Lattice Trees Research Articles

Backbone scaling for critical lattice trees in high dimensions

We prove that for critical (spread-out) lattice trees in dimensions , the unique paths from the origin to vertices of large tree distance converge to Brownian motion. This provides an important ingredient for proving weak convergence of the corresponding historical processes.

Real options approach to explore the effect of organizational change on IoT development project

The emerging information technology of internet of things (IoT) has caught much attention from both theoretical research and production practice. The application of IoT in manufacturing industry can dramatically improve business process efficiency and promote organizational performance. However, firms tend to hesitate in the investment decisions, which poses great challenges to the IoT application progress. Besides the investment nature of long cycle and high risk, the “organizational factors” are also non-ignorable. A general organizational change model is first refined according to the organizational change of Jianghuai Auto Company when applying IoT. Based on the binomial lattice tree framework, we propose a two-stage real options model to solve the optimal investment problems in both stages of internal pilot and product life-cycle management (PLM) expansion. Investment costs and future benefits are considered to investigate the promoting effect of organizational factors in project implementation. This paper demonstrates the internal causes and significances of organizational change for IoT development project, which is also theoretically meaningful for explanation of real options approach in organizational area.

Complex-time singularity and locality estimates for quantum lattice systems

We present and prove a well-known locality bound for the complex-time dynamics of a general class of one-dimensional quantum spin systems. Then we discuss how one might hope to extend this same procedure to higher dimensions using ideas related to the Eden growth process and lattice trees. Finally, we demonstrate with a specific family of lattice trees in the plane why this approach breaks down in dimensions greater than one and prove that there exist interactions for which the complex-time dynamics blows-up in finite imaginary time.

Ring Polymers in the Melt State: The Physics of Crumpling

The conformational statistics of ring polymers in melts or dense solutions is strongly affected by their quenched microscopic topological state. The effect is particularly strong for nonconcatenated unknotted rings, which are known to crumple and segregate and which have been implicated as models for the generic behavior of interphase chromosomes. Here we use a computationally efficient multiscale approach to show that melts of rings of total contour length Lr can be quantitatively mapped onto melts of interacting lattice trees with gyration radii ⟨R(g)(2)(Lr)⟩ ∝ L(r)(2ν) and ν = 0.32 ± 0.01.

Spanning trees of lattices embedded on the Klein bottle.

The problem of enumerating spanning trees in lattices with Klein bottle boundary condition is considered here. The exact closed-form expressions of the numbers of spanning trees for 4.8.8 lattice, hexagonal lattice, and 33 ·42 lattice on the Klein bottle are presented.

Financial Aspects of Interest Rate Derivative

An interest rate derivative is a derivative where the underlying asset is the right to pay or receive a notional amount of money at a given interest rate. These structures are popular for investors with customized cash flow needs or specific views on the interest rate movements (such as volatility movements or simple directional movements) and are therefore usually traded Over The Counter (OTC). The interest rate derivatives market is the largest derivatives market in the world. The Bank for International Settlements estimates that the notional amounts outstanding in June 2009 were US$437 trillion for OTC interest rate contracts, and US$342 trillion for OTC interest rate swaps. According to the International Swaps and Derivatives Association, 80% of the world's top 500 companies as of April 2003 used interest rate derivatives to control their cash flows. This compares with 75% for foreign exchange options, 25% for commodity options and 10% for stock options. Modeling of interest rate derivatives is usually done on a time-dependent multi-dimensional Lattice (tree) built for the underlying risk drivers, usually domestic or foreign short rates and foreign exchange market rates, and incorporating delivery- and day-count conventions. Specialized models are also often used. The trading in Interest Rate Futures was launched on the National Stock Exchange of India (NSE) on 31st August 2009 and will be launched shortly on MCX Stock Exchange (MCX-SX).

