We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (infinite dimensional) coefficient functions from ordinary or partial differential equations, a problem which is typically ill-posed. Regularization of these problems using L2 function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem—namely whether the subjective choice of regularization is compatible with prior knowledge. Using path-integral formalism, Bayesian inference can be explored through various perturbative techniques, such as the semiclassical approximation, which we use in this manuscript. Perturbative path-integral approaches, while offering alternatives to computational approaches like Markov-Chain-Monte-Carlo (MCMC), also provide natural starting points for MCMC methods that can be used to refine approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theories involving the Poisson equation.

Inhomogeneous percolation, for its closer relationship with real-life, can be more useful and reasonable than homogeneous percolation to illustrate the critical phenomena and dynamical behaviour of complex networks. However, due to its intricacy, the theoretical framework of inhomogeneous percolation is far from being complete and many challenging problems are still open. Due to this reasn, in this paper, we investigate inhomogeneous site percolation on Bethe Lattices with two occupation probabilities, and we also extend the result to percolation with m occupation probabilities. The critical behaviour of this inhomogeneous percolation is shown clearly by formulating the percolation probability with given occupation probability, the critical occupation probability, and the average cluster size. Moreover, using the above theory, we discuss in detail the diffusion behaviour of an infectious disease (SARS) and present specific disease-control strategies in consideration of groups with different infection probabilities.

Fractional quantum Hall (FQH) system at Landau level filling fraction $\nu=5/2$ has long been suggested to be non-Abelian, either Pfaffian (Pf) or antiPfaffian (APf) states by numerical studies, both with quantized Hall conductance $\sigma_{xy}=5e^2/2h$. Thermal Hall conductances of the Pf and APf states are quantized at $\kappa_{xy}=7/2$ and $\kappa_{xy}=3/2$ respectively in a proper unit. However, a recent experiment shows the thermal Hall conductance of $\nu=5/2$ FQH state is $\kappa_{xy}=5/2$. It has been speculated that the system contains random Pf and APf domains driven by disorders, and the neutral chiral Majorana modes on the domain walls may undergo a percolation transition to a $\kappa_{xy}=5/2$ phase. In this work, we do perturbative and non-perturbative analyses on the domain walls between Pf and APf. We show the domain wall theory possesses an emergent SO(4) symmetry at energy scales below a threshold $\Lambda_1$, which is lowered to an emergent U(1)$\times$U(1) symmetry at energy scales between $\Lambda_1$ and a higher value $\Lambda_2$, and is finally lowered to the composite fermion parity symmetry $\mathbb{Z}_2^F$ above $\Lambda_2$. Based on the emergent symmetries, we propose a specific phase diagram of the disordered $\nu=5/2$ FQH system, and show that a $\kappa_{xy}=5/2$ phase arises at disorder energy scales $\Lambda>\Lambda_1$. Furthermore, we show the gapped double-semion sector of $N_D$ closed domain walls contributes non-local topological degeneracy $2^{N_D-1}$, causing a low temperature peak in the heat capacity. We also implement a non-perturbative method to bootstrap generic topological 1+1D domain walls (2-surface defects) applicable to any 2+1D non-Abelian topological order.