Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. Whereas large deviation principles identify the asymptotic exponential decay rate of probabilities, sharp large deviation estimates also provide the “prefactor” in front of the exponentially decaying term. For fixed $p\in (1,\mathrm{\infty })$, consider independent sequences ${(X^{(n,p)})}_{n\in \mathbb{N}}$ and ${({\mathrm{\Theta }}^n)}_{n\in \mathbb{N}}$ of random vectors with ${\mathrm{\Theta }}^n$ distributed according to the normalized cone measure on the unit ${\ell }_2^n$ sphere, and $X^{(n,p)}$ distributed according to the normalized cone measure on the unit ${\ell }_p^n$ sphere. For almost every realization ${({\mathit{\theta }}^n)}_{n\in \mathbb{N}}$ of ${({\mathrm{\Theta }}^n)}_{n\in \mathbb{N}}$, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of $X^{(n,p)}$ onto ${\mathit{\theta }}^n$, that are asymptotically exact (as the dimension n tends to infinity). Furthermore, the case when ${(X^{(n,p)})}_{n\in \mathbb{N}}$ is replaced with ${(X^{(n,p)})}_{n\in \mathbb{N}}$, where $X^{(n,p)}$ is distributed according to the uniform (or normalized volume) measure on the unit ${\ell }_p^n$ ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions ${({\mathit{\theta }}^n)}_{n\in \mathbb{N}}$ that encodes additional geometric information that enables one to distinguish between projections of balls and spheres. Moreover, comparison with numerical estimates obtained by direct computation and importance sampling shows that the obtained analytical expressions for tail probabilities provide good approximations even for moderate values of n. The results on the one hand provide more accurate quantitative estimates of tail probabilities of random projections of ${\ell }_p^n$ spheres than logarithmic asymptotics, and on the other hand, generalize classical sharp large deviation estimates in the spirit of Bahadur and Ranga Rao to a geometric setting. The proofs combine Fourier analytic and probabilistic techniques. Along the way, several results of independent interest are obtained including a simpler representation for the quenched large deviation rate function that shows that it is strictly convex, a central limit theorem for random projections under a certain family of tilted measures, and multi-dimensional generalized Laplace asymptotics.

A Hawkes process on $\mathbb{R}$ is a point process whose intensity function at time t is a functional of its past activity before time t. It is defined by its activation function Φ and its memory function h. In this paper, the Hawkes property is expressed as an operator on the sub-space of non-negative sequences associated to distances between its points. By using the classical correspondence between a stationary point process and its Palm measure, we establish a characterization of the corresponding Palm measure as an invariant distribution of a Markovian kernel. We prove that if Φ is continuous and its growth rate is at most linear with a rate below some constant, then there exists a stationary Hawkes point process. The classical Lipschitz condition of the literature for an unbounded function Φ is relaxed. Our proofs rely on a combination of coupling methods, monotonicity properties of linear Hawkes processes and classical results on Palm distributions. An investigation of the Hawkes process starting from the null measure, the empty state, on ${\mathbb{R}}_{-}$ plays also an important role. The linear case of Hawkes and Oakes is revisited at this occasion.

If the memory function h is an exponential function, under a weak condition it is shown that there exists a unique stationary Hawkes point process. In this case, its Palm measure is expressed in terms of the invariant distribution of a one-dimensional Harris ergodic Markov chain. When the activation function is a polynomial Φ with degree $>1$, there does not exist a stationary Hawkes process and if the Hawkes process starts from the empty state, a scaling result for the accumulation of its points is obtained.

A Choquet-type integral representation result for non-negative subharmonic functions of a one-dimensional regular diffusion is established. The representation allows in particular an integral equation for strictly positive subharmonic functions that is driven by the Revuz measure of the associated continuous additive functional. Moreover, via the aforementioned integral equation, one can construct an Itô-Watanabe pair $(g,A)$ that consist of a subharmonic function g and a continuous additive functional A is with Revuz measure ${\mathrm{\mu }}_A$ such that $g(X)exp(-A)$ is a local martingale. Changes of measures associated with Itô-Watanabe pairs are studied and shown to modify the long term behaviour of the original diffusion process to exhibit transience.

We consider the Hamiltonians of mean-field spin glasses, which are certain random functions $H_N$ defined on high-dimensional cubes or spheres in ${\mathbb{R}}^N$. The asymptotic maximum values of these functions were famously obtained by Talagrand and later by Panchenko and by Chen. The landscape of approximate maxima of $H_N$ is described by various forms of replica symmetry breaking exhibiting a broad range of behaviors. We study the problem of efficiently computing an approximate maximizer of $H_N$.

