful to regard $\omega$ as consisting of the directed edges $(x_{i-1},x_{i})$, $1\leq i\leq n$, rather than vertices. Let $\mathcal{W}_{a,b}$ denote the set of all walks from $a$ to $b$, of any length.

Let $J$ be a $\Lambda\times\Lambda$ complex matrix with zero diagonal part (i.e., $J_{x,x}=0$ for all $x\in\Lambda$). Let $D$ be a diagonal matrix with nonzero entries $D_{x,x}=d_{x}\in\mathbb{C}$. We assume that $D-J$ is diagonally dominant; this means that

Given $\omega\in\mathcal{W}_{a,b}$, let

Here we regard $\omega$ as a set of labeled edges $e=(\omega(i-1),\omega(i))$ (the empty product is $1$ if $|\omega|=0$). The simple random walk two-point function is defined by

The assumption that $D-J$ is diagonally dominant ensures that the sum in (2.16) converges absolutely. The following theorem was proved in BFS82 .

Next, we suppose that $d_{x}>0$, $J_{x,y}\geq 0$, and give two alternate representations for $G^{\,\rm srw}_{a,b}$ in terms of continuous-time Markov chains. For the first, which appeared in Dynk83 , we consider the continuous-time Markov chain $X$ defined as follows. The state space of $X$ is $\Lambda\cup\{\partial\}$, where $\partial$ is an absorbing state called the cemetery. When $X$ arrives at state $x$ it waits for an ${\rm Exp}(d_{x})$ holding time and then jumps to $y$ with probability $\pi_{x,y}=d_{x}^{-1}J_{x,y}$ and jumps to the cemetery with probability $\pi_{x,\partial}=1-\sum_{y\in\Lambda}d_{x}^{-1}J_{x,y}$. The holding times are independent of each other and of the jumps. Let $\zeta$ denote the time at which the process arrives in the cemetery. Note that if $D-J$ is diagonally dominant then $\zeta<\infty$ with probability $1$, and by right-continuity of the sample paths the last state visited by $X$ before arriving in the cemetery is $X(\zeta^{-})$. For $x\in\Lambda$, let $L_{x}$ denote the total (continuous) time spent by $X$ at $x$. We denote the expectation for $X$, started from $a\in\Lambda$, by $\mathbb{E}_{a}$.

Next, we derive a third representation for $G^{\,\rm srw}_{a,b}(v)$, which is more general than Theorem 2.5 as it does not require diagonal dominance of $D-J$ (it does require ${\rm Re}\,v_{x}>0$ when $d_{x}=\overline{d}_{x}$). This representation was obtained in BEI92 using the Feynman–Kac formula, but we give a different proof based on Theorem 2.5. The representation involves a second continuous-time Markov process, with generator $\overline{D}-J$ where we set $\overline{d}_{x}=\sum_{y\in\Lambda}J_{x,y}$ and assume $\overline{d}_{x}>0$ for each $x\in\Lambda$. This process is like the one described above, but has no cemetery site and continues for all time. Let $\overline{\mathbb{E}}_{a}$ denote the expectation for this process started at $a\in\Lambda$. Let

denote the time spent by $X$ at $x$ during the time interval $[0,T]$.

The two representations for $G^{\,\rm srw}_{a,b}$ in Theorems 2.5–2.6 show that the right-hand sides of (2.20) and (2.25) are equal. The following proposition generalizes this equality.

Next, we derive a representation for a model of a self-avoiding walk in a background of loops. This requires the introduction of some terminology and notation.

Given not necessarily distinct points $a,b\in\Lambda$, a self-avoiding walk $\omega$ from $a$ to $b$ is a sequence $x_{0}=a,x_{1},x_{2},\ldots,x_{n}=b$, for some $n\geq 1$, where $x_{1},x_{2},\ldots,x_{n-1}$ are distinct points in $\Lambda\setminus\{a,b\}$. In other words, for $a\not=b$, $\omega$ is a non-intersecting path from $a$ to $b$ on the complete graph on $M$ vertices and for $a=b$ it is non-intersecting except at $a=b$. We again write $|\omega|$ for the length $n$ of $\omega$, and sometimes regard $\omega$ as consisting of directed edges rather than vertices. Let $\mathcal{S}_{a,b}$ denote the set of all self-avoiding walks from $a$ to $b$. For $X\subset\Lambda$, we write $\mathcal{S}_{a,b}(X)$ for the subset of $\mathcal{S}_{a,b}$ consisting of walks with $x_{0}=a$, $x_{n}=b$ and $x_{1},x_{2},\ldots,x_{n-1}\in X$. A loop $\gamma$ is an unrooted directed cycle (consisting of distinct vertices) in the complete graph, regarded sometimes as a cyclic list of vertices and sometimes as directed edges. We include the self-loop which joins a vertex to itself by a single edge, as a possible loop (see Remark 2.9 below). We write $\mathcal{L}$ for the set of all loops. We write $\Gamma$ for a subgraph of $\Lambda$ consisting of mutually-avoiding loops, i.e., $\Gamma=\{\gamma_{1},\ldots,\gamma_{m}\}$ with each $\gamma_{i}\in\mathcal{L}$ and $\gamma_{i}\cap\gamma_{j}=\varnothing$ (as sets of vertices) for $i\neq j$. We write $\mathcal{G}$ for the set of all such $\Gamma$ (including $\Gamma=\varnothing$), and $\mathcal{G}(X)$ for the subset of $\mathcal{G}$ which uses only vertices in $X\subset\Lambda$. We write $|\gamma|$ for the length of $\gamma$, and $|\Gamma|=\sum_{i=1}^{m}|\gamma_{i}|$ for the total length of loops in $\Gamma$.