The survival probability and r-point functions in high dimensions

In this paper we investigate the survival probability, n, in high-dimensional statistical physical models, where n denotes the probability that the model survives up to time n. We prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if certain self-repellence and total-population tail-bound conditions are satised, then n n ! 2=(AV ), where A is the asymptotic expected number of particles alive at timen, andV is the vertex factor of the model. Our results apply to spread-out lattice trees above 8 dimensions, spread-out oriented percolation above 4 + 1 dimensions, and the spread-out contact process above 4 + 1 dimensions. In the case of oriented percolation, this reproves a result by the rst author, den Hollander, and Slade (which was proved using heavy lace expansion arguments), at the cost of losing explicit error estimates. We further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n. Our proofs are based on simple weak convergence arguments.

Expansion in High Dimension for the Growth Constants of Lattice Trees and Lattice Animals

We compute the first three terms of the 1/dexpansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice$\mathbb{Z}^d$, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion–exclusion.

Scaling exponents for a monkey on a tree: Fractal dimensions of randomly branched polymers

We study asymptotic properties of diffusion and other transport processes (including self-avoiding walks and electrical conduction) on large, randomly branched polymers using renormalized dynamical field theory. We focus on the swollen phase and the collapse transition, where loops in the polymers are irrelevant. Here the asymptotic statistics of the polymers is that of lattice trees, and diffusion on them is reminiscent of the climbing of a monkey on a tree. We calculate a set of universal scaling exponents including the diffusion exponent and the fractal dimension of the minimal path to two-loop order and, where available, compare them to numerical results.

Resistance Scaling and the Number of Spanning Trees in Self-Similar Lattices

We consider the problem of enumerating spanning trees in self-similar lattices, motivated by recent work of Chang, Chen and Yang, who determined explicit formulae in the case of Sierpinski graphs and some of their generalizations. The aim of this note is to show that their results hold in more generality and that there is a strong relation between this enumeration problem and resistance scaling on self-similar lattices.

The growth constants of lattice trees and lattice animals in high dimensions

We prove that the growth constants for nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice $\mathbb{Z}^d$ are asymptotic to $2de$ as the dimension goes to infinity, and that their critical one-point functions converge to $e$. Similar results are obtained in dimensions $d > 8$ in the limit of increasingly spread-out models; in this case the result for the growth constant is a special case of previous results of M. Penrose. The proof is elementary, once we apply previous results of T. Hara and G. Slade obtained using the lace expansion.

Morphology of technological levels in an innovation propagation model

We study the dynamical properties of the propagation of innovation on a two-dimensional lattice, random network, scale-free network, and Cayley tree. In order to investigate the diversity of technological level, we study the scaling property of width, W(N,t), which represents the root mean square of the technological level of agents. Here, N is the total number of agents. From the numerical simulations, we find that the steady-state value of W(N,t), W(sat)(N), scales as W(sat)(N) ∼ N(-1/2) when the system is in a flat ordered phase for d ≥ 2. In the flat ordered phase, most of the agents have the same technological level. On the other hand, when the system is in a smooth disordered phase, the value of W(sat)(N) does not depend on N. These behaviors are completely different from those observed on a one-dimensional (1D) lattice. By considering the effect of the underlying topology on the propagation dynamics for d ≥ 2, we also provide a mean-field analysis for W(sat)(N), which agrees very well with the observed behaviors of W(sat)(N). This directly shows that the morphological properties in order-disorder transition on a 1D lattice is completely different from that on higher dimensions. It also provides an evidence that the upper critical dimension for the roughening transition of the propagation of innovation is d(u)=2.

On the number of spanning trees on various lattices

We consider the number of spanning trees in lattices; for a lattice , one defines the bulk limit , where NST(G) is the number of spanning trees in a finite section G of . Explicit values for are known in various special cases. In this note we describe a simple yet effective method to deduce relations between the values of for different lattices by means of electrical network theory.

Enumeration of polyominoes inscribed in a rectangle

We develop a number of formulas and generating functions for the enumeration of general polyominoes inscribed in a rectangle of given size according to their area. These formulae are then used for the enumeration of lattice trees inscribed in a rectangle with minimum area plus one.

