Witten Laplacian for an unbounded spin system^{∗}

Ichiro Shigekawa^{†} (Kyoto University)

We consider Witten Laplacians acting on differential forms on an unbounded lattice spin system. The configulation space is ℝ^{ℤ^{d}} and we are given a Gibbs measure on it. The Gibbs measure in this note is expressed formally as

ν = Z^{−1} exp\{−2𝒥 ∑_{{i,j∈ℤ^{d} \atop i∼j}} (x^{i} − x^{j})^{2} − 2 ∑_{i∈ℤ^{d}} U(x^{i})\} ∏_{i∈ℤ^{d}} dx^{i}.  (1)

Here 𝒥 is a positive constnat and U is an ℝ-valued function on ℝ and i ∼ j means that |i − j|^{2} = (i_{1} − j_{1})^{2} + ⋯ + (i_{1} − j_{1})^{2} = 1. Precise characterization is fomulated through Dobrushin-Lanford-Ruelle equation. We show that there is no harmonic p-forms (p ≥ 1) on this space. To do this, we take a finite set Λ ⊂ ℤ^{d} and a boundary condition η and define a finite volume Gibbs measure on ℝ^{Λ} and then take limit.

For this model, the logarithmic Sobolev inequality [4, 3], the spectral gap and the vanishing theorem for 1-fomrs [1] under suitable condition, were already established. So we intend to generalize the vanishing theorem for general p-fomrs.

Finite volume Gibbs measure and Witten Laplacian

For a finite set Λ ⊂ ℤ^{d} and a boundary condition η, we define a Hamiltonian by

Φ_{Λ,η}(x) = ∑_{{i,j∈Λ \atop i∼j}} 𝒥 (x^{i} − x^{j})^{2} + ∑_{i∈Z^{d}} U(x^{i}) + 2 ∑_{{i∈Λ,j∈Λ^{c} \atop i∼j}} 𝒥 (x^{i} − η^{j})^{2}  (2)

and define a measure on ℝ^{Λ} by

ν_{Λ,η} = Z^{−1}e^{−2Φ_{Λ,η}(x)}dx_{Λ}.  (3)

By a standard argument we can define the exterior differentiation d and its adjoint operator d^{∗} with respect to the measure ν_{Λ,η}.

We call the operator dd^{∗} + d^{∗}d as the Witten Laplacian acting on differential forms.

Vanishing theorem and Hodge-Kodaira decomposition

Suppose that the potential U is decomposed as U = V + W so that V is convex and W is bounded. We denote the supremum and the infimum of W by W_{sup} and W_{inf}, respectively. Then we have

Theorem 1. Suppose V′′ ≥ c > 0, and 2(c+8d𝒥)e^{−2(W_{sup}−W_{inf})} > 16d𝒥. Then the lowest eigenvalue of dd^{∗} + d^{∗}d on p-forms is not less than {2(c + 8d𝒥)e^{−2(W_{sup}−W_{inf})} − 16d𝒥 }p. Therefore there is no harmonic p-forms (p ≥ 1).

∗September 11–13, 2005, “Stochastic analysis and related fields” at Osaka University

†E-mail: ichiro@math.kyoto-u.ac.jp URL: http://www.math.kyoto-u.ac.jp/~ichiro

If U has a special form U(t) = at^{4} − bt^{2}, we can give different kind of criterion: Theorem 2. If \sqrt{3a} − b − 4d𝒥 > 0, then the lowest eigenvalue of dd^{∗} + d^{∗}d on p-forms is not less than 2(\sqrt{3a} − b − 4d𝒥)p. Therefore there is no harmonic p-forms (p ≥ 1).

Using these theorems, we can show the Hodge-Kodaira decomposition theorem. Theorem 3. We have the following ： for p = 0,

L^{2}(ν_{Λ,η}) = { constant functions } ⊕ Ran(d^{∗}),  (4)

and for p ≥ 1,

L^{2}(ν_{Λ,η}; ⋀^{p}(ℝ^{Λ})^{∗}) = Ran(d) ⊕ Ran(d^{∗})  (5)

References

[1] B. Helffer, “Semiclassical analysis, Witten Laplacians, and statistical mechanics,” Series on Partial Differential Equations and Applications, 1, World Scientific, River Edge, NJ, 2002.

[2] O. Matte and J. S. Møller On the spectrum of semi-classical Witten-Laplacians and Schrödinger operators in large dimension, J. Funct. Anal., 220 (2005), no. 2, 243–264.

[3] N. Yoshida, The log-Sobolev inequality for weakly coupled lattice fields, Probab. Theory Related Fields, 115 (1999), 1–40.

[4] B. Zegarlinski The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice Comm. Math. Phys., 175 (1996), no. 2, 401–432.