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Register a new userDilute oriented loop models
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Jesper Lykke Jacobsen
Publication date
2015
fields
Physics
and research's language is
English
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We study a model of dilute oriented loops on the square lattice, where each loop is compatible with a fixed, alternating orientation of the lattice edges. This implies that loop strands are not allowed to go straight at vertices, and results in an enhancement of the usual O(n) symmetry to U(n). The corresponding transfer matrix acts on a number of representations (standard modules) that grows exponentially with the system size. We derive their dimension and those of the centraliser by both combinatorial and algebraic techniques. A mapping onto a field theory permits us to identify the conformal field theory governing the critical range, $n le 1$. We establish the phase diagram and the critical exponents of low-energy excitations. For generic n, there is a critical line in the universality class of the dilute O(2n) model, terminating in an SU(n+1) point. The case n=1 maps onto the critical line of the six-vertex model, along which exponents vary continuously.
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We study a model of self propelled particles exhibiting run and tumble dynamics on lattice. This non-Brownian diffusion is characterised by a random walk with a finite persistence length between changes of direction, and is inspired by the motion of bacteria such as E. coli. By defining a class of models with multiple species of particle and transmutation between species we can recreate such dynamics. These models admit exact analytical results whilst also forming a counterpart to previous continuum models of run and tumble dynamics. We solve the externally driven non-interacting and zero-rang
We study a generating function flowing from the one enumerating a set of partitions to the one enumerating the corresponding set of noncrossing partitions; numerical simulations indicate that its limit in the Adjacency random matrix model on bipartite Erdos-Renyi graphs gives a good approximation of the spectral distribution for large average degrees. This model and a Wishart-type random matrix model are described using congruence classes on $k$-divisible partitions. We compute, in the $dto infty$ limit with $frac{Z_a}{d}$ fixed, the spectral distribution of an Adjacency and of a Laplacian random block matrix model, on bipartite Erdos-Renyi graphs and on bipartite biregular graphs with degrees $Z_1, Z_2$; the former is the approximation previously mentioned; the latter is a mean field approximation of the Hessian of a random bipartite biregular elastic network; it is characterized by an isostatic line and a transition line between the one- and the two-band regions.
We introduce a class of generalized Lipkin-Meshkov-Glick (gLMG) models with su$(m)$ interactions of Haldane-Shastry type. We have computed the partition function of these models in closed form by exactly evaluating the partition function of the restriction of a spin chain Hamiltonian of Haldane-Shastry type to subspaces with well-defined magnon numbers. As a byproduct of our analysis, we have obtained strong numerical evidence of the Gaussian character of the level density of the latter restricted Hamiltonians, and studied the distribution of the spacings of consecutive unfolded levels. We have also discussed the thermodynamic behavior of a large family of su(2) and su(3) gLMG models, showing that it is qualitatively similar to that of a two-level system.
In this thesis we present few theoretical studies of the models of self-organized criticality. Following a brief introduction of self-organized criticality, we discuss three main problems. The first problem is about growing patterns formed in the abelian sandpile model (ASM). The patterns exhibit proportionate growth where different parts of the pattern grow in same rate, keeping the overall shape unchanged. This non-trivial property, often found in biological growth, has received increasing attention in recent years. In this thesis, we present a mathematical characterization of a large class of such patterns in terms of discrete holomorphic functions. In the second problem, we discuss a well known model of self-organized criticality introduced by Zhang in 1989. We present an exact analysis of the model and quantitatively explain an intriguing property known as the emergence of quasi-units. In the third problem, we introduce an operator algebra to determine the steady state of a class of stochastic sandpile models.
In our previous works on infinite horizontal Ising strips of width $m$ alternating with layers of strings of Ising chains of length $n$, we found the surprising result that the specific heats are not much different for different values of $N$, the separation of the strings. For this reason, we study here for $N=1$ the spin-spin correlation in the central row of each strip, and also the central row of a strings layer. We show that these can be written as a Toeplitz determinants. Their generating functions are ratios of two polynomials, which in the limit of infinite vertical size become square roots of polynomials whose degrees are $m+1$ where $m$ is the size of the strips. We find the asymptotic behaviors near the critical temperature to be two-dimensional Ising-like. But in regions not very close to criticality the behavior may be different for different $m$ and $n$. Finally, in the appendix we shall present results for generating functions in more general models.
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