We give a two-phase message passing algorithm to approximately maximize $H_N$ when a no overlap gap condition holds. This generalizes the recent works [Sub21, Mon19, AMS21] by allowing a non-trivial external field. For even Ising spin glasses with constant external field, our algorithm succeeds exactly when existing methods fail to rule out approximate maximization for a wide class of algorithms. Moreover we give a branching variant of our algorithm which constructs a full ultrametric tree of approximate maxima.

Let $F(\mathit{\sigma })={\sum }_{n=1}^{\mathrm{\infty }}\frac{X_n}{n^{\mathit{\sigma }}}$ be a random Dirichlet series where ${(X_n)}_{n\in \mathbb{N}}$ are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of $F(\mathit{\sigma })$ in the interval $[T,\mathrm{\infty })$, say $\mathbb{E}N(T,\mathrm{\infty })$, as $T\to 1∕2^{+}$. We also estimate higher moments and with this we derive exponential tails for the probability that the number of zeros in the interval $[T,1]$, say $N(T,1)$, is large. We also consider almost sure lower and upper bounds for $N(T,\mathrm{\infty })$. And finally, we also prove results for another class of random Dirichlet series, e.g., when the summation is restricted to prime numbers.

In this paper, we establish the Hausdorff dimensions of inverse images and collision time sets for a large class of symmetric Markov processes on metric measure spaces. We apply the approach in the works by Hawkes and Jain–Pruitt, and make full use of heat kernel estimates. In particular, the results efficiently apply to symmetric diffusion processes, symmetric stable-like processes, and symmetric diffusion processes with jumps in d-sets.

We study the quenched behaviour of a perturbed version of the simple symmetric random walk on the set of integers. The random walker moves symmetrically with an exception of some randomly chosen sites where we impose a random drift. We show that if the gaps between the marked sites are i.i.d. and regularly varying with a sufficiently small index, then there is no strong quenched limit laws for the position of the random walker. As a consequence we study the quenched limit laws in the context of weak convergence of random measures.

We introduce a model of Poisson random waves in ${\mathbb{S}}^2$ and we study Quantitative Central Limit Theorems when both the rate of the Poisson process and the energy (i.e., frequency) of the waves (eigenfunctions) diverge to infinity. We consider finite-dimensional distributions, harmonic coefficients and convergence in law in functional spaces, and we investigate carefully the interplay between the rate of divergence of eigenvalues and Poisson governing measures.

We study the Langevin dynamics corresponding to the $\nabla \mathit{\varphi }$ (or Ginzburg-Landau) interface model with a uniformly convex interaction potential. We interpret these Langevin dynamics as a nonlinear parabolic equation forced by white noise, which turns the problem into a nonlinear homogenization problem. Using quantitative homogenization methods, we prove a quantitative hydrodynamic limit, obtain the $C^2$ regularity of the surface tension, prove a large-scale $C^{1,\mathit{\alpha }}$-type estimate for the trajectories of the dynamics, and show that the fluctuation-dissipation relation can be seen as a commutativity of homogenization and linearization. Finally, we explain why we believe our techniques can be adapted to the setting of degenerate (non-uniformly) convex interaction potentials.

We consider first passage percolation on certain isotropic random graphs in ${\mathbb{R}}^d$. We assume exponential concentration of passage times $T(x,y)$, on some scale $\mathit{\sigma }(r)$ whenever $|y-x|$ is of order r, with $\mathit{\sigma }(r)$ “growning like $r^{\mathit{\chi }}$” for some $0<\mathit{\chi }<1$. Heuristically this means transverse wandering of geodesics should be at most of order $\mathrm{\Delta }(r)={(r\mathit{\sigma }(r))}^{1∕2}$. We show that in fact uniform versions of exponential concentration and wandering bounds hold: except with probability exponentially small in t, there are no $x,y$ in a natural cylinder of length r and radius $K\mathrm{\Delta }(r)$ for which either (i) $|T(x,y)-ET(x,y)|\ge t\mathit{\sigma }(r)$, or (ii) the geodesic from x to y wanders more than distance $\sqrt{t}\mathrm{\Delta }(r)$ from the cylinder axis. We also establish that for the time constant $\mathrm{\mu }={lim}_{n}ET(0,n{e}_1)∕n$, the “nonrandom error” $ET(0,x)-\mathrm{\mu }|x|$ is at most a constant multiple of $\mathit{\sigma }(|x|)$.

