The Integrable Bootstrap Program at Large N and its Applications in Gauge Theory

We present results for the large-$N$ limit of the (1+1)-dimensional principal chiral sigma model. This is an asymptotically-free $N\times N$ matrix-valued field with massive excitations. All the form factors and the exact correlation functions of the Noether-current operator and the energy-momentum tensor are found, from Smirnov's form-factor axioms. We consider (2+1)-dimensional $SU(\infty)$ Yang-Mills theory as an array of principal chiral models with a current-current interaction. We discuss how to use our new form factors to calculate physical quantities in this gauge theory.Comment: Presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany. Some references added in the updated versio

Longitudinal Rescaling of Quantum Electrodynamics

We investigate quantum longitudinal rescaling of electrodynamics, transforming coordinates as $x^{0,3}\to\lambda x^{0,3}$ and $x^{1,2}\to x^{1,2}$, to one loop. We do this by an aspherical Wilsonian renormalization, which was applied earlier to pure Yang-Mills theory. We find the anomalous powers of $\lambda$ in the renormalized couplings. Our result is only valid for $\lambda \lesssim 1$, because perturbation theory breaks down for $\lambda \ll 1$.Comment: Version to appear in Phys. Rev.

Correlation Functions of the SU(infinity) Principal Chiral Model

We obtain exact matrix elements of physical operators of the (1+1)-dimensional nonlinear sigma model of an SU(N)-valued bare field, in the 't Hooft limit N goes to infinity. Specifically, all the form factors of the Noether current and the stress-energy-momentum tensor are found with an integrable bootstrap method. These form factors are used to find vacuum expectation values of products of these operators.Comment: Further typographical errors corrected to conform to the published version (which will appear in Physical Review D). Still revtex, 12 page

Accounting for Uncertainty When Estimating Counts Through an Average Rounded to the Nearest Integer

In practice, the use of rounding is ubiquitous. Although researchers have looked at the implications of rounding continuous random variables, rounding may be applied to functions of discrete random variables as well. For example, to infer on suicide difference between two time periods, authorities may provide a rounded average of deaths for each period. Suicide rates tend to be relatively low around the world and such rounding may seriously affect inference on the change of suicide rate. In this paper, we study the scenario when a rounded to nearest integer average is used to estimate a non-negative discrete random variable. Specifically, our interest is in drawing inference on a parameter from the pmf of Y, when we get U=n[Y/n]as a proxy for Y. The probability generating function of U, E(U), and Var(U) capture the effect of the coarsening of the support of Y. Also, moments and estimators of distribution parameters are explored for some special cases. Under certain conditions, there is little impact from rounding. However, we also find scenarios where rounding can significantly affect statistical inference as demonstrated in two applications. The simple methods we propose are able to partially counter rounding error effects

Smoothening block rewards: How much should miners pay for mining pools?

The rewards a blockchain miner earns vary with time. Most of the time is spent mining without receiving any rewards, and only occasionally the miner wins a block and earns a reward. Mining pools smoothen the stochastic flow of rewards, and in the ideal case, provide a steady flow of rewards over time. Smooth block rewards allow miners to choose an optimal mining power growth strategy that will result in a higher reward yield for a given investment. We quantify the economic advantage for a given miner of having smooth rewards, and use this to define a maximum percentage of rewards that a miner should be willing to pay for the mining pool services.Comment: 15 pages, 1 figur

An Agent-Based Model Framework for Utility-Based Cryptoeconomies

In this paper, we outline a framework for modeling utility-based blockchain-enabled economic systems using Agent Based Modeling (ABM). Our approach is to model the supply dynamics based on metrics of the cryptoeconomy. We then build autonomous agents that make decisions based on those metrics. Those decisions, in turn, impact the metrics in the next time-step, creating a closed loop that models the evolution of cryptoeconomies over time. We apply this framework as a case-study to Filecoin, a decentralized blockchain-based storage network. We perform several experiments that explore the effect of different strategies, capitalization, and external factors to agent rewards, that highlight the efficacy of our approach to modeling blockchain based cryptoeconomies.Comment: 14 pages, 5 figure

Duality and form factors in the thermally deformed two-dimensional tricritical Ising model

The thermal deformation of the critical point action of the 2D tricritical Ising model gives rise to an exact scattering theory with seven massive excitations based on the exceptional $E_7$ Lie algebra. The high and low temperature phases of this model are related by duality. This duality guarantees that the leading and sub-leading magnetisation operators, $\sigma(x)$ and $\sigma'(x)$, in either phase are accompanied by associated disorder operators, $\mu(x)$ and $\mu'(x)$. Working specifically in the high temperature phase, we write down the sets of bootstrap equations for these four operators. For $\sigma(x)$ and $\sigma'(x)$, the equations are identical in form and are parameterised by the values of the one-particle form factors of the two lightest $\mathbb{Z}_2$ odd particles. Similarly, the equations for $\mu(x)$ and $\mu'(x)$ have identical form and are parameterised by two elementary form factors. Using the clustering property, we show that these four sets of solutions are eventually not independent; instead, the parameters of the solutions for $\sigma(x)/\sigma'(x)$ are fixed in terms of those for $\mu(x)/\mu'(x)$. We use the truncated conformal space approach to confirm numerically the derived expressions of the matrix elements as well as the validity of the $\Delta$-sum rule as applied to the off-critical correlators. We employ the derived form factors of the order and disorder operators to compute the exact dynamical structure factors of the theory, a set of quantities with a rich spectroscopy which may be directly tested in future inelastic neutron or Raman scattering experiments.Comment: v2: typos corrected, some details clarified, references added, tables and figures updated. v3: some more details clarified, typos correcte