We prove a central limit theorem for the log determinant of a Gaussian Pearson sample correlation matrix as the dimension diverges. Only two conditions are imposed: the population correlation matrix is positive definite, and the sample degrees of freedom are at least the dimension. Both are necessary for the ordinary log determinant to be finite. To the best of our knowledge, no previous central limit theorem covers this full nonsingular domain. It covers every aspect ratio from dilute growth to the square hard edge. No uniform lower or upper bound is imposed on the eigenvalues of the population correlation matrices: the smallest may approach zero and the largest may diverge. The proof develops a coordinatewise Wiener chaos reduction for the random diagonal normalization and combines it with an exact Wishart transform comparison. Geometrically, the statistic is twice the log volume of a random parallelotope spanned by standardized Gaussian coordinate vectors.
We prove the full BirchâSwinnertonâDyer conjecture for all elliptic curves over Q. Using a fundamentally new approach that extends the method developed for the Riemann Hypothesis, we construct a sequence of finite-dimensional self-adjoint matrices from the Euler product of the elliptic curve L-function. We establish a strict spectral correspondence between the eigenvalues of these matrices and the squares of the distances from the critical point s=1 to the zeros of L (E, s), with no prior knowledge of zero locations required in the construction. Using mathematical induction, perturbation bounds and the monotone convergence theorem for self-adjoint operators, we extend these results to the infinite-dimensional case, proving that the order of vanishing of L (E, s) at s=1 equals the rank of the MordellâWeil group E (Q). We then prove the exact leading-term formula relating the first non-vanishing coefficient of the Taylor expansion of L (E, s) at s=1 to the arithmetic invariants of the elliptic curve, including the period, regulator, Tamagawa numbers, and the order of the TateâShafarevich group, which we prove is finite. We also embed this result into the broader universal self-adjoint integral operator framework. Keywords: BirchâSwinnertonâDyer conjecture; elliptic curve; L-function; self-adjoint operator; spectral correspondence; MordellâWeil rank; Tate-Shafarevich group MSC 2020 Classification: 11G05; 11M41; 47A10; 14H52; 11G40
The Connesâvan Suijlekom truncated Weil quadratic form, indexed by a cutoff parameter c that controls the primes p †c entering the operator, produces a ground state whose FourierâMellin zeros provably lie on the critical line; whether they converge to the Riemann zeros as c â â is open (Connes 2026; ConnesâConsaniâMoscovici 2025). We present, to our knowledge, the first independent public implementation of the Connesâvan Suijlekom Galerkin matrix at sixteen cutoffs (c = 13 through 67, plus c = 100). Across the in-sample window c = 13 through c = 67 at N = 100, the first-zero absolute error |Îł1 â Îł1Riemann| shrinks monotonically from âŒ2Ă10â55 to âŒ1.5Ă10â168, a 113-OOM convergence across fifteen cutoffs. The smallest-positive even-sector eigenvalue λmineven separately reaches âŒ10â334 at c = 100, N = 250 (275-OOM span from c = 13). Out-of-sample test at c = 100. On the four-point N-sweep N â {100, 150, 200, 250} at dps = 500, consecutive first-difference ratios 0.837 and 0.836 match to two decimal places. Aitken-Î2 on the two overlapping triples yields log10|λâeven| â â536.8 and â â533.7, approaching the Connes 2026 §6.4 heuristic prediction (â â530.4) monotonically with N (6.4 and 3.3 OOM gaps out of |xâ| ⌠530). The same eigenvector recovers Îł1, âŠ, Îł10 to 307â329 matching digits at N = 250, dps = 500. Under the unitary equivalence with ConnesâConsaniâMoscovici Lemma 5.1, this is the deepest such Galerkin-truncation recovery in the public Connesâvan Suijlekom / ConnesâConsaniâMoscovici literature, subject to a hypothesis-status caveat. The raw finite-N matrix carries a small block of negative-sign eigenvalues at the finite archimedean cutoff T = 800; these are an artifact of that cutoff and are absent once T is increased, so the smallest-positive even-sector eigenvalue is the genuine smallest one (continuum positivity of QWλ is RH-equivalent and is not assumed at λ = â100). The fit |log10 λmin| â 13.24 c0.634 on c †67 at N = 100 is shown to be a finite-N rate, falsified at c = 100, N = 200 by 49 OOM in the direction of faster decay. Structural observations include approximate eigenvector c-invariance (overlap â„ 0.9498 on all 105 cutoff pairs despite eigenvalues differing by 113 OOM), multi-zero convergence universality (all ten detectable zeros within 3.8% of each other), an empirical Galerkin-convergence exponent s(c) â 55 log c â 128, un-rescaled Galerkin bulk-spectrum Poisson statistics (ÎČ < 0.05; this is a structural diagnostic of the truncated operator, not a test of Montgomery's conjecture, which applies to locally-rescaled zero spacings), and tight bulk invariants log|det Qc| â â65.6 c + 542 (R2 = 0.997). We make no claim of proof; the contribution is reproducible numerical data and its careful interpretation under the existing CvS / CCM framework. All code, data, and ancillary files are publicly available. Version 3.3 (2026-06-26) correction. The negative-sign eigenvalue blocks reported at c = 100 and for L(s, Ï3) at c = 23, 29 are a finite archimedean-cutoff (T) artifact, not a feature of the operator: they are stable in working precision but vanish once T is increased, so cutoff-free the relevant even sectors are non-negative and the smallest-positive branch is the genuine smallest eigenvalue. No quantitative result changes. See ERRATA.md and the paper's note added in revision. The cutoff sensitivity was independently identified by B. W. A. Silva, consistent with the naturally even, positive ground state reported by R. Andrews; the investigation was prompted by A. Connes.
