A Variational Formula for the Free Energy of the Partially Directed Polymer Collapse

Long linear polymers in dilute solutions are known to undergo a collapse transition from a random coil (expand itself) to a compact ball (fold itself up) when the temperature is lowered, or the solvent quality deteriorates. A natural model for this phenomenon is a 1+1 dimensional self-interacting and partially directed self-avoiding walk. In this paper, we develop a new method to study the partition function of this model, from which we derive a variational formula for the free energy. This variational formula allows us to prove the existence of the collapse transition and to identify the critical temperature in a simple way. We also provide a probabilistic proof of the fact that the collapse transition is of second order with critical exponent 3/2.     

Directed Percolation with Colors and Flavors

A model of directed percolation processes with colors and flavors that is equivalent to a population model with many species near their extinction thresholds is presented. We use renormalized field theory and demonstrate that all renormalizations needed for the calculation of the universal scaling behavior near the multicritical point can be gained from the one-species Gribov process (Reggeon field theory). In addition this universal model shows an instability that generically leads to a total asymmetry between each pair of species of a cooperative society, and finally to unidirectionality of the interspecies couplings. It is shown that in general the universal multicritical properties of unidirectionally coupled directed percolation processes with linear coupling can also be described by the model. Consequently the crossover exponent describing the scaling of the linear coupling parameters is given by =1 to all orders of the perturbation expansion. As an example of unidirectionally coupled directed percolation, we discuss the population dynamics of the tournaments of three species with colors of equal flavor.     

Dimensional Reduction Formulas for Branched Polymer Correlation Functions

In [BI01] we have proven that the generating function for self-avoiding branched polymers in D+2 continuum dimensions is proportional to the pressure of the hard-core continuum gas at negative activity in D dimensions. This result explains why the critical behavior of branched polymers should be the same as that of the i ³ (or Yang–Lee edge) field theory in two fewer dimensions (as proposed by Parisi and Sourlas in 1981). In this article we review and generalize the results of [BI01]. We show that the generating functions for several branched polymers are proportional to correlation functions of the hard-core gas. We derive Ward identities for certain branched polymer correlations. We give reduction formulae for multi-species branched polymers and the corresponding repulsive gases. Finally, we derive the massive scaling limit for the 2-point function of the one-dimensional hard-core gas, and thereby obtain the scaling form of the 2-point function for branched polymers in three dimensions.     

Asymptotic Behavior for a Version of Directed Percolation on the Triangular Lattice

We consider a version of directed bond percolation on the triangular lattice such that vertical edges are directed upward with probability $y$ , diagonal edges are directed from lower-left to upper-right or lower-right to upper-left with probability $d$ , and horizontal edges are directed rightward with probabilities $x$ and one in alternate rows. Let $\tau (M,N)$ be the probability that there is at least one connected-directed path of occupied edges from $(0,0)$ to $(M,N)$ . For each $x \in [0,1]$ , $y \in [0,1)$ , $d \in [0,1)$ but $(1-y)(1-d) \ne 1$ and aspect ratio $\alpha =M/N$ fixed for the triangular lattice with diagonal edges from lower-left to upper-right, we show that there is an $\alpha _c = (d-y-dy)/[2(d+y-dy)] + [1-(1-d)^2(1-y)^2x]/[2(d+y-dy)^2]$ such that as $N \rightarrow \infty $ , $\tau (M,N)$ is $1$ , $0$ and $1/2$ for $\alpha > \alpha _c$ , $\alpha < \alpha _c$ and $\alpha =\alpha _c$ , respectively. A corresponding result is obtained for the triangular lattice with diagonal edges from lower-right to upper-left. We also investigate the rate of convergence of $\tau (M,N)$ and the asymptotic behavior of $\tau (M_N^-,N)$ and $\tau (M_N^+ ,N)$ where $M_N^-/N\uparrow \alpha _c$ and $M_N^+/N\downarrow \alpha _c$ as $N\uparrow \infty $ .     

