Quaternionic second-order freeness and the fluctuations of large symplectically invariant random matrices

We present a definition for second-order freeness in the quaternionic case. We demonstrate that this definition on a second-order probability space is asymptotically satisfied by independent symplectically invariant quaternionic matrices. This definition is different from the natural definition for complex and real second-order probability spaces, those motivated by the asymptotic behavior of unitarily invariant and orthogonally invariant random matrices respectively. Most notably, because the quaternionic trace does not have the cyclic property of a trace over a commutative field, the asymmetries which appear in the multi-matrix context result in an asymmetric contribution from the terms which appear symmetrically in the complex and real cases.

Intertextual Precision Expectations

In the chapters that follow, the third-order probability design is developed. The third-order probability design revolves around how expectations about second- and first-order predictions are developed through structural patterns yielded by genre (III.1), textual gaps and shadow stories (III.2), and intertextual references to unfamiliar texts (III.3). The final chapter of the section, then, traces the tension between flexibility and constraint in probability designs.

Metafictional Nudging

The chapter makes the argument that the moments when the narrative refers to itself do not necessarily disrupt readers’ immersion and sense of flow. Movement between different diegetic levels in mise-en-abyme can unfold fluently, and the joint attention is usually maintained. Instances of metafiction and metanarration rather serve as ‘nudges’ in the second-order probability designs that redirect readers’ attention while maintaining it. Novels as distant in time from each other as Heliodorus’ Ethiopian Adventures and Margaret Atwood’s The Blind Assassin deploy metafictional nudges in their second-order probability design, provoking readers without relinquishing sense of flow.

Plots and Probability Transformations

In the chapters that follow, the first-order probability design around narrative plot is developed. I.1: Plot and Probability Transformations concerns itself with plot events and prediction errors. I.2: Probability Designs discusses the links between design, the creative process, and the author’s intentionality. Finally, I.3: The Height of Drop addresses how readers’ perception of probabilities is manipulated.

Pleasures and Precision Shifts

The chapter completes previous discussions of readers’ sense of agency (I.3) and sense of presence (II.2) by introducing the sense of flow. Readers’ sense of flow is modelled on Csikszentmihalyi’s notion of the ‘flow channel’, where a trade-off between challenges and enabling factors develops dynamically. The second-order probability design, with its embodied cues, provides a similar environment for readers, creating a sense of flow and readerly pleasure while reading. Through examples from Tolstoy’s Anna Karenina, Spufford’s Golden Hill, and Austen’s Persuasion, the chapter outlines different ways in which the sense of flow can be eased, complicated, and disrupted. It also links to traditional formalist arguments about defamiliarisation in literary texts.

An Elopement!

In the chapters that follow, the second-order probability design is developed. The second-order probability design revolves around second-order predictions about predictions, and it is traced in embodied aspects of reading (II.1), readers’ sense of presence (II.2), sense of flow (II.3), and the metafictional nudges that the text provides (II.4).

Reading by Proxy

The chapter addresses how unfamiliar references can do important interpretive work in the third-order probability design. It begins with the observation that readers often do not need to have read the texts that characters read or hold in their libraries to nevertheless get a clear sense of what kind of precision expectation these references provide. The process of discerning what predictions are likely to be relevant without the necessary textual knowledge is introduced as ‘reading by proxy’. It is supported by the text’s probability design. The chapter goes on to discuss the implications of reading by proxy for the analysis of texts from a different historical period and relates predictive processing to schema theory.