*Published Paper*

**Inserted:** 17 jul 2021

**Last Updated:** 11 may 2022

**Journal:** Nonlinear Analysis

**Year:** 2022

**Doi:** 10.1016/j.na.2022.112951

**Links:**
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**Abstract:**

We consider graphs parameterized on a portion $X\subset\mathbb Z^d\times \{1,\ldots, M\}^k$ of a cylindrical subset of the lattice $\mathbb Z^d\times \mathbb Z^k$, and perform a discrete-to-continuum dimension-reduction process for energies defined on $X$ of quadratic type. Our only assumptions are that $X$ be connected as a graph and periodic in the first $d$-directions. We show that, upon scaling of the domain and of the energies by a small parameter $\varepsilon$, the scaled energies converge to a $d$-dimensional limit energy. The main technical points are a dimension-lowering coarse-graining process and a discrete version of the $p$-connectedness approach by Zhikov.

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