Multi-soft theorems in Gauge Theory from MHV Diagrams

In this work we employ the MHV technique to show that scattering amplitudes with any number of consecutive soft particles behave universally in the multi-soft limit in which all particles go soft simultaneously. After identifying the diagrams which give the leading contribution we give the general rules for writing down compact expressions for the multi-soft factor of m gluons, k of which have negative helicity. We explicitly consider the cases where k equals 1 and 2. In N =4 SYM, the multi-soft factors of 2 scalars or 2 fermions forming a singlet of SU(4) R-symmetry, and m-2 positive helicity gluons are derived. The special case of the double-soft limit gives an amplitude whose leading divergence is 1/\delta^2 and not 1/\delta as in the case of 2 scalars or 2 fermions that do not form a singlet under SU(4). The construction based on the analytic supervertices allows us to obtain simple expressions for the triple-soft limit of 1 scalar and 2 positive helicity fermions, as well as for the quadrapole-soft limit of 4 positive helicity fermions, in a singlet configuration.Comment: 25 pages, 7 figures,typos correcte

Soft edge results for longest increasing paths on the planar lattice

For two-dimensional last-passage time models of weakly increasing paths, interesting scaling limits have been proved for points close the axis (the hard edge). For strictly increasing paths of Bernoulli($p$) marked sites, the relevant boundary is the line $y=px$. We call this the soft edge to contrast it with the hard edge. We prove laws of large numbers for the maximal cardinality of a strictly increasing path in the rectangle [\fl{p^{-1}n -xn^a}]\times[n] as the parameters $a$ and $x$ vary. The results change qualitatively as $a$ passes through the value 1/2.Comment: 14 pages, 2 figure

On maximal surfaces in the space of oriented geodesics of hyperbolic 3-space

We study area-stationary, or maximal, surfaces in the space ${\mathbb L}({\mathbb H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral K\"ahler structure. We prove that every holomorphic curve in ${\mathbb L}({\mathbb H}^3)$ is a maximal surface. We then classify Lagrangian maximal surfaces $\Sigma$ in ${\mathbb L}({\mathbb H}^3)$ and prove that the family of parallel surfaces in ${\mathbb H}^3$ orthogonal to the geodesics $\gamma\in\Sigma$ form a family of equidistant tubes around a geodesic.Comment: 16 pages, AMS-Late

Random structures for partially ordered sets

This thesis is presented in two parts. In the first part, we study a family of models of random partial orders, called classical sequential growth models, introduced by Rideout and Sorkin as possible models of discrete space-time. We analyse a particular model, called a random binary growth model, and show that the random partial order produced by this model almost surely has infinite dimension. We also give estimates on the size of the largest vertex incomparable to a particular element of the partial order. We show that there is some positive probability that the random partial order does not contain a particular subposet. This contrasts with other existing models of partial orders. We also study "continuum limits" of sequences of classical sequential growth models. We prove results on the structure of these limits when they exist, highlighting a deficiency of these models as models of space-time. In the second part of the thesis, we prove some correlation inequalities for mappings of rooted trees into complete trees. For T a rooted tree we can define the proportion of the total number of embeddings of T into a complete binary tree that map the root of T to the root of the complete binary tree. A theorem of Kubicki, Lehel and Morayne states that, for two binary trees with one a subposet of the other, this proportion is larger for the larger tree. They conjecture that the same is true for two arbitrary trees with one a subposet of the other. We disprove this conjecture by analysing the asymptotics of this proportion for large complete binary trees. We show that the theorem of Kubicki, Lehel and Morayne can be thought of as a correlation inequality which enables us to generalise their result in other directions

On the optimality of ternary arithmetic for compactness and hardware design

In this paper, the optimality of ternary arithmetic is investigated under strict mathematical formulation. The arithmetic systems are presented in generic form, as the means to encode numeric values, and the choice of radix is asserted as the main parameter to assess the efficiency of the representation, in terms of information compactness and estimated implementation cost in hardware. Using proper formulations for the optimization task, the universal constant 'e' (base of natural logarithms) is proven as the most efficient radix and ternary is asserted as the closest integer choice.Comment: 10 pages, 3 figure