It takes a while for a newly planted shrub to adjust so this browning may simply be the result of not having a well-established root system going into winter. Over or under watering could also cause root rot which in turn results in brown foliage or die back. 

It's difficult to tell for sure from these photos, but the damage does not look like what's typical of a disease, though you could do a closer inspection, or send us another picture that is close-up.

https://ag.umass.edu/landscape/fact-sheets/arborvitae-needle-blight

Since the brown foliage is not helping the shrub, I recommend that you cut it out, then monitor the shrub for further browning and water levels. Make sure that your shrub is sighted well and planted properly. If there is a drainage issue or the plants are not receiving proper care (supplemental water if the soil drains quickly), the same thing might happen to the next shrub you try to plant in that location. If you don't think watering might be the issue, consider protecting with a wind block next winter (screen with some burlap) to help prevent desiccation.

https://extension.psu.edu/the-mighty-arborvitae