Let $ξ$ be the stationary occupation field generated by a Poisson system of independent simple symmetric random walks on $\mathbb Z$ in space--time dimension $1+1$. For a finite set $A\subset\mathbb Z$, we consider the classical fixed-region observables $W_N(A)$, the cumulative occupation of $A$ up to time $N$, and $D_N(A)$, the number of distinct particles visiting $A$ up to time $N$. We prove quantitative central limit theorems for both observables, with Wasserstein rate of order $N^{-1/4}$. In addition, we introduce an independent nearest-neighbour random walk $S=(S_n,\,n\ge 0)$ on $\mathbb Z$ with non-zero drift and sample the field along this ballistic path. For a fixed polynomial observable $φ(x)=\sum_{j=0}^k β_j x^j, β_k\neq 0$, of degree $k\in \mathbb N$, we consider the partial sums $Y_{N,φ}=\sum_{n=1}^N φ(ξ(n,S_n)).$ We prove a Wasserstein bound of order $N^{-1/2}$ for the normal approximation of the standardized $Y_{N,φ}$. To the best of our knowledge, this is the first quantitative normal approximation result for polynomial functionals of the Poisson occupation field sampled along a random walk path. The drift induces an effective decorrelation of the sampled environment, leading to a substantial improvement over fixed-region sampling. The proofs rely on a representation of $ξ$ as a Poisson functional on path space and on the Malliavin--Stein method for Poisson functionals.
Cascading liquidations across decentralized finance (DeFi) lending protocols represent a systemic risk that standard empirical models often fail to capture. To quantify this phenomenon, we apply a 3-variate Hawkes process to model crossprotocol liquidation clustering among Aave V3, Compound V3, and Morpho on Ethereum. Using 7,500 on-chain liquidation events spanning 2023-01-01 through 2025-12-31, we estimate exponential triggering kernels via maximum likelihood estimation and validate the approach against nonparametric spectral estimates. The results indicate a stable, subcritical regime (ρ = 0.725) characterized by statistically significant off-diagonal excitation. The strongest cross-protocol channel runs from Morpho to Compound V3 (branching ratio Γ = 0.418), while selfexcitation ratios range from 0.28 to 0.32. Directional predictive dependence tests confirm asymmetric spillover effects. Furthermore, likelihood-based comparisons demonstrate that crossprotocol excitation significantly outperforms self-excitation-only and common-factor baselines, including models with ETH return controls. Placebo permutations verify that this off-diagonal structure is not an artifact of shared timing. Ultimately, while the findings document robust cross-protocol clustering consistent with spillover channels, we emphasize that Hawkes crossexcitation captures directional predictive dependence rather than strict structural causation.
We propose a novel approach to marked Hawkes kernel inference which we name the moment-based neural Hawkes estimation method. Hawkes processes are fully characterized by their first- and second-order statistics through a Fredholm integral equation of the second kind. Using recent advances in solving partial differential equations with physics-informed neural networks, we provide a numerical procedure to solve this integral equation in high dimension. Together with an adapted training pipeline, we give a generic set of hyperparameters that produces robust results across a wide range of kernel shapes. We conduct an extensive numerical validation on simulated data. We finally propose two applications of the method to the analysis of the microstructure of cryptocurrency markets. In a first application, we extract the influence of volume on the arrival rate of BTC-USD trades and in a second application we analyze the causality relationships and their directions amongst a universe of 15 cryptocurrency pairs in a centralized exchange.
In this paper we consider nonnegatively curved finite dimensional Alexandrov spaces with a non-collapsing condition, i.e., such that unit balls have volumes uniformly bounded from below away from zero. We study the relation between the isoperimetric profile, the existence of isoperimetric sets, and the asymptotic structure at infinity of such spaces. In this setting, we prove that the following conditions are equivalent: the space has linear volume growth; it is Gromov--Hausdorff asymptotic to one cylinder at infinity; it has uniformly bounded isoperimetric profile; the entire space is a tubular neighborhood of either a line or a ray. Moreover, on a space satisfying any of the previous conditions, we prove existence of isoperimetric sets for sufficiently large volumes, and we characterize the geometric rigidity at the level of the isoperimetric profile. Specializing our study to the $2$-dimensional case, we prove that unit balls have always volumes uniformly bounded from below away from zero, and we prove existence of isoperimetric sets for every volume, characterizing also their topology when the space has no boundary. The proofs exploit a variational approach, and in particular apply to Riemannian manifolds with nonnegative sectional curvature and to Euclidean convex bodies. Up to the authors' knowledge, most of the results are new even in these smooth cases.
