  The approximate uniform sampling of graph realizations with a given degree
sequence is an everyday task in several social science, computer science,
engineering etc. projects. One approach is using Markov chains. The best
available current result about the well-studied switch Markov chain is that it
is rapidly mixing on P-stable degree sequences (see
DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any
degree sequence. However, there are cases where degree intervals are specified
rather than a single degree sequence. (A natural scenario where this problem
arises is in hypothesis testing on social networks that are only partially
observed.) Rechner, Strowick, and M\"uller-Hannemann introduced in 2018 the
notion of degree interval Markov chain which uses three (separately
well-studied) local operations (switch, hinge-flip and toggle), and employing
on degree sequence realizations where any two sequences under scrutiny have
very small coordinate-wise distance. Recently Amanatidis and Kleer published a
beautiful paper (arXiv:2110.09068), showing that the degree interval Markov
chain is rapidly mixing if the sequences are coming from a system of very thin
intervals which are centered not far from a regular degree sequence. In this
paper we extend substantially their result, showing that the degree interval
Markov chain is rapidly mixing if the intervals are centred at P-stable degree
sequences.
