SRQ seminar – December 3rd, 2018
Notes from a talk in the SRQ series.
SRQ Seminar 20181203 Tyler Helmut
Random walks, spin systems, isomorphisms theorems
(joint work with R. Bauerschmidt and A. Swan)
1Classic theorem: the
Brydges–Fröhlich–Spencer-Dynkin isomorphism
\(\Lambda\) finite set of vertices (i.e. the vertices of a finite box in
\(\mathbb{Z}^d\)) \(\{ \beta_{i, j} \}_{i, j}\) edge weights,
\(\beta_{i, j} = \beta_{j, i} \geqslant 0\) and \(\beta_{i, j} > 0\)
means that there is an edge between \(i\) and \(j\).
\(\mathbb{R}^2\)-valued Gaussian free field (GFF): measure on \(\varphi
\in (\mathbb{R}^2)^{\Lambda}\) \(\varphi = (\varphi_i)_{i \in \Lambda}\)
\(\varphi_i = (x_i, y_i)\).
\(\displaystyle \langle g (\varphi) \rangle_{\beta, \Lambda, m^2} =
\langle g (\varphi)
\rangle \propto \int_{(\mathbb{R}^2)^{\Lambda}} g
(\varphi) e^{- \sum_{i, j}
\beta_{i, j} | \varphi_i - \varphi_j |^2 -
\frac{1}{2} \sum_i m^2 | \varphi_i
|^2} \mathrm{d} \varphi,\)
\(m^2 > 0\) for convergence (translation symmetry needs to be broken).
Laplacian: \((\Delta_{\beta} f) (i) := \sum_{j \in \Lambda} \beta_{i, j}
(f (j) - f (i))\). Calculate
\(\displaystyle H_{\beta} (\varphi) = \frac{1}{2} \sum_{i, j} \beta_{i,
j} | \varphi_i -
\varphi_j |^2 = \sum_i \varphi_i (- \Delta_{\beta}
\varphi)_i .\)
Random walk: \(X_t\) is a continuous time random walk with jump rates
\((\beta_{i, j})_{i, j}\) meaning that
\(\displaystyle \mathbb{P} [X_{t + \delta t} = j|X_t = i] = \beta_{i, j}
\delta t + o (\delta
t)\)
The random walk has a local time \(L_t (i) = \int_0^t \mathbb{I}_{X_s =
i} \mathrm{d} s\).
Theorem 1. (BFS-Dynkin) If \(g
: [0, \infty)^{\Lambda} \rightarrow \mathbb{R}\) has rapid decay,
then
\(\displaystyle \int_{(\mathbb{R}^2)^{\Lambda}} \mathrm{d} \varphi
e^{- H_{\beta} (\varphi)} g
\left( \frac{1}{2} \varphi^2 \right) x_i
x_j = \int_{(\mathbb{R}^2)^{\Lambda}}
\mathrm{d} \varphi e^{-
H_{\beta} (\varphi)} \mathbb{E}_i \left[ g \left(
\frac{1}{2}
\varphi^2 + L_t \right) \mathbb{I}_{X_t = j} \right] .\)
Remark 2. The expectation is
unnormalized. A more familiar version comes from taking
\(\displaystyle g \left( \frac{1}{2} \varphi^2 \right) = \exp \left( -
\frac{m^2}{2} \sum_i
\varphi_i^2 \right) f \left( \frac{1}{2} \varphi^2
\right)\)
and normalizing the measure to get
\(\displaystyle \left\langle x_i x_j f \left( \frac{1}{2} \varphi^2
\right)
\right\rangle_{m^2} = \left\langle \int_0^{\infty} e^{- m^2 t}
\mathbb{E}_i
\left[ \mathbb{I}_{X_t = j} f \left( \frac{1}{2} \varphi^2
+ L_t \right)
\right] \mathrm{d} t \right\rangle_{m^2} .\)
This last formula is the Dynkin formulation of this
“isomorphism” because can be interpreted as an identity in
law.
Remark 3. Following ideas from
Disertori's talk we can replace the GFF with a supersymmetric GFF and
get
\(\displaystyle \left\langle x_i x_j f \left( \frac{1}{2} \Phi^2
\right)
\right\rangle^{\text{SUSY}} = \int_0^{\infty} e^{- m^2 t}
\mathbb{E}_i
[\mathbb{I}_{X_t = j} f (L_t)] \mathrm{d} t\)
where the SUSY \(\)GFF \(\Phi\) has a two additional fermionic
components.