For any integer $m<n$, where m can depend on n, we study the rate of convergence of $\frac{1}{\sqrt{m}}Tr{\mathbf{U}}^m$ to its limiting Gaussian as $n\to \mathrm{\infty }$ for orthogonal, unitary and symplectic Haar distributed random matrices U of size n. In the unitary case, we prove that the total variation distance is less than $\mathrm{\Gamma }{(\lfloor n∕m\rfloor +2)}^{-1}{m}^{-\lfloor n∕m\rfloor }{\lfloor n∕m\rfloor }^{1∕4}\sqrt{logn}$ times a constant. This result interpolates between the super-exponential bound obtained for fixed m and the $1∕n$ bound coming from the Berry–Esseen theorem applicable when $m\ge n$ by a result of Rains. We obtain analogous results for the orthogonal and symplectic groups. In these cases, our total variation upper bound takes the form $\mathrm{\Gamma }{(2\lfloor n∕m\rfloor +1)}^{-1∕2}{m}^{-\lfloor n∕m\rfloor +1}{(logn)}^{1∕4}$ times a constant and the result holds provided $n>2m$. For $m=1$, we obtain complementary lower bounds and precise asymptotics for the $L^2$-distances as $n\to \mathrm{\infty }$, which show how sharp our results are.

In this work, we study general Dirichlet coalescents, which are a family of Ξ-coalecents constructed from i.i.d mass partitions, and are an extension of the symmetric coalescent. This class of models is motivated by population models with recurrent demographic bottlenecks. We study the short time behavior of the multidimensional block counting process whose ith component counts the number of blocks of size i. Compared to standard coalescent models (such as the class of Λ-coalescents coming down from infinity), our process has no deterministic speed of coming down from infinity. In particular, we prove that, under appropriate re-scaling, it converges to a stochastic process which is the unique solution of a martingale problem. We show that the multivariate Lamperti transform of this limiting process is a Markov Additive Process (MAP). This allows us to provide some asymptotics for the n-Site Frequency Spectrum, which is a statistic widely used in population genetics. In particular, the rescaled number of mutations converges to the exponential functional of a subordinator.

We consider the deviation inequalities for the sums of independent d by d random matrices, as well as rank one random tensors. Our focus is on the non-isotropic case and the bounds that do not depend explicitly on the dimension d, but rather on the effective rank. In an elementary and unified manner, we show the following results:

Our approach is based on the duality formula between entropy and moment generating functions. In contrast to the known proofs of dimension-free bounds, we avoid Talagrand’s majorizing measure theorem, as well as generic chaining bounds for empirical processes. Some of our tools were pioneered by O. Catoni and co-authors in the context of robust statistical estimation.

This work studies the averaging principle for a fully coupled two time-scale system, whose slow process is a diffusion process and fast process is a purely jumping process on an infinitely countable state space. The ergodicity of the fast process has important impact on the limit system and the averaging principle. We show that under strongly ergodic condition, the limit system admits a unique solution, and the slow process converges in the $L^1$-norm to the limit system. However, under certain weaker ergodicity condition, the limit system admits a solution, but not necessarily unique, and the slow process can be proved to converge weakly to a solution of the limit system.

In [10], Cerf and Gorny constructed a model of self-organized criticality, by introducing an automatic control of the temperature parameter in the generalized Ising Curie-Weiss model. The fluctuations of the magnetization of this spin model are of order $n^{3∕4}$ with a limiting law of the form $Cexp(-x^4)$, as in the critical regime of the Curie-Weiss model.

In this article, we build upon this model by replacing the mean-field interaction with a one-dimensional interaction with a certain range $r_n$ which varies as a function of the number n of particles. In the Gaussian case, we show that the self-critical behaviour observed in the mean-field case extends to interaction ranges $r_n\gg n^{3∕4}$ and we show that this threshold is sharp, with different fluctuations when the interaction range is of order of $n^{3∕4}$ or smaller than $n^{3∕4}$.

We prove a probabilistic generalization of the classic result that infinite power towers, $c^{c^{⋰}}$, converge if and only if $c\in [e^{-e},e^{1∕e}]$. Given an i.i.d. sequence ${\{A_i\}}_{i\in \mathbb{N}}$, we find that convergence of the power tower $A_1^{A_2^{⋰}}$ is determined by the bounds of $A_1$’s support, $a=inf(\mathrm{supp}(A_1))$ and $b=sup(\mathrm{supp}(A_1))$. When $b\in [e^{-e},e^{1∕e}]$, $a<1<b$, or $a=0$, the power tower converges almost surely. When $b<e^{-e}$, we define a special function B such that almost sure convergence is equivalent to $a<B(b)$. Only in the case when $a=1$ and $b>e^{1∕e}$ are the values of a and b insufficient to determine convergence. We show a rather complicated necessary and sufficient condition for convergence when $a=1$ and b is finite.