Mean-Field Theory for Percolation Models of the Ising Type

The q=2 random cluster model is studied in the context of two mean-field models: the Bethe lattice and the complete graph. For these systems, the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high-density side, which vindicates the results of earlier studies. In particular, the exponent ~ which characterizes the divergence of the average size of finite clusters is 1/2, and ~, the exponent associated with the length scale of finite clusters, is 1/4. The full collection of exponents indicates an upper critical dimension of 6. The standard mean field exponents of the Ising system are also present in this model (=1/2, =1), which implies, in particular, the presence of two diverging length-scales. Furthermore, the finite cluster exponents are stable to the addition of disorder, which, near the upper critical dimension, may have interesting implications concerning the generality of the disordered system/correlation length bounds.     

Maximal Clusters in Non-Critical Percolation and Related Models

We investigate the maximal non-critical cluster in a big box in various percolation-type models. We investigate its typical size, and the fluctuations around this typical size. The limit law of these fluctuations is related to maxima of independent random variables with law described by a single cluster.     

Rigidity of the Interface in Percolation and Random-Cluster Models

We study conditioned random-cluster measures with edge-parameter p and cluster-weighting factor q satisfying q1. The conditioning corresponds to mixed boundary conditions for a spin model. Interfaces may be defined in the sense of Dobrushin, and these are proved to be rigid in the thermodynamic limit, in three dimensions and for sufficiently large values of p. This implies the existence of non-translation-invariant (conditioned) random-cluster measures in three dimensions. The results are valid in the special case q=1, thus indicating a property of three-dimensional percolation not previously noted.     

Percolation on Inhomogeneous Bethe Lattice and Forest Fire Models

The inhomogeneous Bethe lattice (IBL) is definedand studied. It is used to study the random neighbor forforest fire model, and we show that it is more realisticthan the Bethe lattice, and gives large probability for the subcritical case.     

Bootstrap Percolation and Kinetically Constrained Models on Hyperbolic Lattices

We study bootstrap percolation (BP) on hyperbolic lattices obtained by regular tilings of the hyperbolic plane. Our work is motivated by the connection between the BP transition and the dynamical transition of kinetically constrained models, which are in turn relevant for the study of glass and jamming transitions. We show that for generic tilings there exists a BP transition at a nontrivial critical density, 0<ρ _c <1. Thus, despite the presence of loops on all length scales in hyperbolic lattices, the behavior is very different from that on Euclidean lattices where the critical density is either zero or one. Furthermore, we show that the transition has a mixed character since it is discontinuous but characterized by a diverging correlation length, similarly to what happens on Bethe lattices and random graphs of constant connectivity.     

PAC-Bayes Bounds on Variational Tempered Posteriors for Markov Models

Datasets displaying temporal dependencies abound in science and engineering applications, with Markov models representing a simplified and popular view of the temporal dependence structure. In this paper, we consider Bayesian settings that place prior distributions over the parameters of the transition kernel of a Markov model, and seek to characterize the resulting, typically intractable, posterior distributions. We present a Probably Approximately Correct (PAC)-Bayesian analysis of variational Bayes (VB) approximations to tempered Bayesian posterior distributions, bounding the model risk of the VB approximations. Tempered posteriors are known to be robust to model misspecification, and their variational approximations do not suffer the usual problems of over confident approximations. Our results tie the risk bounds to the mixing and ergodic properties of the Markov data generating model. We illustrate the PAC-Bayes bounds through a number of example Markov models, and also consider the situation where the Markov model is misspecified.     

Symmetrized Models of Last Passage Percolation and Non-Intersecting Lattice Paths

It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups U(l), Sp(2l) and O(l). We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.     

Variational solutions for Hermite--Gaussian solitons in nonlocal nonlinear media

The physical features exhibited by Hermite--Gaussian (HG) beams propagating in nonlocal nonlinear media with Gaussian-shaped response are discussed with an approximate variational method. Using direct numerical simulations, we find that the beam properties in the normalized system are different with the change of the degree of nonlocality. It is shown that initial HG profiles break up into several individual beams with propagation when the degree of nonlocality $\alpha$ is small. HG beams can propagate stably when $\alpha$ is large enough.     

Airy Kernel with Two Sets of Parameters in Directed Percolation and Random Matrix Theory

We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices.     

