Natural languages exhibit striking regularities in their statistical structure, including notably the emergence of Zipf's and Heaps' laws. Despite this, it remains broadly unclear how these properties relate to the modern tokenisation schemes used in contemporary transformer models. In this note, we analyse the information content (as measured by the Shannon entropy) of various corpora under the assumption of a Zipfian frequency distribution, and derive a closed-form expression for the slot entropy expectation value. We then empirically investigate how byte--pair encoding (BPE) transforms corpus statistics, showing that recursive applications of BPE drive token frequencies toward a Zipfian power law while inducing a characteristic growth pattern in empirical entropy. Utilizing the ability of transformers to learn context dependent token probability distributions, we train language models on corpora tokenised at varying BPE depths, revealing that the model predictive entropies increasingly agree with Zipf-derived predictions as the BPE depth increases. Attention-based diagnostics further indicate that deeper tokenisation reduces local token dependencies, bringing the empirical distribution closer to the weakly dependent (near IID) regime. Together, these results clarify how BPE acts not only as a compression mechanism but also as a statistical transform that reconstructs key informational properties of natural language.

We study a deliberately simple, fully non-linguistic model of text: a sequence of independent draws from a finite alphabet of letters plus a single space symbol. A word is defined as a maximal block of non-space symbols. Within this symbol-level framework, which assumes no morphology, syntax, or semantics, we derive several structural results. First, word lengths follow a geometric distribution governed solely by the probability of the space symbol. Second, the expected number of words of a given length, and the expected number of distinct words of that length, admit closed-form expressions based on a coupon-collector argument. This yields a critical word length k* at which word types transition from appearing many times on average to appearing at most once. Third, combining the exponential growth of the number of possible strings of length k with the exponential decay of the probability of each string, we obtain a Zipf-type rank-frequency law p(r) proportional to r^{-alpha}, with an exponent determined explicitly by the alphabet size and the space probability. Our contribution is twofold. Mathematically, we give a unified derivation linking word lengths, vocabulary growth, critical length, and rank-frequency structure in a single explicit model. Conceptually, we argue that this provides a structurally grounded null model for both natural-language word statistics and token statistics in large language models. The results show that Zipf-like patterns can arise purely from combinatorics and segmentation, without optimization principles or linguistic organization, and help clarify which phenomena require deeper explanation beyond random-text structure.

Heaps' or Herdan's law characterizes the word-type vs. word-token relation by a power-law function, which is concave in linear-linear scale but a straight line in log-log scale. However, it has been observed that even in log-log scale, the type-token curve is still slightly concave, invalidating the power-law relation. At the next-order approximation, we have shown, by twenty English novels or writings (some are translated from another language to English), that quadratic functions in log-log scale fit the type-token data perfectly. Regression analyses of log(type)-log(token) data with both a linear and quadratic term consistently lead to a linear coefficient of slightly larger than 1, and a quadratic coefficient around -0.02. Using the ``random drawing colored ball from the bag with replacement" model, we have shown that the curvature of the log-log scale is identical to a ``pseudo-variance" which is negative. Although a pseudo-variance calculation may encounter numeric instability when the number of tokens is large, due to the large values of pseudo-weights, this formalism provides a rough estimation of the curvature when the number of tokens is small.

The spectacular success of deep generative models calls for quantitative tools to measure their statistical performance. Divergence frontiers have recently been proposed as an evaluation framework for generative models, due to their ability to measure the quality-diversity trade-off inherent to deep generative modeling.

Here we present a new class of optimality for coding systems. Members of that class are displaced linearly from optimal coding and thus exhibit Zipf's law, namely a power-law distribution of frequency ranks. Within that class, Zipf's law, the size-rank law and the size-probability law form a group-like structure. We identify human languages that are members of the class. All languages showing sufficient agreement with Zipf's law are potential members of the class. In contrast, there are communication systems in other species that cannot be members of that class for exhibiting an exponential distribution instead but dolphins and humpback whales might. We provide a new insight into plots of frequency versus rank in double logarithmic scale. For any system, a straight line in that scale indicates that the lengths of optimal codes under non-singular coding and under uniquely decodable encoding are displaced by a linear function whose slope is the exponent of Zipf's law. For systems under compression and constrained to be uniquely decodable, such a straight line may indicate that the system is coding close to optimality. We provide support for the hypothesis that Zipf's law originates from compression and define testable conditions for the emergence of Zipf's law in compressing systems.