We also briefly discuss the relationship between the distribution of $A_1$ and the corresponding power tower $T=A_1^{A_2^{⋰}}$. For example, when $T\sim \mathrm{Unif}[0,1]$, then the corresponding distribution of $A_1$ is given by $UV$ where $U,V\sim \mathrm{Unif}[0,1]$ are independent. We generalize this example by showing that for $U\sim \mathrm{Unif}[\mathit{\alpha },\mathit{\beta }]$ and $r\in \mathbb{R}$, there exists an i.i.d. sequence ${\{A_i\}}_{i\in \mathbb{N}}$ such that $U^r\stackrel{d}{=}{A}_1^{A_2^{⋰}}$ if and only if $r\in [0,\frac{1}{1+log\mathit{\beta }}]$.

We study the kinetic mean field Langevin dynamics under the functional convexity assumption of the mean field energy functional. Using hypocoercivity, we first establish the exponential convergence of the mean field dynamics and then show the corresponding N-particle system converges exponentially in a rate uniform in N modulo a small error. Finally we study the short-time regularization effects of the dynamics and prove its uniform-in-time propagation of chaos property in both the Wasserstein and entropic sense. Our results can be applied to the training of two-layer neural networks with momentum and we include the numerical experiments.

We develop a variant of rough path theory tailor-made for analyzing a class of financial asset price models known as rough volatility models. As an application, we prove a pathwise large deviation principle (LDP) for a certain class of rough volatility models, which in turn describes the limiting behavior of implied volatility for short maturity under those models. First, we introduce a partial rough path space and an integration map on it and then investigate several fundamental properties including local Lipschitz continuity of the integration map from the partial rough path space to a rough path space. Second, we construct a rough path lift of a rough volatility model. Finally, we prove an LDP on the partial rough path space, and the LDP for rough volatility then follows by the continuity of the solution map of rough differential equations.

In the present article we consider the Schelling model, an agent-based model describing a segregation dynamics when we have a cohabitation of two social groups. As for several social models, the behaviour of the Schelling model was analyzed along several directions, notably by exploiting theoretical physics tools and computer simulations. This approach led to conjecture a phase diagram in which either different social groups were segregated in two large clusters or they were mixed. In this article, we describe and analyze a perturbation of the Schelling model as a particle systems model by adding a Glauber and Kawasaki dynamics to the original Schelling dynamics. As far as the authors know, this is the first rigorous mathematical analysis of the perturbed Schelling model. We prove the existence of an hydrodynamic limit described by a reaction-diffusion equation with a discontinuous non-linear reaction term. The existence and uniqueness of the solution is non trivial and the analysis of the limit PDE is interesting in its own. Based on our results, we conjecture, as in other variations of this model, the existence of a phase diagram in which we have a mixed, a segregated and a metastable segregation phase. We also describe how this phase transition can be viewed as a transition between a relevant and irrelevant disorder regime in the model.

Stein’s method for Gaussian process approximation can be used to bound the differences between the expectations of smooth functionals h of a càdlàg random process X of interest and the expectations of the same functionals of a well understood target random process Z with continuous paths. Unfortunately, the class of smooth functionals for which this is easily possible is very restricted. Here, we provide an infinite dimensional Gaussian smoothing inequality, which enables the class of functionals to be greatly expanded — examples are Lipschitz functionals with respect to the uniform metric, and indicators of arbitrary events — in exchange for a loss of precision in the bounds. Our inequalities are expressed in terms of the smooth test function bound, an expectation of a functional of X that is closely related to classical tightness criteria, a similar expectation for Z, and, for the indicator of a set K, the probability $\mathbb{P}(Z\in K^{\mathit{\theta }}\setminus K^{-\mathit{\theta }})$ that the target process is close to the boundary of K.

We derive limit laws for the empirical spectral distributions of random band and block matrices with correlated entries. In the first part of the paper, we study band matrices with approximately uncorrelated entries. We strengthen previously obtained results while requiring weaker assumptions, which is made possible by a refined application of the method of moments. In the second part of the paper, we introduce a new two-layered correlation structure we call SSB-HKW correlated, which enables the study of structured random matrices with correlated entries. Our results include semicircle laws in probability and almost surely, but we also obtain other limiting spectral distributions depending on the conditions. Simple necessary and sufficient conditions for the limit law to be the semicircle are provided.