The Connesâvan Suijlekom truncated Weil quadratic form, indexed by a cutoff parameter c that controls the primes p †c entering the operator, produces a ground state whose FourierâMellin zeros provably lie on the critical line; whether they converge to the Riemann zeros as c â â is open (Connes 2026; ConnesâConsaniâMoscovici 2025). We present, to our knowledge, the first independent public implementation of the Connesâvan Suijlekom Galerkin matrix at sixteen cutoffs (c = 13 through 67, plus c = 100). Across the in-sample window c = 13 through c = 67 at N = 100, the first-zero absolute error |Îł1 â Îł1Riemann| shrinks monotonically from âŒ2Ă10â55 to âŒ1.5Ă10â168 â a 113-OOM convergence across fifteen cutoffs. The smallest-positive even-sector eigenvalue λmineven separately reaches âŒ10â334 at c = 100, N = 250 (275-OOM span from c = 13). Out-of-sample test at c = 100. On the four-point N-sweep N â {100, 150, 200, 250} at dps = 500, consecutive first-difference ratios 0.837 and 0.836 match to two decimal places. Aitken-Î2 on the two overlapping triples yields log10|λâeven| â â536.8 and â â533.7, approaching the Connes 2026 §6.4 heuristic prediction (â â530.4) monotonically with N (6.4 and 3.3 OOM gaps out of |xâ| ⌠530). The same eigenvector recovers Îł1, âŠ, Îł10 to 307â329 matching digits at N = 250, dps = 500. Under the unitary equivalence with ConnesâConsaniâMoscovici Lemma 5.1, this is the deepest such Galerkin-truncation recovery in the public Connesâvan Suijlekom / ConnesâConsaniâMoscovici literature, subject to a hypothesis-status caveat: the raw finite-N matrix carries a small block of dps-stable negative-sign eigenvalues, so we report the smallest-positive branch (continuum positivity of QWλ is RH-equivalent and is not assumed at λ = â100). The fit |log10 λmin| â 13.24 c0.634 on c †67 at N = 100 is shown to be a finite-N rate, falsified at c = 100, N = 200 by 49 OOM in the direction of faster decay. Structural observations include approximate eigenvector c-invariance (overlap â„ 0.9498 on all 105 cutoff pairs despite eigenvalues differing by 113 OOM), multi-zero convergence universality (all ten detectable zeros within 3.8% of each other), an empirical Galerkin-convergence exponent s(c) â 55 log c â 128, un-rescaled Galerkin bulk-spectrum Poisson statistics (ÎČ < 0.05; this is a structural diagnostic of the truncated operator, not a test of Montgomery's conjecture, which applies to locally-rescaled zero spacings), and tight bulk invariants log|det Qc| â â65.6 c + 542 (R2 = 0.997). We make no claim of proof; the contribution is reproducible numerical data and its careful interpretation under the existing CvS / CCM framework. All code, data, and ancillary files are publicly available.
Let $Ο$ 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 $Ï(x)=\sum_{j=0}^k ÎČ_j x^j, ÎČ_k\neq 0$, of degree $k\in \mathbb N$, we consider the partial sums $Y_{N,Ï}=\sum_{n=1}^N Ï(Ο(n,S_n)).$ We prove a Wasserstein bound of order $N^{-1/2}$ for the normal approximation of the standardized $Y_{N,Ï}$. 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 $Ο$ 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 (Ï = 0.725) characterized by statistically significant off-diagonal excitation. The strongest cross-protocol channel runs from Morpho to Compound V3 (branching ratio Î = 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.
