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Site Monotonicity and Uniform Positivity for Interacting Random Walks and the Spin O(N) Model with Arbitrary N

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Abstract

We provide a uniformly-positive point-wise lower bound for the two-point function of the classical spin O(N) model on the torus of \(\mathbb {Z}^d\), \(d \ge 3\), when \(N \in \mathbb {N}_{>0}\) and the inverse temperature \(\beta \) is large enough. This is a new result when \(N>2\) and extends the classical result of Fröhlich et al. (Commun Math Phys 50:79–95, 1976). Our bound follows from a new site-monotonicity property of the two-point function which is of independent interest and holds not only for the spin O(N) model with arbitrary \(N \in \mathbb {N}_{>0}\), but for a wide class of systems of interacting random walks and loops, including the loop O(N) model, random lattice permutations, the dimer model, the double-dimer model, and the loop representation of the classical spin O(N) model.

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Acknowledgements

Benjamin Lees acknowledges support from the Alexander von Humboldt foundation. Lorenzo Taggi acknowledges support from the DFG German Research Foundation BE 5267/1 and from the EPSRC Early Career Fellowship EP/N004566/1. The authors thank the two anonymous referees for carefully reading the paper and their useful suggestions.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, UK

    Benjamin Lees

  2. Weierstrass-Institut fur Angewandte Analysis und Stochastik, Berlin, Germany

    Lorenzo Taggi

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  1. Benjamin Lees
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  2. Lorenzo Taggi
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Correspondence to Lorenzo Taggi.

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Communicated by H. Duminil-Copin

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A Proof of Proposition 2.3

A Proof of Proposition 2.3

Proof

For any \(A \subset \mathcal {V}_x\), define,

$$\begin{aligned} Z_{N, \beta }^{spin}(A) = \Big ( \, \prod _{x \in \mathcal {V}} \int _{\mathbb {S}^{N-1}} d \varphi _x \, \Big ) \, \big ( \, \prod _{x \in A} \varphi ^1_x \, \big ) \, e^{- \beta H_{N}(\varphi )}. \end{aligned}$$
(A.1)

We will prove that, for any \(N \in \mathbb {N}_{>0}\), \(A \subset \mathcal {V}\), \(\beta \ge 0\), under the choice of the weight function U as in Proposition 2.3, we have that,

$$\begin{aligned} Z_{N, \beta }^{spin}(A) = Z_{N, \beta , U}(A). \end{aligned}$$
(A.2)

Thus, by the definition of point-to-point function, Definition 2.2, we will deduce Proposition 2.3. The starting point of the expansion is the following identity, proved in [7, Appendix A], which holds for any \(N \in \mathbb {N}_{>0}\) and \(n_1, n_2, \ldots n_N \in \mathbb {N}\),

$$\begin{aligned} \int _{\mathbb {S}^{N-1}} (\varphi ^{1})^{n_{1}} \ldots (\varphi ^{N})^{n_{N}} d \varphi = {\left\{ \begin{array}{ll} 0 &{}\quad { \textit{if } } n_{i} \in 2 \mathbb {N} + 1 { \textit{for some} } i \in \{1, \ldots N \},\\ \frac{\Gamma (\frac{N}{2}) \prod _{i=1}^{N} (n_{i} - 1)!! }{ 2^{\frac{n}{2}} \Gamma \big ((n+N)/2 \big )} &{} \quad { \textit{otherwise}}, \end{array}\right. }\nonumber \\ \end{aligned}$$
(A.3)

where \(d \varphi \) denotes the normalised uniform measure on \(\mathbb {S}^{N-1}\), \(n= n_1 + \cdots +n_N\), and \((n_i-1)!!\) is the double factorial, i.e. the number of ways to pair \(n_i\) objects (hence \((-1)!!=1\)). Below, we will omit all sub-scripts to lighten the notation. To begin, we re-write the exponential as follows,

$$\begin{aligned} \exp \Big \{ \beta \sum \limits _{\{x,y\} \in \mathcal {E}} \varphi _x \cdot \varphi _y \Big \} \, = \, \prod _{ \{x,y\} \in \mathcal {E}} \, \, \prod _{i=1}^{N} \, e^{ \beta \varphi _x^{i} \varphi _x^{i}}. \end{aligned}$$
(A.4)

For any \(A \subset \mathcal {V}\), define

$$\begin{aligned}&\mathcal {M}_{\mathcal {G}} (A) : = \{ m \in \mathcal {M}_{\mathcal {G}} \, \, : \, \, \forall x \in A, \sum _{e \in \mathcal {E} : x \in e} m_e \, \, \in 2 \mathbb {N}+1 \, \, \text{ and } \\&\quad \forall z \in \mathcal {V} {\setminus } A, \sum _{e \in \mathcal {E} : z \in e} m_e \, \, \in 2 \mathbb {N} \}. \end{aligned}$$