We study the characteristic polynomials of both the Gaussian Orthogonal and Symplectic Ensembles. We show that for both ensembles, powers of the absolute value of the characteristic polynomials converge in law to Gaussian multiplicative chaos measures after normalization for sufficiently small real powers. The main tool is a new asymptotic relation between the fractional moments of the absolute characteristic polynomials of Gaussian Orthogonal, Unitary, and Symplectic Ensembles.

Recently Ren et al. [Stoch. Proc. Appl., 137 (2021)] have proved that the extremal process of the super-Brownian motion converges in distribution in the limit of large times. Their techniques rely heavily on the study of the convergence of solutions to the Kolmogorov-Petrovsky-Piscounov equation along the lines of [M. Bramson, Mem. Amer. Math. Soc., 44 (1983)]. In this paper we take a different approach. Our approach is based on the skeleton decomposition of super-Brownian motion. The skeleton may be interpreted as immortal particles that determine the large time behaviour of the process. We exploit this fact and carry asymptotic properties from the skeleton over to the super-Brownian motion. Some new results concerning the probabilistic representations of the limiting process are obtained, which cannot be directly obtained through the results of [Y.-X. Ren et al., Stoch. Proc. Appl., 137 (2021)]. Apart from the results, our approach offers insights into the driving force behind the limiting process for super-Brownian motions.

We compute the limiting measure for the Feynman loop representation of the Bose gas for a non mean-field energy. As predicted in previous works, for high densities the limiting measure gives positive weight to random interlacements, indicating the quantum Bose–Einstein condensation. We prove that in many cases there is a shift in the critical density compared to the free/mean-field case, and that in these cases the density of the random interlacements has a jump-discontinuity at the critical point.

We prove a new result relating solutions of the scaled fractional Allen–Cahn equation to motion by mean curvature flow, motivated by the motion of hybrid zones in populations that exhibit long range dispersal. Our proof is purely probabilistic and takes inspiration from Etheridge et al. [30] to describe solutions of the fractional Allen–Cahn equation in terms of ternary branching α-stable motions. To overcome technical difficulties arising from the heavy-tailed nature of the stable distribution, we couple ternary branching stable motions to ternary branching Brownian motions subordinated by truncated stable subordinators.

The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. The average mixing time is defined to be the mixing time starting at a uniformly random vertex and hence is not sensitive to the slow diffusion caused by these bottlenecks.

In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order ${(logn)}^2$, speeding up the mixing to order $logn$. First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erdős-Rényi graph is logarithmic.

We study the number of connected components of non-Gaussian random spherical harmonics on the two dimensional sphere ${\mathbb{S}}^2$. We prove that the expectation of the nodal domains count is independent of the distribution of the coefficients provided it has a finite second moment.

We introduce a broad class of mechanistic spatial models to describe how spatially heterogeneous populations live, die, and reproduce. Individuals are represented by points of a point measure, whose birth and death rates can depend both on spatial position and local population density, defined at a location to be the convolution of the point measure with a suitable non-negative integrable kernel centred on that location. We pass to three different scaling limits: an interacting superprocess, a nonlocal partial differential equation (PDE), and a classical PDE. The classical PDE is obtained both by a two-step convergence argument, in which we first scale time and population size and pass to the nonlocal PDE, and then scale the kernel that determines local population density; and in the important special case in which the limit is a reaction-diffusion equation, directly by simultaneously scaling the kernel width, timescale and population size in our individual based model.

A novelty of our model is that we explicitly model a juvenile phase. The number of juveniles produced by an individual depends on local population density at the location of the parent; these juvenile offspring are thrown off in a (possibly heterogeneous, anisotropic) Gaussian distribution around the location of the parent; they then reach (instant) maturity with a probability that can depend on the local population density at the location at which they land. Although we only record mature individuals, a trace of this two-step description remains in our population models, resulting in novel limits in which the spatial dynamics are governed by a nonlinear diffusion.

Using a lookdown representation, we are able to retain information about genealogies relating individuals in our population and, in the case of deterministic limiting models, we use this to deduce the backwards in time motion of the ancestral lineage of an individual sampled from the population. We observe that knowing the history of the population density is not enough to determine the motion of ancestral lineages in our model. We also investigate (and contrast) the behaviour of lineages for three different deterministic models of a population expanding its range as a travelling wave: the Fisher-KPP equation, the Allen-Cahn equation, and a porous medium equation with logistic growth.

We give a new derivation of the finite N master loop equation for lattice Yang-Mills theory with structure group $SO(N)$, $U(N)$ or $SU(N)$. The $SO(N)$ case was initially proved by Chatterjee in [6], and $SU(N)$ was analyzed in a follow-up work by Jafarov [23]. Our approach is based on the Langevin dynamic, an SDE on the manifold of configurations, and yields a simple proof via Itô’s formula.