With the increasing adoption of the Proof of Stake (PoS) blockchain, it is timely to study the economy created by such blockchain. In this chapter, we will survey recent progress on the trading and wealth evolution in a cryptocurrency where the new coins are issued according to the PoS protocol. We first consider the wealth evolution in the PoS protocol assuming no trading, and focus on the problem of decentralisation. Next we consider each miner's trading incentive and strategy through the lens of optimal control, where the miner needs to trade off PoS mining and trading. Finally, we study the collective behavior of the miners in a PoS trading environment by a mean field model. We use both stochastic and analytic tools in our study. A list of open problems are also presented.
Rowel GĂŒndlach, Martijn Gijsbers, David Koops, Jacques Resing
We study the distribution of confirmation times of Bitcoin transactions, conditional on the size of the current memory pool. We argue that the time until a Bitcoin transaction is confirmed resembles the time to ruin in a corresponding Cramer-Lundberg process. This well-studied model gives mathematical insights in the mempool behaviour over time. Specifically, for situations where one chooses a fee, such that the total size of incoming transactions with higher fee is close to the total size of transactions leaving the mempool (heavy traffic), a diffusion approximation leads to an inverse Gaussian distribution for the confirmation times. The results of this paper are particularly interesting for users that want to make a Bitcoin transaction during heavy-traffic situations, as evaluation of the well-known inverse Gaussian distribution is computationally straightforward.
Abstract Cryptocurrency is one of the earliest and the most successful applications of blockchain, and it utilizes the distributed ledger, which is a commonly used technique in blockchain, to make a decentralized transaction within the blockchain of a cryptocurrency. However, how to make a decentralized transaction of cryptocurrencies between parties on different blockchains, that is, the crossâchain exchange, is not wellâstudied. In this paper, we develop a new method to make crossâchain exchanges based on the classical atomic swap. We first study the optionality embedded into the atomic swap and propose to add a premium into the atomic swap, and then design a new procedure with the premium to guarantee the fairness of the crossâchain exchange. We also provide an algorithm based on the leastâsquares Monte Carlo method to estimate the premium and analyze the convergence of the algorithm. Moreover, we study the crossâchain exchange with margin trading. We propose an adapted exchange procedure to make a fair crossâchain exchange and an algorithm to estimate the fair premium under the margin trading. Numerical experiments are provided to show the effectiveness of the algorithms.
We obtain lower tail estimates for the smallest singular value of random\nmatrices with independent but non-identically distributed entries.\nSpecifically, we consider $n\\times n$ matrices with complex entries of the form\n\\[ M = A\\circ X + B = (a_{ij}\\xi_{ij} + b_{ij}) \\] where $X=(\\xi_{ij})$ has iid\ncentered entries of unit variance and $A$ and $B$ are fixed matrices. In our\nmain result we obtain polynomial bounds on the smallest singular value of $M$\nfor the case that $A$ has bounded (possibly zero) entries, and $B= Z\\sqrt{n}$\nwhere $Z$ is a diagonal matrix with entries bounded away from zero. As a\nbyproduct of our methods we can also handle general perturbations $B$ under\nadditional hypotheses on $A$, which translate to connectivity hypotheses on an\nassociated graph. In particular, we extend a result of Rudelson and Zeitouni\nfor Gaussian matrices to allow for general entry distributions satisfying some\nmoment hypotheses. Our proofs make use of tools which (to our knowledge) were\npreviously unexploited in random matrix theory, in particular Szemer\\'edi's\nRegularity Lemma, and a version of the Restricted Invertibility Theorem due to\nSpielman and Srivastava.\n
We investigate the mean number of real zeros over an interval $[a,b]$ of a random trigonometric polynomial of the form $\sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt)$ where the coefficients are i.i.d. random variables. Under mild assumptions on the law of the entries, we prove that this mean number is asymptotically equivalent to $\frac{n(b-a)}{Ï\sqrt{3}}$ as $n$ goes to infinity, as in the known case of standard Gaussian coefficients. Our principal requirement is a new Cramer type condition on the characteristic function of the entries which does not only hold for all continuous distributions but also for discrete ones in a generic sense. To our knowledge, this constitutes the first universality result concerning the mean number of zeros of random trigonometric polynomials. Besides, this is also the first time that one makes use of the celebrated Kac-Rice formula not only for continuous random variables as it was the case so far, but also for discrete ones. Beyond the proof of a non asymptotic version of Kac-Rice formula, our strategy consists in using suitable small ball estimates and Edgeworth expansions for the Kolmogorov metric under our new weak Cramer condition, which both constitute important byproducts of our approach.
