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Register a new userCriticality at absolute zero from Ising model on two-dimensional dynamical triangulations
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We reconsider the criticality of the Ising model on two-dimensional dynamical triangulations based on the N-by-N hermitian two-matrix model with the introduction of a loop-counting parameter and linear terms in the potential. We show that in the large-N limit even though the Ising model is classical, the critical temperature can reach absolute zero by tuning the loop-counting parameter, and the corresponding continuum theory turns out to be the quantised theory of neither pure gravity nor gravity coupled to conformal matter with central charge being 1/2.
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We study the zero-temperature criticality of the Ising model on two-dimensional dynamical triangulations to contemplate its physics. As it turns out, an inhomogeneous nature of the system yields an interesting phase diagram and the physics at the zero temperature is quite sensitive about how we cool down the system. We show the existence of a continuous parameter that characterizes the way we approach the zero-temperature critical point and it may enter in a critical exponent.
Vacuum Einstein equations when projected on to a black hole horizon is analogous to the dynamics of fluids. In this work we address the question, whether certain properties of semi-classical black holes could be holographically mapped into properties of (2 + 1)-dimensional fluid living on the horizon. In particular, we focus on the statistical mechanical description of the horizon-fluid that leads to Bekenstein-Hawking entropy. Within the paradigm of Landau mean field theory and existence of a condensate at a critical temperature, we explicitly show that Bekenstein-Hawking entropy and other features of black hole thermodynamics can be recovered from the statistical modelling of the fluid. We also show that a negative cosmological constant acts like an external magnetic field that induces order in the system leading to the appearance of a tri-critical point in the phase diagram.
We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of +1 or -1 to each site in ${bf Z}^2$, is the zero-temperature limit of the stochastic homogeneous Ising ferromagnet (with Glauber dynamics): the initial state is chosen uniformly at random and then each site, at rate one, polls its 4 neighbors and makes sure it agrees with the majority, or tosses a fair coin in case of a tie. Among the main results (almost sure, with respect to both the process and initial state) are: clusters (maximal domains of constant sign) are finite for times $t< infty$, but the cluster of a fixed site diverges (in diameter) as $t to infty$; each of the two constant states is (positive) recurrent. We also present other results and conjectures concerning positive and null recurrence and the role of absorbing states.
We estimate thermal one-point functions in the 3d Ising CFT using the operator product expansion (OPE) and the Kubo-Martin-Schwinger (KMS) condition. Several operator dimensions and OPE coefficients of the theory are known from the numerical bootstrap for flat-space four-point functions. Taking this data as input, we use a thermal Lorentzian inversion formula to compute thermal one-point coefficients of the first few Regge trajectories in terms of a small number of unknown parameters. We approximately determine the unknown parameters by imposing the KMS condition on the two-point functions $langle sigmasigma rangle$ and $langle epsilonepsilon rangle$. As a result, we estimate the one-point functions of the lowest-dimension $mathbb Z_2$-even scalar $epsilon$ and the stress-energy tensor $T_{mu u}$. Our result for $langle sigmasigma rangle$ at finite-temperature agrees with Monte-Carlo simulations within a few percent, inside the radius of convergence of the OPE.
The dynamical triangulations approach to quantum gravity is investigated in detail for the first time in five dimensions. In this case, the most general action that is linear in components of the f-vector has three terms. It was suspected that the corresponding space of couplings would yield a rich phase structure. This work is primarily motivated by the hope that this new viewpoint will lead to a deeper understanding of dynamical triangulations in general. Ultimately, this research programme may give a better insight into the potential application of dynamical triangulations to quantum gravity. This thesis serves as an exploratory study of this uncharted territory. The five dimensional (k,l) moves used in the Monte Carlo algorithm are proven to be ergodic in the space of combinatorially equivalent simplicial 5-manifolds. A statement is reached regarding the possible existence of an exponential upper bound on the number of combinatorially equivalent triangulations of the 5-sphere. Monte Carlo simulations reveal non-trivial phase structure which is analysed in some detail. Further investigations deal with the geometric and fractal nature of triangulations. This is followed by a characterisation of the weak coupling limit in terms of stacked spheres. Simple graph theory arguments are used to reproduce and generalise a well-known result in combinatorial topology. Finally, a comprehensive study of singular structures in dynamical triangulations is presented. It includes a new understanding of their existence, which appears to be consistent with the non-existence of singular vertices in three dimensions. The thesis is concluded with an overview of results, general discussion and suggestions for future work.
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