We consider the asymptotic behavior of the KPZ fixed point ${\{\mathsf{H}(x,t)\}}_{x\in \mathbb{R},t>0}$ conditioned on $\mathsf{H}(0,T)=L$ as L goes to infinity. The main result is a conditional limit theorem for the fluctuations of $\mathsf{H}$ in the region near the line segment connecting the origin $(0,0)$ and $(0,T)$ for both step and flat initial conditions. The limit random field can be represented as a functional of two independent Brownian bridges, and in addition the limit random field depends also on the initial law of the KPZ fixed point. In particular for temporal fluctuations, the limit process indexed by line segment between $(0,0)$ and $(0,T)$, when the KPZ is with step initial condition, has the law of the minimum of two independent Brownian bridges; and when the KPZ is with flat initial condition the limit process has the law of the minimum of two independent Brownian bridges, each in addition perturbed by a common Gaussian random variable. For spatial-temporal fluctuations, the conditional limit theorem sheds light on the asymptotic behaviors of the point-to-point geodesic of the directed landscape conditioned on its length and as the length tends to infinity.

We introduce a biologically natural, mathematically tractable model of random phylogenetic network to describe evolution in the presence of hybridization. One of the features of this model is that the hybridization rate of the lineages correlates negatively with their phylogenetic distance. We give formulas / characterizations for quantities of biological interest that make them straightforward to compute in practice. We show that the appropriately rescaled network, seen as a metric space, converges to the Brownian continuum random tree, and that the uniformly rooted network has a local weak limit, which we describe explicitly.

The Skorokhod reflection was used in 1961 to create a reflected diffusion on the half-line. Later, it was used for processes with jumps such as reflected Lévy processes. Like a Brownian motion, which is a weak limit of random walks, reflected processes on the half-line serve as weak limits of random walks with switching regimes at zero: one regime away from zero, the other around zero. In this article, we develop a general theory of this regime change and prove convergence to a function with generalized reflection. Our results are deterministic and can be applied to a wide class of stochastic processes. Applications include storage processes, heavy traffic limits, diffusion on a half-line with a combination of continuous reflection, jump exit, and a delay at 0.

In this article we prove existence of the asymptotic capacity of the range of random walks on free products of graphs. In particular, we will show that the asymptotic capacity of the range is almost surely constant and strictly positive. Furthermore, we provide a central limit theorem for the capacity of the range and show that it varies real-analytically in terms of finitely supported probability measures of constant support.

We consider the model of random planar maps of size n biased by a weight $u>0$ per 2-connected block, and the closely related model of random planar quadrangulations of size n biased by a weight $u>0$ per simple component. We exhibit a phase transition at the critical value $u_C=9∕5$. If $u<u_C$, a condensation phenomenon occurs: the largest block is of size $\mathrm{\Theta }(n)$. Moreover, for quadrangulations we show that the diameter is of order $n^{1∕4}$, and the scaling limit is the Brownian sphere. When $u>u_C$, the largest block is of size $\mathrm{\Theta }(log(n))$, the scaling order for distances is $n^{1∕2}$, and the scaling limit is the Brownian tree. Finally, for $u=u_C$, the largest block is of size $\mathrm{\Theta }(n^{2∕3})$, the scaling order for distances is $n^{1∕3}$, and the scaling limit is the stable tree of parameter $3∕2$.

We consider slowly time-dependent singular stochastic partial differential equations on the two-dimensional torus, driven by weak space-time white noise, and renormalised in the Wick sense. Our main results are concentration results on sample paths near stable equilibrium branches of the equation without noise, measured in appropriate Besov and Hölder norms. We also discuss a case involving a pitchfork bifurcation. These results extend to the two-dimensional torus those obtained in [4] for finite-dimensional SDEs, and in [9] for SPDEs on the one-dimensional torus.

For a Haar-distributed element H of a compact Lie group L, Eric Rains proved in [10] that there is a natural number $D=D_L$ such that, for all $d\ge D$, the eigenvalue distribution of $H^d$ is fixed, and Rains described this fixed eigenvalue distribution explicitly. In the present paper we consider random elements U of a compact Lie group with general distribution. In particular, we introduce a mild absolute continuity condition under which the eigenvalue distribution of powers of U converges to that of $H^D$. Then, rather than the eigenvalue distribution, we consider the limiting distribution of $U^d$ itself.

