This thesis concerns the generation and characterisation of optical fractals. Chapter 1 presents a brief introduction to fractals and fractal dimensions, and then a review of contexts where fractal concepts have arisen in optics. These contexts are classified in terms of whether the fractal-generating mechanisms at work are linear or nonlinear.

Dendrites are tree-like topological spaces, and in this thesis, the physical characteristics of various random fractal versions of this type of set are investigated. This work will contribute to the development of analysis on fractals, an area which has grown considerably over the last twenty years. First, a collection of random self-similar dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have often relied on the scaling factors being bounded uniformly away from zero. However, using a percolative argument, and taking advantage of the tree-like structure of the sets considered here, it is shown that this condition is not necessary; a simple condition on the tail of the distribution of the scaling factors at zero is all that is assumed. The scaling factors of these recursively defined structures form what is known as a multiplicative cascade, and results about the height of this random object are also obtained. With important physical and probabilistic applications, the heat equation has justifiably received a substantial amount of attention in a variety of settings. For certain types of fractals, it has become clear that a key factor in estimating the heat kernel is the volume growth with respect to the resistance metric on the space. In particular, uniform polynomial volume growth, which occurs for many deterministic self-similar fractals, immediately implies uniform (on-diagonal) heat kernel behaviour. However, in the random fractal setting, this is frequently not the case, and volume fluctuations are often observed. Motivated by this, an analysis of how volume fluctuations lead to corresponding heat kernel fluctuations for measure-metric spaces equipped with a resistance form is conducted here. These results apply to the aforementioned random self-similar dendrites, amongst other examples. The continuum random tree (CRT) of Aldous is an important random example of a measure-metric space, and fits naturally into the framework of the previous paragraph. In this thesis, quenched (almost-sure) volume growth asymptotics for the CRT are deduced, which show that the behaviour in almost-every realisation is not uniform. Applying the results introduced above, these yield heat kernel bounds for the CRT, demonstrating that heat kernel fluctuations occur almost-surely. Finally, a new representation of the CRT as a random self-similar dendrite is presented.

Soit f une instance du whole-plane $\SLE_\kappa$ : on sait que pour certaines valeurs de κ, p les moments dérivés $\mathbb{E}(\vert f'(z) \vert^p)$ peuvent être écrits sous une forme fermée, étude qui a permis de mettre au jour une nouvelle phase du spectre des moyennes intégrales. Le but de cette thèse est une étude des moments généralisés $\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}$ : cette étude permet de confirmer la structure algébrique riche du whole-plane SLE. On montre que les formes fermées des moments mixtes $\mathbb{E}\big(\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}\big)$ apparaissent sur une famille dénombrable de paraboles du plan (p, q), en étendant les équations de Beliaev-Smirnov à ce cas. Nous introduisons également le spectre généralisé β(p, q; κ), correspondant au comportement asymptotiques des moyennes intégrales mixtes. Le spectre généralisé moyen du whole-plane SLE prend quatre formes possibles, séparés par cinq séparatrices dans $\R^2$. Nous proposons également une approche semblable pour la dérivée Schwarziene S(f)(z) de l’application de SLE. Les calculs sur les équations de Beliaev-Smirnov d’une certaine générale forme de moment mène à une formulation explicite de $\mathbb{E}(S(f)(z))$ . Nous étudions finalement la variance asymptotique de McMullen et démontrons une relation entre la croissance infinitésimale du spectre de la moyenne intégrale et la variance asymptotique pour SLE₂. / Let f an instance of the whole-plane $\SLE_\kappa$ conformal map from the unit disk D to the slit plane: We know that for certain values of κ, p the derivative moments $\mathbb{E}(\vert f'(z) \vert^p)$ can be written in a closed form, study that has updated a new phase of the integral means spectrum. The goal of this thesis is a study on generalized moments $\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}$ : ΒββThis study permit confirm the rich algebraic structure of the whole-plane version of SLE. It will be showed that closed forms of the mixed moments E mixtes $\mathbb{E}\big(\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}\big)$ can be obtained on a countable family of parabolas in the moment plane (p, q), by extending the so-called Beliaev–Smirnov equation to this case. We also introduce the generalized integral means spectrum, β(p, q; κ), corresponding to the singular behavior of the mixed moments. The average generalized spectrum of whole-plane SLE takes four possible forms, separated by five phase transition lines in $\R^2$. We also propose a similar approach for the Schwarzian derivative S(f)(z) of SLE maps. Computations on the Beliaev–Smirnov equation of a certain general form of moment lead to an explicit formula of $\mathbb{E}(S(f)(z))$ . We finally study the McMullen asymptotic variance and prove a relation between the infinitesimal growth of the integral mean spectrum and the asymptotic variance in an expectation sense for SLE₂.