429 research outputs found
Higher Order Analogues of Tracy-Widom Distributions via the Lax Method
No full textAkemann G, Atkin M. Higher Order Analogues of Tracy-Widom Distributions via the Lax Method. J.Phys. A. 2013;46(1): 15202.We study the distribution of the largest eigenvalue in formal Hermitianone-matrix models at multicriticality, where the spectral density acquires anextra number of k-1 zeros at the edge. The distributions are directly expressedthrough the norms of orthogonal polynomials on a semi-infinite interval, as analternative to using Fredholm determinants. They satisfy non-linear recurrencerelations which we show form a Lax pair, making contact to the stringliterature in the early 1990's. The technique of pseudo-differential operatorsallows us to give compact expressions for the logarithm of the gap probabilityin terms of the Painleve XXXIV hierarchy. These are the higher order analoguesof the Tracy-Widom distribution which has k=1. Using known Backlundtransformations we show how to simplify earlier equivalent results that arederived from Fredholm determinant theory, valid for even k in terms of thePainleve II hierarchy
CDT coupled to dimer matter: An analytical approach via tree bijections
No full textAtkin M, Zohren S. CDT coupled to dimer matter: An analytical approach via tree bijections. AIP Conference Proceedings. 2012;1483(1):330-335.We review a recently obtained analytical solution of a restricted so-calledhard dimers model coupled to two-dimensional CDT. The combinatorial solution isobtained via bijections of causal triangulations with dimers and decoratedtrees. We show that the scaling limit of this model can also be obtained from amulti-critical point of the transfer matrix for dynamical triangulations oftriangles and squares when one disallows for spatial topology changes to occur
An analytical analysis of CDT coupled to dimer-like matter
No full textAtkin MR, Zohren S. An analytical analysis of CDT coupled to dimer-like matter. Physics Letters B. 2012;712(4-5):445-450.We consider a model of restricted dimers coupled to two-dimensional Causal Dynamical Triangulations (CDT), where the dimer configurations are restricted in the sense that they do not include dimers in regions of high curvature. It is shown how the model can be solved analytically using bijections with decorated trees. At a negative critical value for the dimer fugacity the model undergoes a phase transition at which the critical exponent associated to the geometry changes. This represents the first account of an analytical study of a matter model with two-dimensional interactions coupled to CDT. (C) 2012 Elsevier B.V. All rights reserved
Instantons and Extreme Value Statistics of Random Matrices
No full textAtkin M, Zohren S. Instantons and Extreme Value Statistics of Random Matrices. Journal of High Energy Physics. 2014;2014(4): 118.We discuss the distribution of the largest eigenvalue of a random N x NHermitian matrix. Utilising results from the quantum gravity and string theoryliterature it is seen that the orthogonal polynomials approach, firstintroduced by Majumdar and Nadal, can be extended to calculate both the leftand right tail large deviations of the maximum eigenvalue. This framework doesnot only provide computational advantages when considering the left and righttail large deviations for general potentials, as is done explicitly for thefirst multi-critical potential, but it also offers an interestinginterpretation of the results. In particular, it is seen that the left taillarge deviations follow from a standard perturbative large N expansion of thefree energy, while the right tail large deviations are related to thenon-perturbative expansion and thus to instanton corrections. Considering thestandard interpretation of instantons as tunnelling of eigenvalues, we see thatthe right tail rate function can be identified with the instanton action whichin turn can be given as a simple expression in terms of the spectral curve.From the string theory point of view these non-perturbative correctionscorrespond to branes and can be identified with FZZT branes
Modular symbols over number fields
No full textLet K be a number field, R its ring of integers. For some classes of fields, spaces of cusp forms of weight 2 for GL(2;K) have been computed using methods based on modular symbols. J.E. Cremona [9] began the programme of extending the classical methods over Q to the case of imaginary quadratic fields. This work was continued by some of his Ph.D. students [35, 6, 22], and results have been obtained for some imaginary quadratic fields with small class number. More recently, P. Gunnells and D. Yasaki [18] have developed related algorithms for real quadratic fields.
The aim of this thesis is to contribute to the extension of the modular symbols method, when possible developing algorithms and implementations for effective computations. Some parts of the theory are purely algebraic and can be extended to all number fields. We generalise the theory for cusps and Manin symbols; we also describe a generalisation of Atkin-Lehner involutions and study other normaliser elements. On the other hand, all previous explicit computations for the imaginary quadratic field case were done only for specific fields. In the last part of this thesis we begin work towards a general implementation of the techniques used in this case. In particular, we are able to compute a fundamental domain of the hyperbolic 3-space for any imaginary quadratic field.