Variational Corner Transfer Matrix Renormalization Group Method for Classical Statistical Models

In the context of tensor network states, we for the first time reformulate the corner transfer matrix renormalization group(CTMRG) method into a variational bilevel optimization algorithm. The solution of the optimization problem corresponds to the fixed-point environment pursued in the conventional CTMRG method, from which the partition function of a classical statistical model, represented by an infinite tensor network, can be efficiently evaluated. The validity of this variational idea is dem...     

Energy Formulas for Three Quantum Brownian Motion Models Obtained with the IEO Method

For three quantum Brownian motion models: a material particle immersed in environment; two entangled particles coupled to an environment with position coupling; and two entangled particles coupled to an environment involving both position and momentum coupling, we employ the Invariant Eigenoperator Method (IEO) to successfully derive their energy formulas.     

Studying Asymmetric Structure in Directed Networks by Overlapping and Non-Overlapping Models

We consider the problem of modeling and estimating communities in directed networks. Models to this problem in the previous literature always assume that the sending clusters and the receiving clusters have non-overlapping property or overlapping property simultaneously. However, previous models cannot model the directed network in which nodes in sending clusters have overlapping property, while nodes in receiving clusters have non-overlapping property, especially for the case when the number of sending clusters is no larger than that of the receiving clusters. This kind of directed network exists in the real world for its randomness, and by the fact that we have little prior knowledge of the community structure for some real-world directed networks. To study the asymmetric structure for such directed networks, we propose a flexible and identifiable Overlapping and Non-overlapping model (ONM). We also provide one model as an extension of ONM to model the directed network, with a variation in node degree. Two spectral clustering algorithms are designed to fit the models. We establish a theoretical guarantee on the estimation consistency for the algorithms under the proposed models. A small scale computer-generated directed networks are designed and conducted to support our theoretical results. Four real-world directed networks are used to illustrate the algorithms, and the results reveal the existence of highly mixed nodes and the asymmetric structure for these networks.     

Collapsed and Adsorbed States of a Directed Polymer Chain in Two Dimensions

A phase diagram for a surface-interacting long flexible partially-directed polymer chain in a two-dimensional poor solvent, where the possibility of collapse in the bulk exists, is determined using exact enumeration methods. We used a model of self-attracting self-avoiding walks and evaluated 30 steps in series. An intermediate phase between the desorbed collapsed and adsorbed expanded phases, having the conformation of a surface-attached globule, is found. The four phases, viz ., (i) desorbed expanded (DE), (ii) desorbed collapsed (DC), (iii) adsorbed expanded (AE), (iv) surface-attached globule (SAG), are found to meet at a multicritical point. These features are in agreement with those of an isotropic (or non-directed) polymer chain.     

Variational Data Assimilation Methods in Geophysical Hydrodynamics Models and Their Application

Radiophysics and Quantum Electronics - We consider direct and inverse problems of geophysical hydrodynamics, associated with prediction, posterior analysis, and variational assimilation of...     

Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A ^–1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that . We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.     

Variational Beta Process Hidden Markov Models with Shared Hidden States for Trajectory Recognition

Hidden Markov model (HMM) is a vital model for trajectory recognition. As the number of hidden states in HMM is important and hard to be determined, many nonparametric methods like hierarchical Dirichlet process HMMs and Beta process HMMs (BP-HMMs) have been proposed to determine it automatically. Among these methods, the sampled BP-HMM models the shared information among different classes, which has been proved to be effective in several trajectory recognition scenes. However, the existing BP-HMM maintains a state transition probability matrix for each trajectory, which is inconvenient for classification. Furthermore, the approximate inference of the BP-HMM is based on sampling methods, which usually takes a long time to converge. To develop an efficient nonparametric sequential model that can capture cross-class shared information for trajectory recognition, we propose a novel variational BP-HMM model, in which the hidden states can be shared among different classes and each class chooses its own hidden states and maintains a unified transition probability matrix. In addition, we derive a variational inference method for the proposed model, which is more efficient than sampling-based methods. Experimental results on a synthetic dataset and two real-world datasets show that compared with the sampled BP-HMM and other related models, the variational BP-HMM has better performance in trajectory recognition.