We start with a random polynomial PN(z) of degree N with independent coefficients. We then consider a new polynomial P-t(N) obtained by [Nt1 applications of a fractional differential operator of the form za(d/dz)b, where a and b are real numbers. When b > 0, we compute the limiting root distribution & micro;t of P-t(N) as N -> infinity. We show that & micro;t is the push-forward of the limiting root distribution of PN under a transport map Tt. The map Tt is defined by flowing along the characteristic curves of a partial differential equation satisfied by the log potential of & micro;t. In the special case of repeated differentiation, our results may be interpreted as saying that the roots evolve radially with constant speed until they hit the origin, at which point, they cease to exist. For general a and b, the transport map Tt has a free probability interpretation as multiplication of an R-diagonal operator by an R-diagonal "transport operator." As an application, we obtain a push-forward characterization of the free self-convolution semigroup W of radial measures on C. We also consider the case b < 0, which includes the case of repeated integration. More complicated behavior of the roots can occur in this case.

We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial P(z), we prove that the empirical zero distribution of its heat-evolved n-th power converges to a distribution on the complex plane as n tends to infinity. We describe this limit distribution μ_t as a function of the time parameter t of the heat evolution: For small time, zeros start to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a widespread semicircle law through the initial center of mass. The Stieltjes transform of the limit distribution μ_t satisfies a self-consistent equation and a Burgers' equation. The present paper deals with general complex-rooted polynomials for which, in contrast to the real-rooted case, no free-probabilistic representation for μ_t is available.

In this work, we present a complete characterization of the covariance structure of number statistics in boxes for hyperuniform point processes. Under a standard integrability assumption, the covariance depends solely on the overlap of the faces of the box. Beyond this assumption, a novel interpolating covariance structure emerges. This enables us to identify a limiting Gaussian “coarse-grained” process, counting the number of points in large boxes as a function of the box position. Depending on the integrability assumption, this process may be continuous or discontinuous, e.g. in d=1 it is given by an increment process of a fractional Brownian motion.

We study a block mean-field Ising model with N spins split into s_N blocks, with Curie-Weiss interaction within blocks and nearest-neighbor coupling between blocks. While previous models deal with the block magnetization for a fixed number of blocks, we study the the simultaneous limit N→∞ and s_N→∞. The model interpolates between Curie-Weiss model for s_N=1, multi-species mean field for fixed s_N=s, and the 1D Ising model for each spin in its own block at s_N=N. Under mild growth conditions on s_N, we prove a law of large numbers and a multivariate CLT with covariance given by the lattice Green's function. For instance, the high temperature CLT essentially covers the optimal range up to s_N=o(N/(log N)^c) and the low temperature regime is new even for fixed number of blocks s > 2. In addition to the standard competition between entropy and energy, a new obstacle in the proofs is a curse of dimensionality as s_N →∞.

We establish basic properties of the heat flow on entire holomorphic functions that have order at most 2. We then look specifically at the action of the heat flow on the Gaussian analytic function (GAF). We show that applying the heat flow to a GAF and then rescaling and multiplying by an exponential of a quadratic function gives another GAF. It follows that the zeros of the GAF are invariant in distribution under the heat flow, up to a simple rescaling. We then show that the zeros of the GAF evolve under the heat flow approximately along straight lines, with an error whose distribution is independent of the starting point. Finally, we connect the heat flow on the GAF to the metaplectic representation of the double cover of the group $SL(2;\mathbb{R}).$

We develop a theory of optimal transport for stationary random measures with a focus on stationary point processes and construct a family of distances on the set of stationary random measures. These induce a natural notion of interpolation between two stationary random measures along a shortest curve connecting them. In the setting of stationary point processes we leverage this transport distance to give a geometric interpretation for the evolution of infinite particle systems with stationary distribution. Namely, we characterise the evolution of infinitely many Brownian motions as the gradient flow of the specific relative entropy w.r.t.~the Poisson point process. Further, we establish displacement convexity of the specific relative entropy along optimal interpolations of point processes and establish an stationary analogue of the HWI inequality, relating specific entropy, transport distance, and a specific relative Fisher information.

We investigate the evolution of the empirical distribution of the complex roots of high-degree random polynomials, when the polynomial undergoes the heat flow. In one prominent example of Weyl polynomials, the limiting zero distribution evolves from the circular law into the elliptic law until it collapses to the Wigner semicircle law, as was recently conjectured for characteristic polynomials of random matrices by Hall and Ho, 2022. Moreover, for a general family of random polynomials with independent coefficients and isotropic limiting distribution of zeros, we determine the zero distribution of the heat-evolved polynomials in terms of its logarithmic potential. Furthermore, we explicitly identify two critical time thresholds, at which singularities develop and at which the limiting distribution collapses to the semicircle law. We completely characterize the limiting root distribution of the heat-evolved polynomials before singularities develop as the push-forward of the initial distribution under a transport map. Finally, we discuss the results from the perspectives of partial differential equations (in particular Hamilton-Jacobi equation and Burgers' equation), optimal transport, and free probability. The theory is accompanied by explicit examples, simulations, and conjectures.