Division of Psychology, Communication & Human Neuroscience, The University of Manchester Author Note Hugo Rydel - Johnston https://orcid.org/0009 - 0006 - 1103 - 1015 Alex Ka fkas https://orcid.org/0000 - 0001 - 5133 - 8827 We have no conflict s of interest to disclose. Correspondence concerning this article should be addressed to Hugo Rydel - Johnston, Division of Psychology, Communication & Human Neuroscience, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK . DYSLEXIC READING TAKES LONGER 2 Abstract We ask where, and under what conditions, dyslexic reading costs arise in a large - scale naturalistic reading dataset. Using eye - tracking aligned to word - level properties -- word length, frequency, and predictability -- we model the influence of each of these feat ures on dyslexic time costs. We find that all three properties robustly change reading times in both typical and dyslexic readers, but dyslexic readers show stronger sensitivities to each of the three features, especially predictability. Counterfactual man ipulations of these features substantially narrow the dyslexic - control gap -- by about one - third -- with predictability showing the strongest effect, followed by length, and frequency. These patterns align with existing dyslexia theories suggesting heightened de mands on linguistic working memory and phonological encoding in dyslexic reading and directly motivate further research into lexical complexity and preview benefits to further explain the quantified gap. In effect, these findings break down when extra dysl exic costs arise, how large they are, and provide actionable guidance for the development of interventions and computational models for dyslexic readers. Keywords: e ye movements, r eading time, w ord length, l exical f requency, p redictability, s kipping, t otal reading time DYSLEXIC READING TAKES LONGER 3 Why Dyslexic Reading Takes Longer - And When Dyslexia is characterized by persistent difficulty in accurate and/or fluent word recognition and decoding (Lyon et al., 2003) and affects between 4 - 8% of individuals (Yang et al., 2022; Doust et al., 2022).

This study presents a comparative analysis of the statistical and linguistic properties of neural audio codecs (NACs). We investigate discrete speech tokens produced by various NAC models, examining their adherence to linguistic statistical laws such as Zipf's law and Heaps' law, as well as their entropy and redundancy. To assess how these token-level properties relate to semantic and acoustic preservation in synthesized speech, we evaluate intelligibility using error rates of automatic speech recognition, and quality using the UTMOS score. Our results reveal that NAC tokens, particularly 3-grams, exhibit language-like statistical patterns. Moreover, these properties, together with measures of information content, are found to correlate with improved performances in speech recognition and resynthesis tasks. These findings offer insights into the structure of NAC token sequences and inform the design of more effective generative speech models.

Game-theoretic interactions between agents with Large Language Models (LLMs) have revealed many emergent capabilities, yet the linguistic diversity of these interactions has not been sufficiently quantified. In this paper, we present the Conversational Robustness Evaluation Score: CORE, a metric to quantify the effectiveness of language use within multi-agent systems across different game-theoretic interactions. CORE integrates measures of cluster entropy, lexical repetition, and semantic similarity, providing a direct lens of dialog quality. We apply CORE to pairwise LLM dialogs across competitive, cooperative, and neutral settings, further grounding our analysis in Zipf's and Heaps' Laws to characterize word frequency distributions and vocabulary growth. Our findings show that cooperative settings exhibit both steeper Zipf distributions and higher Heap exponents, indicating more repetition alongside greater vocabulary expansion. In contrast, competitive interactions display lower Zipf and Heaps exponents, reflecting less repetition and more constrained vocabularies. These results provide new insights into how social incentives influence language adaptation, and highlight CORE as a robust diagnostic for measuring linguistic robustness in multi-agent LLM systems. Our code is available at https://github.com/psyonp/core.

The spectacular success of deep generative models calls for quantitative tools to measure their statistical performance. Divergence frontiers have recently been proposed as an evaluation framework for generative models, due to their ability to measure the quality-diversity trade-off inherent to deep generative modeling.