In theories that propose the existence of extra dimensions, such as string theory and certain versions of quantum gravity, there can indeed be the notion of overlapping interactions between dimensions. These theories suggest that the universe may have more than the four dimensions (three spatial dimensions and one time dimension) that we are familiar with.
In some models, the extra dimensions are thought to be compactified, meaning that they are "curled up" or "hidden" at extremely small scales. While we don't directly perceive these extra dimensions, their presence can have indirect effects on our observable universe. The specific configuration of the extra dimensions and the way in which they interact with the familiar four dimensions can influence the behavior of particles and the physical laws we observe.
One possibility is that the physical laws in our four-dimensional spacetime arise as a result of the properties and dynamics of the higher-dimensional space. In this case, the extra dimensions could manifest themselves through the cross-over effect you mentioned, where different physical laws might be observed in different regions or at different energy scales. This concept is often referred to as "brane worlds" or the "brane concept."
According to some versions of string theory, our universe could be considered a three-brane embedded within a higher-dimensional spacetime. The interactions and dynamics of the particles and fields on our brane would be influenced by the presence of the extra dimensions. This could potentially lead to phenomena that appear as novel particles, modified forces, or even different physical laws within our observable universe.
However, it's important to note that the existence and nature of extra dimensions, as well as the specifics of their interactions with our four-dimensional world, are still active areas of research and not yet experimentally confirmed. String theory, for example, is a highly complex and mathematically challenging framework that is still being explored. As our understanding of these theories and their predictions advances, experiments and observations may provide insights into the possible existence and characteristics of extra dimensions and their effects on our universe.