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原文摘要

We develop a quantitative theory of the Random Language Model (RLM), an ensemble of stochastic context-free grammars, in a scaling limit where the number of hidden symbols $N \to \infty$ while the grammar temperature $\tildeε_d \to 0$ at fixed $x = \tildeε_d \log N$. In this limit, the model admits a controlled description based on a large-deviation principle over rule-usage patterns. A semi-annealed approximation maps the problem to a class of Random Energy Models with nontrivial combinatorics. We show that the RLM exhibits a freezing transition at a critical value $x_c=1/8$, below which rule usage condenses on a subset of preferred rules and language statistics acquire a nontrivial dependence on corpus length. A second characteristic scale at $x=1/2$, associated with the breakdown of Gaussian fluctuations, separates transition and babbling regimes. Across regimes, we derive explicit scaling laws for the number of distinct rules, entropy, and energy, identifying distinct frozen and critical regimes controlled by the interplay of grammar size, corpus length, and temperature. The theory resolves previous ambiguities regarding the existence of a thermodynamic transition and explains the slow approach to the large-$N$ limit as a consequence of the dependence on $\log N$. It further provides a unified framework in which universal statistical properties of language emerge from typical realizations of generative grammars, with implications for both natural language statistics and the behavior of large language models. Heaps' law and Zipf's law are both derived, and the dependence of predictive entropy on context length is rationalized. Comparison with empirical measurements of Heaps' and Zipf's laws yields parameter estimates for real languages.