Now we expand as a Taylor series and use (A.3) to restrict the sum to the terms which are not necessarily zero, obtaining

$$\begin{aligned} Z^{spin}(A)= & {} \sum \limits _{m^{1} \in \mathcal {M}_\mathcal {G}(A)} \, \sum \limits _{m^{2} \in \mathcal {M}_\mathcal {G}(\emptyset )} \ldots \sum \limits _{m^{N} \in \mathcal {M}_\mathcal {G}(\emptyset )} \, \Big ( \prod _{e \in \mathcal {E}} \frac{ \beta ^{m_e^{1}+\dots +m_e^{N}}}{m_e^{1}! \ldots m_e^{N}!} \Big ) \nonumber \\&\Big ( \, \prod _{x \in \mathcal {V}} \int _{\mathbb {S}^{N-1}} d \varphi _x \, \Big ) \prod _{x \in \mathcal {V} {\setminus } A} \Big ( \, (\varphi _x^{1})^{q_x^{1}} \ldots (\varphi _x^{N})^{q_x^{N}} \Big )\nonumber \\&\prod _{x \in A } \Big ( \, (\varphi _x^{1})^{q_x^{1}+1} (\varphi _x^{2})^{q_x^{2}} \ldots (\varphi _x^{N})^{q_x^{N}} \Big ) \end{aligned}$$
(A.5)

where we defined for any \(x \in \mathcal {V}\), \( q^i_x(m) : = \sum _{e \in \mathcal {E} : x \in e} m^i_e. \) We now rewrite the expression by first summing over all \(m \in \mathcal {M}_{{{\mathcal {G}}}}(A)\) and \((m^1\), \(m^2\), \(\ldots \), \(m^N)\), such that, \(m^1\in {{\mathcal {M}}}_{{\mathcal {G}}}(A)\), \(m^i\in {{\mathcal {M}}}_{{{\mathcal {G}}}}(\emptyset )\), when \(i\in \{2,\dots ,N\}\), \(m = \sum _{i=1}^{N} m^i\), and \(q_x = \sum _{i=1}^{N} q_x^i\), obtaining,

$$\begin{aligned} Z^{spin}(A)= & {} \sum \limits _{m \in \mathcal {M}_{\mathcal {G}}(A)} \, \prod _{e \in \mathcal {E}} \Big ( \frac{\beta ^{m_e}}{m_e!} \Big ) \sum \limits _{ \begin{array}{c} m^1 \in \mathcal {M}_\mathcal {G}(A) \\ m^{i} \in \mathcal {M}_\mathcal {G}(\emptyset ), i \ge 2 : \\ \sum _{i=1}^{N} m^i = m \end{array}} \prod _{e \in \mathcal {E}} \Big (\frac{m_e!}{m_e^1! \ldots m_e^N!} \, \Big ) \nonumber \\&\prod _{x \in \mathcal {V} {\setminus } A } \Bigg ( \frac{\Gamma (\frac{N}{2})}{2^{\frac{q_x}{2}} \Gamma ( (q_x+N)/2) } \prod _{i=1}^{N} ( q^i_x - 1) !! \Bigg )\nonumber \\&\prod _{x \in A } \Bigg ( \frac{\Gamma (\frac{N}{2})}{2^{\frac{q_x+1}{2}} \Gamma ( (q_x+1 + N)/2) } q_x^1!! \, \, \prod _{i=2}^{N} ( q^i_x - 1) !! \Bigg ) . \end{aligned}$$
(A.6)

Above, the product right after the second sum can be interpreted as the number of colour assignments to the \(m_e\) links which are parallel to the edge e such that precisely \(m_e^i\) links have colour i, for each \(i=1, \ldots , N\). Moreover, note that, if \(q^i_x\) is an odd integer, then \( q^i_x !! \) is the number of ways \(q^i_x\) links which are incident to x can be “paired" in such a way that only one link is unpaired and the remaining \((q^i_x-1)\) links are paired, while, if \(q^i_x\) is an even integer, then \((q^i_x-1)!!\) is the number of ways such \(q^i_x\) links can be paired. Thus, in the next step, we replace the sum over \((m^i)_{i=1, \ldots , N}\) by the sum over N possible colours for each link and the double factorial terms by the sum over all possible pairings of the links which are incident to each vertex. Recalling the definition of \(n^i_x(m,c, \pi )\), which was given in (2.2), and putting \(n_x(m,c, \pi ) = \sum _{i=1}^{N} n^i_x(m, c, \pi )\), we obtain that,

$$\begin{aligned} Z^{spin}(A)= & {} \sum \limits _{ m \in \mathcal {M}_{\mathcal {G}}(A)} \, \prod _{e \in \mathcal {E}} \bigg ( \frac{\beta ^{m_e}}{m_e!} \bigg ) \nonumber \\&\sum _{c\in {{\mathcal {C}}}_{{{\mathcal {G}}}}(m)}\sum \limits _{\pi \in \mathcal {P}_{\mathcal {G}}(m,c) } \prod _{x \in \mathcal {V} {\setminus } A } \Bigg ( \frac{ \Gamma (N/2)}{2^{n_x(m,c,\pi )}\Gamma ( n_x(m, c, \pi ) + N/2) } \Bigg ) \nonumber \\&\prod _{x \in A } \Bigg ( \frac{ \Gamma (N/2)}{ 2^{n_x(m,c, \pi )}\Gamma (n_x(m,c, \pi ) + N/2) } \Bigg ) . \end{aligned}$$
(A.7)

In the previous expression we also used the fact that, if for a realisation \((m, c, \pi ) \in \mathcal {W}_{\mathcal {G}}(A)\), \(q_x\) links touch the vertex x, where \(x \in A\), this means that \(q_x + 1 = 2 n_x(m,c, \pi )\). Similarly, if for a realisation \((m, c, \pi ) \in \mathcal {W}_{\mathcal {G}}(A)\), \(q_x\) links touch the vertex x, where \(x \not \in A\), then \(q_x = 2 n_x(m, c, \pi )\). Plugging in the definition of the weight function \(U_x\) (recall Definition 2.1 and the assumption of Proposition 2.3), the proof of Proposition 2.3 is concluded. \(\square \)

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Lees, B., Taggi, L. Site Monotonicity and Uniform Positivity for Interacting Random Walks and the Spin O(N) Model with Arbitrary N. Commun. Math. Phys. 376, 487–520 (2020). https://doi.org/10.1007/s00220-019-03647-6

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