Given a $\Lambda\times\Lambda$ real matrix $C$, $\omega\in\mathcal{W}_{a,b}$ and $\Gamma\in\mathcal{G}$, let

where here we regard self-avoiding walks and loops as collections of directed edges and write, e.g., $e=(\omega(i-1),\omega(i))$. An empty product is equal to $1$. We define the two-point function

The representation for $G^{\,\rm loop}_{a,b}$ is elementary and we derive it now.

We define the two-point function:

When $a=b$, the walks are self-avoiding except for the fact that the walk begins and ends at the same site. In this case, there is, in particular, a contribution due to the one-step walk that steps from $a$ to $a$, which has weight $C_{a,a}\neq 0$. The only new result in this paper is the integral representation for $G^{\,\rm saw}_{a,b}$. The representation for the loop model (2.39) is easier than for (3.1), as (2.39) is in terms of a bosonic (ordinary) Gaussian integral. To eliminate the loops and obtain a representation for the walk model (3.1), we will need fermionic (Grassmann) integrals involving anticommuting variables. The necessary mathematical background for this is developed in Section 4, and the representation is stated and derived in Section 5.2. This representation is the point of departure for the analysis of the 4-dimensional self-avoiding walk in BS10 , for a convenient particular choice of $C$.

The two-point functions (2.39) and (3.1) are for strictly self-avoiding walks and loops. We also consider the continuous-time weakly self-avoiding walk, which is defined as follows.

Let $D$ have diagonal entries $d_{x}>0$, $J$ have zero diagonal entries and $J_{x,y}\geq 0$, and suppose that $D-J$ is diagonally dominant. Let $X$ and $\mathbb{E}_{a}$ be the continuous-time Markov process and corresponding expectation, as in Theorem 2.5. In particular, the process dies at the random time $\zeta$ at which it makes a transition to the cemetery state. The local time at $x$ is given by $L_{x}=\int_{0}^{\infty}{\mathbb{I}}_{X(s)=x}ds$ (note that the integral effectively terminates at $\zeta<\infty$). By definition,

so $\sum_{x\in\Lambda}L_{x}^{2}$ is a measure of the amount of self-intersection of $X$ up to time $\zeta$. The continuous-time weakly self-avoiding walk two-point function is defined by

where $g>0$, and $\lambda$ is a parameter (possibly negative) which is chosen in such a way that the integral converges. In (3.4), self-intersections are suppressed by the factor $\exp[-g\sum_{x\in\Lambda}L_{x}^{2}]$. We will derive a representation for (3.4) in Section 5.1.

It follows from Proposition 2.7 that there is also the alternate representation:

In the homogeneous case, in which $d_{x}-\overline{d}_{x}=a$ is independent of $x$, the second exponential can be written as $e^{-\lambda^{\prime}T}$ where $\lambda^{\prime}=\lambda+a$. This representation is the starting point for the analysis of the weakly self-avoiding walk on a 4-dimensional hierarchical lattice in BEI92 , BI03c , BI03d , on ${\mathbb{Z}}^{4}$ in BS10 , and for a model on ${\mathbb{Z}}^{3}$ in MS08 .

We recall and extend the formalism introduced in Section 2. Let $\Lambda=\{1,\ldots,M\}$ be a finite set of cardinality $M$. Let $u_{1},v_{1},\ldots,u_{M},v_{M}$ be standard coordinates on ${\mathbb{R}}^{2N}$, so that $du_{1}\wedge dv_{1}\wedge\cdots\wedge du_{M}\wedge dv_{M}$ is the standard volume form on ${\mathbb{R}}^{2M}$, where $\wedge$ denotes the usual anticommuting wedge product (see [Rudi76, , Chapter 10] for an introduction). We will drop the wedge from the notation and write simply $du_{i}dv_{j}$ in place of $du_{i}\wedge dv_{j}$. The one-forms $du_{i}$, $dv_{j}$ generate the Grassmann algebra of differential forms on ${\mathbb{R}}^{2M}$. A form which is a function of $u,v$ times a product of $p$ differentials is said to have degree $p$, for $p\geq 0$.

The integral of a differential form over ${\mathbb{R}}^{2M}$ is defined to be zero unless the form has degree $2M$. A form $K$ of degree $2M$ can be written as $K=f(u,v)du_{1}dv_{1}\cdots du_{M}dv_{M}$, and we define