Executive Stock-Based Compensation: A Case Study in Valuing Employee Stock Options and Restricted Stock Issues in a Breach of Contract Case

In this case study, the former CEO of a large telecommunication company filed a lawsuit against her former employer that alleged that the company’s contract breach resulted in the loss ESO grants and diminished the value of the vested ESOs that she held at the time of the alleged contract breach. The former company executive also alleges that the breach prevented her from receiving restricted company stock shares that she believes she was entitled to under her contract with the employer. In this paper, I present an application of the Shapiro and O’Connor (2001) modified Black-Scholes (B-S) and the Hull-White (2002) binomial lattice tree employee stock option (ESO) valuation models to the valuation of breach of employment contract damages. In addition to ESO valuation, this paper also illustrates the valuation of restricted company stock issues and provides a discussion of the approaches used to account for the unique valuation related issues that arise in litigation. The case study illustrates the importance of incorporating case specific factors, such as the plaintiff's historical ESO exercise rates and firm specific ESO forfeiture probabilities, into the valuation models.

Enumeration of inscribed polyominos

We introduce a new family of polyominos that are inscribed in a rectangle of given size for which we establish a number of exact formulas and generating functions. In particular, we study polyominos inscribed in a rectangle with minimum area and minimum area plus one. These results are then used for the enumeration of lattice trees inscribed in a rectangle with minimum area plus one. Nous introduisons une nouvelle famille de polyominos inscrits dans un rectangle de format donné pour lesquels des formules exactes et des séries génératrices sont présentées. Nous étudions en particulier les polyominos inscrits d'aire minimale et ceux d'aire minimale plus un. Ces résultats sont ensuite utilisés pour l'énumération de polyominos arbres inscrits dans un rectangle d'aire minimum plus un.

Trees, Animals, and Percolation on Hyperbolic Lattices

We study lattice trees, lattice animals, and percolation on non-Euclidean lattices that correspond to regular tessellations of two- and three-dimensional hyperbolic space. We prove that critical exponents of these models take on their mean field values. Our methods are mainly combinatorial and geometric.

Lattice Tree Methods for Strongly Path Dependent Options

This review article summarizes the applications of the forward shooting grid method to pricing of various types of strongly path dependent options. The forward shooting grid approach is characterized by augmenting an auxiliary state vector at each node in the usual lattice tree, which serves to capture the path dependent feature of the option. Examples of path dependent derivatives considered include the lookback options, Asian options, callable convertible bonds, and call options with strike reset feature.

ON UNFOLDING LATTICE POLYGONS/TREES AND DIAMETER-4 TREES

We consider the problems of straightening polygonal trees and convexifying polygons by continuous motions such that rigid edges can rotate around vertex joints and no edge crossings are allowed. A tree can be straightened if all its edges can be aligned along a common straight line such that each edge points "away" from a designated leaf node. A polygon can be convexified if it can be reconfigured to a convex polygon. A lattice tree (resp. polygon) is a tree (resp. polygon) containing only edges from a square or cubic lattice. We first show that a 2D lattice chain or a 3D lattice tree can be straightened efficiently in O(n) moves and time, where n is the number of tree edges. We then show that a 2D lattice tree can be straightened efficiently in O(n2) moves and time. Furthermore, we prove that a 2D lattice polygon or a 3D lattice polygon with simple shadow can be convexified efficiently in O(n) moves and in O(n log n) time. Finally, we show that two special classes of diameter-4 trees in two dimensions can always be straightened.

Survival and coexistence for a multitype contact process

We study the ergodic theory of a multitype contact process with equal death rates and unequal birth rates on the d-dimensional integer lattice and regular trees. We prove that for birth rates in a certain interval there is coexistence on the tree, which by a result of Neuhauser is not possible on the lattice. We also prove a complete convergence result when the larger birth rate falls outside of this interval.