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We present both numerical and analytical study of graphene roughness
with a crystal structure including $500 \times 500$ atoms. The
roughness can effectively result in a random gauge field and has
important consequences for its electronic structure. Our results
show that its height fluctuations in small scales have scaling
behavior with a temperature dependent roughness exponent in the
interval of $ 0.6 < \chi < 0.7 $. The correlation function of
height fluctuations depends upon temperature with characteristic
length scale of $ \approx 90 {\AA}$ (at room temperature). We show
that the correlation function of the induced gauge field has a
short-range nature with correlation length of about $\simeq 2-3
{\AA}$. We also treat the problem analytically by using the
Martin-Siggia-Rose method. The renormalization group flows did not
yield any delocalized-localized transition arising from the graphene
roughness. Our results are in good agreement with recent
experimental observations.
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