We first show that the discounted cost, cost up to an exit time, and ergodic cost involving controlled non-degenerate diffusions are continuous on the space of stationary control policies when the policies are given a topology introduced by Borkar [V. S. Borkar, A topology for Markov controls, Applied Mathematics and Optimization 20 (1989), 55–62]. The same applies for finite horizon problems when the control policies are Markov and the topology is revised to include time also as a parameter. We then establish that finite action/piecewise constant stationary policies are dense in the space of stationary Markov policies under this topology and the same holds for continuous policies. Using the above mentioned continuity and denseness results we establish that finite action/piecewise constant policies approximate optimal stationary policies with arbitrary precision. This gives rise to the applicability of many numerical methods such as policy iteration and stochastic learning methods for discounted cost, cost up to an exit time, and ergodic cost optimal control problems in continuous-time. For the finite-horizon setup, we establish additionally near optimality of time-discretized policies by an analogous argument. We thus present a unified and concise approach for approximations directly applicable under several commonly adopted cost criteria.

We describe the rate of growth of the derivative $C^{\prime }$ of the convex minorant of a Lévy path at times where $C^{\prime }$ increases continuously. Since the convex minorant is piecewise linear, $C^{\prime }$ may exhibit such behaviour either at the vertex time ${\mathit{\tau }}_s$ of finite slope $s=C_{{\mathit{\tau }}_s}^{\prime }$ or at time 0 where the slope is $-\mathrm{\infty }$. While the convex hull depends on the entire path, we show that the local fluctuations of the derivative $C^{\prime }$ depend only on the fine structure of the small jumps of the Lévy process and are the same for all time horizons. In the domain of attraction of a stable process, we establish sharp results essentially characterising the modulus of continuity of $C^{\prime }$ up to sub-logarithmic factors. As a corollary we obtain novel results for the growth rate at 0 of meanders in a wide class of Lévy processes.

We prove that the Loop $O(1)$ model, a well-known graphical expansion of the Ising model, is a factor of i.i.d. on unimodular random rooted graphs under various conditions, including in the presence of a non-negative external field. As an application we show that the gradient of the free Ising model is a factor of i.i.d. on simply connected unimodular planar maps having a locally finite dual. The key idea is to develop an appropriate theory of local limits of uniform even subgraphs with various boundary conditions and prove that they can be sampled as a factor of i.i.d. Another key tool we exploit is that the wired uniform spanning tree on a unimodular transient graph is a factor of i.i.d. This partially answers some questions posed by Hutchcroft [33].

We study and develop multilevel methods for the numerical approximation of a log-concave probability π on ${\mathbb{R}}^d$, based on (over-damped) Langevin diffusion. In the continuity of [12] concentrated on the uniformly log-concave setting, we here study the procedure in the absence of the uniformity assumption. More precisely, we first adapt an idea of [8] by adding a penalization term to the potential to recover the uniformly convex setting. Such approach leads to an ε-complexity of the order ${\mathit{\epsilon }}^{-5}\mathrm{\pi }{(|.{|}^2)}^{9∕2}d$ (up to logarithmic terms). Then, in the spirit of [14], we propose to explore the robustness of the method in a weakly convex parametric setting where the lowest eigenvalue of the Hessian of the potential U is controlled by the function ${U(x)}^{-r}$ for $r\in (0,1)$. In this intermediary framework between the strongly convex setting ($r=0$) and the “Laplace case” ($r=1$), we show that with the help of the control of exponential moments of the Euler scheme, we can adapt some fundamental properties for the efficiency of the method. In the “best” setting where U is ${\mathcal{C}}^3$ and ${U(x)}^{-r}$ control the largest eigenvalue of the Hessian, we obtain an ε-complexity of the order $c_{\mathit{\rho },\mathit{\delta }}{\mathit{\epsilon }}^{-2-\mathit{\rho }}{d}^{1+\frac{\mathit{\rho }}{2}+(4-\mathit{\rho }+\mathit{\delta })r}$ for any $\mathit{\rho }>0$ (but with a constant $c_{\mathit{\rho },\mathit{\delta }}$ which increases when ρ and δ go to 0).

We provide a many-to-few formula in the general setting of non-local branching Markov processes. This formula allows one to compute expectations of k-fold sums over functions of the population at k different times. The result generalises [13] to the non-local setting, as introduced in [11] and [8]. As an application, we consider the case when the branching process is critical, and conditioned to survive for a large time. In this setting, we prove a general formula for the limiting law of the death time of the most recent common ancestor of two particles selected uniformly from the population at two different times, as $t\to \mathrm{\infty }$. Moreover, we describe the limiting law of the population sizes at two different times, in the same asymptotic regime.

We derive the leading order asymptotics of the logarithmic potential of a two dimensional Coulomb gas at arbitrary positive temperature. The proof is based on precise evaluation of exponential moments, and the theory of Gaussian multiplicative chaos.