Given a sequence of polynomials (P_n)_n ∈ℕ with only nonpositive zeros, the aim of this article is to present a user-friendly approach for determining the limiting zero distribution of P_n as deg P_n →∞. The method is based on establishing an equivalence between the existence of a limiting empirical zero distribution μ and the existence of an exponential profile g associated with the coefficients of the polynomials (P_n)_n ∈ℕ. The exponential profile g, which can be roughly described by [z^k]P_n(z) ≈exp(n g(k/n)), offers a direct route to computing the Cauchy transform G of μ: the functions t ↦ tG(t) and α↦exp(-g'(α)) are mutual inverses. This relationship, in various forms, has previously appeared in the literature, most notably in the paper [Van Assche, Fano and Ortolani, SIAM J. Math. Anal., 1987]. As a first contribution, we present a self-contained probabilistic proof of this equivalence by representing the polynomials as generating functions of sums of independent Bernoulli random variables. This probabilistic framework naturally lends itself to tools from large deviation theory, such as the exponential change of measure. The resulting theorems generalize and unify a range of previously known results, which were traditionally established through analytic or combinatorial methods. Secondly, using the profile-based approach, we investigate how the exponential profile and the limiting zero distribution behave under certain operations on polynomials, including finite free convolutions, Hadamard products, and repeated differentiation. In particular, our approach yields new proofs of the convergence results `⊞_n →⊞' and `⊠_n →⊠', extending them to cases where the distributions are not necessarily compactly supported.

We compare a mean-field Gibbs distribution on a finite state space on N spins to that of an explicit simple mixture of product measures. This illustrates the situation beyond the so-called increasing propagation of chaos introduced by Ben Arous and Zeitouni [3], where marginal distributions of size k=o(N) are compared to product measures.

In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to ∞. In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little q-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices.

We investigate the Wasserstein distance between the empirical spectral distribution of non-Hermitian random matrices and the Circular Law. For general entry distributions, we obtain a nearly optimal rate of convergence in 1-Wasserstein distance of order $n^{-1/2+\epsilon}$ and we prove that the optimal rate $n^{-1/2}$ is attained by Ginibre matrices. This shows that the expected transport cost of complex eigenvalues to the uniform measure on the unit disk decays faster compared to that of i.i.d. points, which is known to include a logarithmic factor.

We investigate increasing propagation of chaos for the mean-field Ising model of ferromagnetism (also known as the Curie-Weiss model) with $N$ spins at inverse temperature $\beta>0$ and subject to an external magnetic field of strength $h\in\mathbb{R}$. Using a different proof technique than in [Ben Arous, Zeitouni; 1999] we confirm the well-known propagation of chaos phenomenon: If $k=k(N)=o(N)$ as $N\to\infty$, then the $k$'th marginal distribution of the Gibbs measure converges to a product measure at $\beta <1$ or $h \neq 0$ and to a mixture of two product measures, if $\beta >1$ and $h =0$. More importantly, we also show that if $k(N)/N\to \alpha\in (0,1]$, this property is lost and we identify a non-zero limit of the total variation distance between the number of positive spins among any $k$-tuple and the corresponding binomial distribution.

In this note we study the block spin mean-field Potts model, in which the spins are divided into s blocks and can take $$q\ge 2$$ different values (colors). Each block is allowed to contain a different proportion of vertices and behaves itself like a mean-field Ising/Potts model which also interacts with other blocks according to different temperatures. Of particular interest is the behavior of the magnetization, which counts the number of colors appearing in the distinct blocks. We prove central limit theorems for the magnetization in the generalized high-temperature regime and provide a moderate deviation principle for its fluctuations on lower scalings. More precisely, the magnetization concentrates around the uniform vector of all colors with an explicit, but singular, Gaussian distribution. In order to remove the singular component, we will also consider a rotated magnetization, which enables us to compare our results to various related models.

We consider one infinite path of a Random Walk in Random Environment (RWRE, for short) in an unknown environment. This environment consists of either i.i.d. site or bond randomness. At each position the random walker stops and tells us the environment it sees at the point where it is, without telling us, where it is. These observations chi' are spoiled by reading errors that occur with probability p < 1. We show: If the RWRE is recurrent and satisfies the standard assumptions on such RWREs, then with probability one in the environment, the errors, and the random walk we are able reconstruct the law of the environment. For most situations this result is even independent of the value of p. If the distribution of the environment has a non-atomic part, we can even reconstruct the environment itself, up to translation.

The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by [Formula: see text]. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.

We study the rate of convergence of the empirical spectral distribution of products of independent non-Hermitian random matrices to the power of the Circular Law. The distance to the deterministic limit distribution will be measured in terms of a uniform Kolmogorov-like distance. First, we prove that for products of Ginibre matrices, the optimal rate is given by $\mathcal O (1/\sqrt n)$, which is attained with overwhelming probability up to a logarithmic correction. Avoiding the edge, the rate of convergence of the mean empirical spectral distribution is even faster. Second, we show that also products of matrices with independent entries attain this optimal rate in the bulk up to a logarithmic factor. In the case of Ginibre matrices, we apply a saddlepoint approximation to a double contour integral representation of the density and in the case of matrices with independent entries we make use of techniques from local laws.