Compactness by Coarse-Graining in Long-Range Lattice Systems
Abstract We consider energies on a periodic set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">ℒ</m:mi> </m:math> {\mathcal{L}} of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∑</m:mo> <m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>,</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi mathvariant="script">ℒ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:msubsup> <m:mi>a</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mi>ε</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>-</m:mo> <m:msub> <m:mi>u</m:mi> <m:mi>j</m:mi> </m:msub> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {\sum_{i,j\in\mathcal{L}}a^{\varepsilon}_{ij}\lvert u_{i}-u_{j}\rvert} , defined on spin functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {u_{i}\in\{0,1\}} , and we suppose that the typical range of the interactions is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>R</m:mi> <m:mi>ε</m:mi> </m:msub> </m:math> {R_{\varepsilon}} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>R</m:mi> <m:mi>ε</m:mi> </m:msub> <m:mo>→</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:mrow> </m:math> {R_{\varepsilon}\to+\infty} , i.e., if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>-</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:msub> <m:mi>R</m:mi> <m:mi>ε</m:mi> </m:msub> </m:mrow> </m:math> {\lvert i-j\rvert\leq R_{\varepsilon}} , then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msubsup> <m:mi>a</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mi>ε</m:mi> </m:msubsup> <m:mo>≥</m:mo> <m:mi>c</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {a^{\varepsilon}_{ij}\geq c&gt;0} . In a discrete-to-continuum analysis, we prove that the overall behavior as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ε</m:mi> <m:mo>→</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {\varepsilon\to 0} of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ε</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">ℒ</m:mi> </m:mrow> </m:math> {\varepsilon\mathcal{L}} with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>R</m:mi> <m:mi>ε</m:mi> </m:msub> </m:math> {R_{\varepsilon}} and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">ℒ</m:mi> <m:mo>=</m:mo> <m:msup> <m:mi>ℤ</m:mi> <m:mi>d</m:mi> </m:msup> </m:mrow> </m:math> {\mathcal{L}=\mathbb{Z}^{d}} .