Stablecoins, digital assets pegged to a specific currency or commodity value, are heavily involved in transactions of major cryptocurrencies. The effects of deviations from their desired fixed values (depeggings) on the cryptocurrencies for which they are frequently used in transactions are therefore of interest to study. We propose a model for this phenomenon using a multivariate mutually-exciting Hawkes process, and present a numerical example applying this model to Tether (USDT) and Bitcoin (BTC).
The paper constructs a multi-variate Hawkes process model of Bitcoin block arrivals and price jumps. Hawkes processes are selfexciting point processes that can capture the self- and cross-excitation effects of block mining and Bitcoin price volatility. We use publicly available blockchain datasets to estimate the model parameters via maximum likelihood estimation. The results show that Bitcoin price volatility boost block mining rate and Bitcoin investment return demonstrates mean reversion. Quantile-Quantile plots show that the proposed Hawkes process model is a better fit to the blockchain datasets than a Poisson process model.
We consider the zero-range process with arbitrary bounded monotone rates on the complete graph, in the regime where the number of sites diverges while the density of particles per site converges. We determine the asymptotics of the mixing time from any initial configuration, and establish the cutoff phenomenon. The intuitive picture is that the system separates into a slowly evolving solid phase and a quickly relaxing liquid phase: as time passes, the solid phase dissolves into the liquid phase, and the mixing time is essentially the time at which the system becomes completely liquid. Our proof uses the path coupling technique of Bubley and Dyer, and the analysis of a suitable hydrodynamic limit. To the best of our knowledge, even the order of magnitude of the mixing time was unknown, except in the special case of constant rates.
We construct a ‘reflexivity’ index to measure the activity generated endogenously within a market for cryptocurrencies. For this purpose, we fit a univariate self-exciting Hawkes process with two classes of parametric kernels to high-frequency trading data. A parsimonious model of both endogenous and exogenous dynamics enables a direct comparison with exchanges for traditional asset classes, in terms of identified branching ratios. We also formulate a ‘Hawkes disorder problem,’ as generalization of the established Poisson disorder problem, and provide a simulation-based approach to determining an optimal observation horizon. Our analysis suggests that Bitcoin mid-price dynamics feature long-memory properties, well explained by the power-law kernel, at a level of criticality similar to fiat-currency markets.
Cryptocurrencies have recently experienced a new wave of price volatility and interest; activity within social media communities relating to cryptocurrencies has increased significantly. There is currently limited documented knowledge of factors which could indicate future price movements. This paper aims to decipher relationships between cryptocurrency price changes and topic discussion on social media to provide, among other things, an understanding of which topics are indicative of future price movements. To achieve this a well-known dynamic topic modelling approach is applied to social media communication to retrieve information about the temporal occurrence of various topics. A Hawkes model is then applied to find interactions between topics and cryptocurrency prices. The results show particular topics tend to precede certain types of price movements, for example the discussion of 'risk and investment vs trading' being indicative of price falls, the discussion of 'substantial price movements' being indicative of volatility, and the discussion of 'fundamental cryptocurrency value' by technical communities being indicative of price rises. The knowledge of topic relationships gained here could be built into a real-time system, providing trading or alerting signals.
In recent years, there has been a surge in demand for statistical tools for analyzing dynamic networks involving point processes. This has been driven largely by the availability of high-dimensional data in a number of application areas such as systems neuroscience and stochastic finance. Given some network data, an extremely challenging problem is to infer the causal dependencies between nodes to reconstruct the network. The high volume and decentralized storage of such data poses new challenges for analysis. Here we develop for the first time a distributed optimization algorithm based on the point process likelihood for dynamic networks of interacting Hawkes processes. We demonstrate the algorithm using genomic data to construct a transcriptional regulatory network in embryonic stem cells and tick-level financial data to construct an influence map of major currencies.