Condensed Matter Theory

Abstract:

We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.

Abstract:

Unravelling the structure of genotype–phenotype (GP) maps is an important problem in biology. Recently, arguments inspired by algorithmic information theory (AIT) and Kolmogorov complexity have been invoked to uncover simplicity bias in GP maps, an exponentially decaying upper bound in phenotype probability with the increasing phenotype descriptional complexity. This means that phenotypes with many genotypes assigned via the GP map must be simple, while complex phenotypes must have few genotypes assigned. Here, we use similar arguments to bound the probability P(x → y) that phenotype x, upon random genetic mutation, transitions to phenotype y. The bound is P(x→y)≲2^{−aK~(y|x)−b} , where K~(y|x) is the estimated conditional complexity of y given x, quantifying how much extra information is required to make y given access to x. This upper bound is related to the conditional form of algorithmic probability from AIT. We demonstrate the practical applicability of our derived bound by predicting phenotype transition probabilities (and other related quantities) in simulations of RNA and protein secondary structures. Our work contributes to a general mathematical understanding of GP maps and may facilitate the prediction of transition probabilities directly from examining phenotype themselves, without utilizing detailed knowledge of the GP map.

Abstract:

We present a computational study of the pairwise interactions between defects in the recently introduced non-reciprocal Cahn-Hilliard model. The evolution of a defect pair exhibits dependence upon their corresponding topological charges, initial separation, and the non-reciprocity coupling constant $α$. We find that the stability of isolated topologically neutral targets significantly affects the pairwise defect interactions. At large separations, defect interactions are negligible and a defect pair is stable. When positioned in relatively close proximity, a pair of oppositely charged spirals or targets merge to form a single target. At low $α$, like-charged spirals form rotating bound pairs, which are however torn apart by spontaneously formed targets at high $α$. Similar preference for charged or neutral solutions is also seen for a spiral target pair where the spiral dominates at low $α$, but concedes to the target at large $α$. Our work sheds light on the complex phenomenology of non-reciprocal active matter systems when their collective dynamics involves topological defects