Abstract
This work is based on the idea that the observed cosmological constant is preferred for a stable vacuum, where a running gravitational coupling is considered. The most probable value of the cosmological constant is determined by the minimum of the vacuum energy density, and it is just proportional to the matter density. When the cosmological constant is connected with a modified dispersion relation (MDR) via an effective Newton constant, it may impose a constraint on the latter: the exponent of the leading correction should be n = 3.
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