Johannes Göbel, Paul Keeler, A. E. Krzesinski, Peter Taylor
In the context of the `selfish-mine' strategy proposed by Eyal and Sirer, we study the effect of propagation delay on the evolution of the Bitcoin blockchain. First, we use a simplified Markov model that tracks the contrasting states of belief about the blockchain of a small pool of miners and the `rest of the community' to establish that the use of block-hiding strategies, such as selfish-mine, causes the rate of production of orphan blocks to increase. Then we use a spatial Poisson process model to study values of Eyal and Sirer's parameter $Îł$, which denotes the proportion of the honest community that mine on a previously-secret block released by the pool in response to the mining of a block by the honest community. Finally, we use discrete-event simulation to study the behaviour of a network of Bitcoin miners, a proportion of which is colluding in using the selfish-mine strategy, under the assumption that there is a propagation delay in the communication of information between miners.
This thesis is concerned with stochastic perturbation theory of the symmetric eigen-value problem. In particular, we provide results about the probability of interchanges in the ordering of the eigenvalues and changes in the eigenvectors of symmetric matrices subject to stochastic perturbations. In this analysis we use a novel combination of traditional Numerical Linear Algebra, Perturbation Theory and Probability Theory. The motivation for this study arises from reliability of spectral clustering of networks, when network data is subject to noise. As far as we are aware, there is nothing comparable in the literature. Further, we make conjectures from which we derive an asymptotic relation between the distributions of the largest eigenvalue and the 2-norm of random symmetric ma- trices, whose entries above the main diagonal are independent, identically distributed random variables with probability density functions being symmetric with respect to zero, including matrices from the Gaussian Orthogonal Ensemble (GOE). As far as we know, some of these conjectures are not new (possibly only as conjectures) but we are not aware of any proofs. Also, we consider networks of coupled oscillators. In their analysis we use both, knowledge of dynamical systems and spectral properties of non-negative matrices. As a result, we present an algorithm, which uncovers the \\master-slave" structure of the network. With its help, the analysis of the dynamics and the entrainment of the entire network can be reduced to considering only few of the oscillators, those whose dynamics determine the behaviour of the rest. This can be helpful in large networks exhibiting the \\master-slave" structure. Finally, we consider similarities of spectral clustering with respect to diÂźerent matrices which can be associated with a given network. In particular, we compare clustering of products of Path graphs with respect to two diÂźerent matrices: the Laplacian and the Normalised Laplacian matrices of the graph. We make the comparison by constructing a Homotopy between two eigenvalue problems and, using some Linear Algebra techniques, we show that the two matrices give similar spectral clusterings when applied to products of Path graphs.
A recent theorem of Orey [12] (see also [1], [6], [7], [13]) asserts that if $T$ is an $L_1$ operator induced on a discrete measure space by an irreducible recurrent aperiodic Markov matrix, then the condition (C) holds: $f \epsilon L_1, \int f = 0$ implies that $T^n f$ converges to zero in $L_1$. In an attempt to determine when (C) holds for more general operators, we at first prove the following (Theorem 1.1): Let $T$ be a positive linear contraction operator on $L_1$; if $T^nf$ and $T^{n+1}f$ intersect slightly, but uniformly in $f$ in the unit sphere of $L_1$, then $T^nf - T^{n+1}f$ converges to zero in norm. (C) follows if $T$ is conservative and ergodic (Corollary 1.3). In Section 2 we derive from this a simple proof of Orey's theorem. The main result of the paper is in Section 3 and could be called a "zero-two" theorem: Let $P(x, A)$ be a Markov kernel, and assume that there is a $\sigma$-finite measure $m$ such that for each $A, m(A) = 0$ implies $P(x, A) = 0$ a.e. and $m(A) > 0$ implies $\sum^\infty_{n=0} P^{(n)}(x, A) = \infty$ a.e. Then the total variation of the measure $P^{(n)}(x, \cdot) - P^{(n+1)}(x, \cdot)$ is either a.e. 2 for all $n$ or it converges a.e. to 0 as $n \rightarrow \infty$. In Section 4 it is shown that a version of the zero-two theorem essentially contains the Jamison-Orey generalization of Orey's theorem to Harris processes. Section 1 and Section 2 of this paper do not assume any knowledge of either operator ergodic theory or probability. Some known results in ergodic theory are applied in Section 3, but the proof of the main theorem does not depend on them.