We performed the renormalization group analysis of the quantum Einstein gravity in the deep infrared regime for different types of extensions of the model. It is shown that an attractive infrared point exists in the broken symmetric phase of the model. It is also shown that due to the Gaussian fixed point the IR critical exponent $ u$ of the correlation length is 1/2. However, there exists a certain extension of the model which gives finite correlation length in the broken symmetric phase. It typically appears in case of models possessing a first order phase transitions as is demonstrated on the example of the scalar field theory with a Coleman-Weinberg potential.
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