Rigidity of interfaces in the Falicov-Kimball model

We analyze the thermodynamic properties of interfaces in the three-dimensional Falicov Kimball model, which can be viewed as a primitive quantum lattice model of crystalline matter. In the strong coupling limit, the ionic subsystem of this model is governed by the Hamiltonian of an effective classical spin model whose leading part is the Ising Hamiltonian. We prove that the 100 interface in this model, at half-filling, is rigid, as in the three-dimensional Ising model. However, despite the above similarities with the Ising model, the thermodynamic properties of its 111 interface are very different. We prove that even though this interface is expected to be unstable for the Ising model, it is stable for the Falicov Kimball model at sufficiently low temperatures. This rigidity results from a phenomenon of "ground state selection" and is a consequence of the Fermi statistics of the electrons in the model.Comment: 79 pages, 9 figures included as ps-files, appendix added in revisio

The Sphere Packing Bound via Augustin's Method

A sphere packing bound (SPB) with a prefactor that is polynomial in the block length $n$ is established for codes on a length $n$ product channel $W_{[1,n]}$ assuming that the maximum order $1/2$ Renyi capacity among the component channels, i.e. $\max_{t\in[1,n]} C_{1/2,W_{t}}$, is $\mathit{O}(\ln n)$. The reliability function of the discrete stationary product channels with feedback is bounded from above by the sphere packing exponent. Both results are proved by first establishing a non-asymptotic SPB. The latter result continues to hold under a milder stationarity hypothesis.Comment: 30 pages. An error in the statement of Lemma 2 is corrected. The change is inconsequential for the rest of the pape

Coding against synchronisation and related errors

In this thesis, we study aspects of coding against synchronisation errors, such as deletions and replications, and related errors. Synchronisation errors are a source of fundamental open problems in information theory, because they introduce correlations between output symbols even when input symbols are independently distributed. We focus on random errors, and consider two complementary problems: We study the optimal rate of reliable information transmission through channels with synchronisation and related errors (the channel capacity). Unlike simpler error models, the capacity of such channels is unknown. We first consider the geometric sticky channel, which replicates input bits according to a geometric distribution. Previously, bounds on its capacity were known only via numerical methods, which do not aid our conceptual understanding of this quantity. We derive sharp analytical capacity upper bounds which approach, and sometimes surpass, numerical bounds. This opens the door to a mathematical treatment of its capacity. We consider also the geometric deletion channel, combining deletions and geometric replications. We derive analytical capacity upper bounds, and notably prove that the capacity is bounded away from the maximum when the deletion probability is small, meaning that this channel behaves differently than related well-studied channels in this regime. Finally, we adapt techniques developed to handle synchronisation errors to derive improved upper bounds and structural results on the capacity of the discrete-time Poisson channel, a model of optical communication. Motivated by portable DNA-based storage and trace reconstruction, we introduce and study the coded trace reconstruction problem, where the goal is to design efficiently encodable high-rate codes whose codewords can be efficiently reconstructed from few reads corrupted by deletions. Remarkably, we design such n-bit codes with rate 1-O(1/log n) that require exponentially fewer reads than average-case trace reconstruction algorithms.Open Acces

On a Generalised Typicality and Its Applications in Information Theory

Typicality lemmas have been successfully applied in many information theoretical problems. The conventional strong typicality is only defined for finite alphabets. Conditional typicality and Markov lemmas can be obtained for strong typicality. Weak typicality can be defined based on a measurable space without additional constraints, and can be easily defined based on a general stochastic process. However, to the best of our knowledge, no conditional typicality or strong Markov lemmas have been obtained for weak typicality in classic works. As a result, some important coding theorems can only be proved by strong typicality lemmas and using the discretisation-and-approximation-technique. In order to solve the aforementioned problems, we will show that the conditional typicality lemma can be obtained for a generic typicality. We will then define a multivariate typicality for general alphabets and general probability measures on product spaces, based on the relative entropy, which can be a measure of the relevance between multiple sources. We will provide a series of multivariate typicality lemmas, including conditional and joint typicality lemmas, packing and covering lemmas, as well as the strong Markov lemma for our proposed generalised typicality. These typicality lemmas can be used to solve source and channel coding problems in a unified way for finite, continuous, or more general alphabets. We will present some coding theorems with general settings using the generalised multivariate typicality lemmas without using the discretisation-and-approximation technique. Generally, the proofs of the coding theorems in general settings are simpler by using the generalised typicality, than using strong typicality with the discretisation-and-approximation technique

Hidden Markov Models

Hidden Markov Models (HMMs), although known for decades, have made a big career nowadays and are still in state of development. This book presents theoretical issues and a variety of HMMs applications in speech recognition and synthesis, medicine, neurosciences, computational biology, bioinformatics, seismology, environment protection and engineering. I hope that the reader will find this book useful and helpful for their own research