# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021100
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## Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients

 Department of Mathematics, Drexel University, 3141 Chestnut St., Philadelphia, PA 19104

Received  March 2021 Revised  July 2021 Early access August 2021

We consider a linear Fermi-Pasta-Ulam-Tsingou lattice with random spatially varying material coefficients. Using the methods of stochastic homogenization we show that solutions with long wave initial data converge in an appropriate sense to solutions of a wave equation. The convergence is strong and both almost sure and in expectation, but the rate is quite slow. The technique combines energy estimates with powerful classical results about random walks, specifically the law of the iterated logarithm.

Citation: Joshua A. McGinnis, J. Douglas Wright. Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021100
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##### References:
Figure 1 is a log-log plot of the relative error $\rho$ divided by $\sqrt{\log\log(1/\epsilon)}.$
Figure 2 is 10 box plots of 40 different realization of masses at 10 various epsilons. It is also log-log
Figure 3 is a log-log plot of the relative error masses chosen periodically
In Figure 4 masses are chosen so that $\chi(j)$ will grow like $\sqrt{j}$
In Figure 5 $\epsilon$ is fixed and small while $\sigma$ is varied and the absolute error is measured. When $\sigma$ is smallest, the data is concentrated near the error for the constant coefficient case
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