Implementations of the algorithms described in this thesis have been written by the author in the open-source mathematics software Sage [31]
FZZT Brane Relations in the Presence of Boundary Magnetic Fields
No full textAtkin M, Zohren S. FZZT Brane Relations in the Presence of Boundary Magnetic Fields. Journal of High Energy Physics. 2012;2012(11): 163.We show how a boundary state different from the (1,1) Cardy state may berealised in the (m,m+1) minimal string by the introduction of an auxiliarymatrix into the standard two hermitian matrix model. This boundary is a naturalgeneralisation of the free spin boundary state in the Ising model. Theresolvent for the auxiliary matrix is computed using an extension of thesaddle-point method of Zinn-Justin to the case of non-identical potentials. Thestructure of the saddle-point equations result in a Seiberg-Shih like relationbetween the boundary states which is valid away from the continuum limit, inaddition to an expression for the spectral curve of the free spin boundarystate. We then show how the technique may be used to analyse boundary statescorresponding to a boundary magnetic field, thereby allowing us to generalisethe work of Carroll et al. on the boundary renormalisation flow of the Isingmodel, to any (m,m+1) model
Applications of random graphs to 2D quantum gravity
No full textThe central topic of this thesis is two dimensional Quantum Gravity and its properties. The term Quantum Gravity itself is ambiguous as there are many proposals for its correct formulation and none of them have been verified experimentally. In this thesis we consider a number of closely related approaches to two dimensional quantum gravity that share the property that they may be formulated in terms of random graphs. In one such approach known as Causal Dynamical Triangulations, numerical computations suggest an interesting phenomenon in which the effective spacetime dimension is reduced in t he UV. In this thesis we first address whether such a dynamical reduction in the number of dimensions may be understood in a simplified model. vVe introduce a continuum limit where t his simplified model exhibits a reduction in the effective dimension of spacetime in the UV, in addition to having rich cross-over behaviour. In the second part of this thesis we consider an approach closely related to causal dynamical triangu lation; namely dynamical triangulation. Although this theory is less well-behaved than causal dynamical triangulation) it is known how to couple it to matter) t herefore allowing for potentially multiple boundary states to appear in the theory. We address t he conjecture of Seiberg and Shih which states that all these boundary states are degenerate and may be constructed from a single principal boundary state. By use of the random graph formulation of the theory we compute the higher genus amplitudes with a single boundary and find that they violate the Seiberg-Shih conjecture. Finally we discuss whether this result prevents the replacement of boundary states by local operators as proposed by Seiberg.EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Random matrix ensembles with singularities and a hierarchy of Painlevé III equations
No full textWe study unitary invariant random matrix ensembles with singular potentials. We obtain asymptotics for the partition functions associated to the Laguerre and Gaussian Unitary Ensembles perturbed with a pole of order k at the origin, in the double scaling limit where the size of the matrices grows, and at the same time the strength of the pole decreases at an appropriate speed. In addition, we obtain double scaling asymptotics of the correlation kernel for a general class of ensembles of positive-definite Hermitian matrices perturbed with a pole. Our results are described in terms of a hierarchy of higher order analogues to the Painlev\'e III equation, which reduces to the Painlev\'e III equation itself when the pole is simple
On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble
No full textWe study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the n × n Laguerre unitary ensemble. The probability that this ratio is greater than r > 1 is expressed in terms of an n × n Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as n → ∞ is found as an integral over (0, ∞) containing two functions q1(x) and q2(x). These functions satisfy a system of two coupled Painlevé V equations, which are derived from a Lax pair of a Riemann–Hilbert problem. We compute asymptotic behaviours of these functions as rx → 0+ and (r − 1)x → ∞, as well as large n asymptotics for the associated Hankel determinants in several regimes of r and x
Violations of Bell inequalities from random pure states
No full textWe consider the expected violations of Bell inequalities from random pure states. More precisely, we focus on a slightly generalized version of the Collins-Gisin-Linden-Massar-Popescu inequality, which concerns Bell experiments of two parties, two measurement options, and N outcomes, and analyze their expected quantum violations from random pure states for varying N, assuming the conjectured optimal measurement operators. It is seen that for small N the Bell inequality is not violated on average, while for larger N it is. Both ensembles of unstructured as well as structured random pure states are considered. Using techniques from random matrix theory this is obtained analytically for small and large N and numerically for intermediate N. The results show a beautiful interplay of different aspects of random matrix theory, ranging from the Marchenko-Pastur distribution and fixed-trace ensembles to the O(n) model
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