A conceptual proof of the isomorphism follows (otherwise just compute
Laplace transforms explicitly). \(X_t\) is looked at as a marginal of
the process \((X_t, L_t)_t\) with generator
\(\displaystyle (\mathcal{L}g) (i, \ell) = (\Delta_{\beta} g) (i, \ell)
+
\dfrac{\partial}{\partial_{\ell_i}} g (i, \ell) .\)
Using Lemma 4 below, we can choose \(g_t (i, \ell)
=\mathbb{E}_{i, \ell} (\mathbb{I}_{X_t = j} g (L_t))\) and we can
compute that
\(\displaystyle \partial_t g_t =\mathcal{L}g_t .\)
By integrating we get
\(\displaystyle \mathbb{I}_{i = j} = \int_0^{\infty} \mathcal{L}g_t
\left( i, \frac{1}{2}
\varphi^2 \right) \mathrm{d} t\)
and we plug into the Lemma to get the theorem, indeed
\(\displaystyle \int \mathrm{d} \varphi e^{- H_{\beta} (\varphi)} x_i
x_j g \left( L_t +
\frac{1}{2} \varphi^2 \right) = - \sum_j
\int_0^{\infty} \int \mathrm{d}
\varphi e^{- H_{\beta} (\varphi)} x_i
x_j (\mathcal{L}g_t) \left( j,
\frac{1}{2} \varphi^2 \right) \mathrm{d}
t\)
\(\displaystyle = \int_0^{\infty} \int \mathrm{d} \varphi e^{- H_{\beta}
(\varphi)} g_t \left(
i, \frac{1}{2} \varphi^2 \right) \mathrm{d} t =
\int_0^{\infty} \int
\mathrm{d} \varphi e^{- H_{\beta} (\varphi)}
\mathbb{E}_{i, \frac{1}{2}
\varphi^2} (\mathbb{I}_{X_t = j} g (L_t))
\mathrm{d} t\)
Lemma 4. \(g :
\Lambda \times [0, \infty)^{\Lambda} \rightarrow \mathbb{R}\) with rapid
decay.
\(\displaystyle - \sum_j \int \mathrm{d} \varphi e^{- H_{\beta}
(\varphi)} x_i x_j
(\mathcal{L}g) \left( j, \frac{1}{2} \varphi^2
\right) = \int \mathrm{d}
\varphi e^{- H_{\beta} (\varphi)} g \left(
i, \frac{1}{2} \varphi^2 \right)\)
Proof. Integration by parts. \(T_i = \partial /
\partial_{x_i}\). Let \([f] := \int_{(\mathbb{R}^2)^{\Lambda}}
\mathrm{d} \varphi e^{- H_{\beta}
(\varphi)} f (\varphi)\). Then if
\(f\) has decay \([T_i f] = 0\) and \([(T_i f) g] = [f T_i^{\ast} g]\)
with
\(\displaystyle T_i^{\ast} g (\varphi) = - (T_i g) (\varphi) + (T_i
H_{\beta}) (\varphi) g
(\varphi) .\)
By a summation by parts one check direcly that
\(\displaystyle \sum_j (T_j^{\ast} g) \left( j, \frac{1}{2} \varphi^2
\right) = \sum_j x_j
(\mathcal{L}g) \left( j, \frac{1}{2} \varphi^2
\right) .\)
Indeed
\(\displaystyle \frac{\partial}{\partial x_i} g \left( j, \frac{1}{2}
\varphi^2 \right) =
\frac{\partial}{\partial \ell_i} g \left( j,
\frac{1}{2} \varphi^2 \right)
\frac{\partial}{\partial x_i} \left(
\frac{1}{2} \varphi^2 \right) = x_i
\frac{\partial}{\partial \ell_i} g
\left( j, \frac{1}{2} \varphi^2 \right) .\)
And this allows to conclude.\(\Box\)
2Applications
\(\Phi^4_d\).
Take \(g (\psi) = \exp \left( - \lambda \frac{1}{4} \psi^2 - \beta
\frac{1}{2} \psi
\right)\), then
\(\displaystyle \int_{(\mathbb{R}^2)^{\Lambda}} \mathrm{d} \varphi e^{-
H_{\beta} (\varphi) -
\lambda \sum_i | \varphi_i |^4 - \beta \sum_i |
\varphi_i |^2} x_i x_j
=\mathbb{E}_i [F (L_t) \mathbb{I}_{X_t = j}]\)
where \(F (L_t)\) is a \(\varphi^4\) partition function depending on the
local time of the random walk:
\(\displaystyle F (L_t) = \int_{(\mathbb{R}^2)^{\Lambda}} e^{- H_{\beta}
(\varphi) -
\frac{\lambda}{4} \sum_i (| \varphi_i |^2 + 2 L_t (i))^2 -
\frac{\beta}{2}
\sum_i (| \varphi_i |^2 + 2 L_t (i))} \mathrm{d} \varphi
.\)
3Vertex reinforced jump process.
VRJP is a Markov process on \(\Lambda\) with generator (for \((X_t,
L_t)\))
\(\displaystyle \mathcal{L} (i, \ell) = \sum_j \beta_{i, j} (1 + \ell_t
(i)) (g (i, \ell) - g
(i, \ell)) + \frac{\partial}{\partial \ell_i} g
(i, \ell) .\)
so
\(\displaystyle \mathbb{P} (X_{t + \delta t} = j|X_t = i) = \beta_{i, j}
(1 + L_t (i)) + o
(\delta t) .\)
Discovery due to Sabot-Tarres: VRJP is related to the
\(\mathbb{H}^{2|2}\) sigma model (a supersymmetric targe space with two
bosonic and two fermionic degrees of freedom).
Let us introduce the \(\mathbb{H}^2\) model, the (purely bosonic)
hyperbolic plane:
\(\displaystyle \mathbb{H}^2 = \{ \varphi = (x, y, z) | \varphi \cdot
\varphi := x^2 + y^2 -
z^2 = - 1 ; z \geqslant 1 \} .\)
So this is the (hyper)hyperboloid with positive \(z\).
Definition 5. (of
\(\mathbb{H}^2\) spin model). We proceed like for the GFF by replacing
the Euclidean inner product with the Minkowski inner product, i.e.:
\(\varphi_i \varphi_j = x_i x_j + y_i y_j - z_i z_j\). We replace the
Lebesgue measure with \(\mathrm{d} \varphi_i \rightarrow \frac{1}{z_i}
\mathrm{d} \varphi_i\) and \(m^2 \varphi^2 \rightarrow h (z_i - 1)\)
with \(h > 0\). Therefore we write
\(\displaystyle \langle f (\varphi) \rangle^{\mathbb{H}^2}_{\Lambda,
\beta, h} :=
\int_{(\mathbb{H}^2)^{\Lambda}} e^{- \frac{1}{2} \sum_{i,
j} \beta_{i, j} |
\varphi_i - \varphi_j |^2_{\mathbb{H}^2} - \sum_i h
(z_i - 1)} f (\varphi)
\prod_i \frac{\mathrm{d} \varphi_i}{z_i} .\)
Theorem 6. (BHS) \(g : \Lambda
\times [0, \infty)^{\Lambda} \rightarrow \mathbb{R}\) smooth and
bounded
\(\displaystyle \langle x_i x_j g (z - 1)
\rangle^{\mathbb{H}^2}_{\Lambda, \beta, h} =
\left\langle z_i
\int_0^{\infty} \mathrm{d} t\mathbb{E}_{i, z - 1}
[\mathbb{I}_{X_t =
j} g (L_t) e^{- h t}] \right\rangle_{\Lambda, \beta,
h}^{\mathbb{H}^2}
.\)
Remark 7. There is a
\(\mathbb{H}^{2|2}\) variant which reads
\(\displaystyle \langle x_i x_j g (z - 1)
\rangle^{\mathbb{H}^{2|2}}_{\Lambda, \beta, h} =
\int_0^{\infty}
\mathrm{d} t\mathbb{E}_{i, 0} [\mathbb{I}_{X_t = j} g (L_t)
e^{- h t}]
.\)
Therefore the correlation structure of the \(\mathbb{H}^{2|2}\) model is
encoded in the VRJP.
An application:
Theorem 8. (BHS) For any
\((\beta_{i, j})_{i, j}\) that is translation invariant and finite range
on \(\mathbb{Z}^2\). Then the VRJP is recurrent, meaning that
\(\displaystyle \mathbb{E}_{0, 0} L_{\infty} (0) = \infty .\)
The proof goes via the Mermin–Wagner theorem for the
\(\mathbb{H}^{2|2}\) model by taking \(g = 1\) in the above theorem and
then taking the infinite volume limit and the \(h \rightarrow 0\) limit.