We study moderate deviations for additive functionals of stochastic lattice gases (Kawasaki dynamics for the Ising model). Under a mixing condition, we prove that the additive functional of any local functions satisfies a moderate deviation principle. The main tool is the logarithmic Sobolev inequality obtained by Yau.

We study the stochastic six-vertex model on a strip

$\left\{(x,y)\in {\mathbb{Z}}^2:0\le y\le x\le y+N\right\}\text{(0.1)}$

with two open boundaries. We develop a ‘matrix product ansatz’ method to solve for its stationary measure, based on the compatibility of three types of local moves of any down-right path. The stationary measure on a horizontal path turns out to be a tilting of the stationary measure of the asymmetric simple exclusion process (ASEP) with two open boundaries. Similar to open ASEP, the statistics of this tilted stationary measure as the number of sites $N\to \mathrm{\infty }$ (with the bulk and boundary parameters fixed) also exhibit a phase diagram, which is a tilting of the phase diagram of open ASEP. We study the limit of mean particle density as an example.

We discuss a natural extension of Gilles Pisier’s approach to the study of measure concentration, isoperimetry, and Poincaré-type inequalities. This approach allows one to explore counterparts of various results about Gaussian measures in the class of rotationally invariant probability distributions on Euclidean spaces, including multidimensional Cauchy measures.

Here we propose the Donsker-Varadhan-type compactness conditions and prove the joint large deviation principle for the empirical measure and empirical flow of Markov renewal processes (semi-Markov processes) with a countable state space, generalizing the relevant results for continuous-time Markov chains with a countable state space obtained in [Ann. Inst. H. Poincaré Probab. Statist. 51, 867-900 (2015)] and [Stoch. Process. Appl. 125, 2786-2819 (2015)], as well as the relevant results for Markov renewal processes with a finite state space obtained in [Adv. Appl. Probab. 48, 648-671 (2016)]. In particular, our results hold when the flow space is endowed with either the bounded weak* topology or the strong $L^1$ topology. Even for continuous-time Markov chains, our compactness conditions are weaker than the ones proposed in previous papers. Furthermore, under some stronger conditions, we obtain the explicit expression of the marginal rate function of the empirical flow.

When the limiting compensator of a sequence of martingales is continuous, we obtain a weak convergence theorem for the martingales; the limiting process can be written as a Brownian motion evaluated at the compensator and we find sufficient conditions for both processes to be independent. As examples of applications, we revisit some known results for the occupation times of Brownian motion and symmetric random walks. In the latter case, our proof is much simpler than the construction of strong approximations. Furthermore, we extend finite dimensional convergence of statistical estimators of financial volatility measures to convergence as stochastic processes.

In this study, we investigate the energy landscape of the Ising and Potts models on fixed and finite but large three-dimensional (3D) lattices where no external field exists and quantitatively characterize the metastable behavior of the associated Glauber dynamics in the very low temperature regime. Such analyses for the models with non-zero external magnetic fields have been extensively performed over the past two decades; however, models without external fields remained uninvestigated. Recently, the corresponding investigation has been conducted for the two-dimensional (2D) model without an external field, and in this study, we further extend these successes to the 3D model, which has a far more complicated energy landscape than the 2D one. In particular, we provide a detailed description of the highly complex plateau structure of saddle configurations between ground states and then analyze the typical behavior of the Glauber dynamics thereon. Thus, we achieve a quantitatively precise analysis of metastability, including the Eyring–Kramers law, the Markov chain model reduction, and a full characterization of metastable transition paths.

We raise a question of whether there might be deep mathematical connections between two properties for bijective functions $F:\mathcal{X}\times \mathcal{X}\to \mathcal{X}\times \mathcal{X}$ for a set $\mathcal{X}$ which are introduced from very different backgrounds, and study the supporting evidence that suggests this. One of the property is that F is a Yang-Baxter map, namely it satisfies the “set-theoretical” Yang-Baxter equation, and the other property is the independence preserving property, which means that there exist independent (non-constant) $\mathcal{X}$-valued random variables $X,Y$ such that $U,V$ are also independent with $(U,V)=F(X,Y)$. Recently in the study of invariant measures for a discrete integrable system, a class of functions having these two properties were found. Motivated by this, we analyze a relationship between the Yang-Baxter maps and the independence preserving property, which has never been studied as far as we are aware. We focus on the case $\mathcal{X}={\mathbb{R}}_{+}$. Our first main result is that all quadrirational Yang-Baxter maps $F:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}$ in an important subclass have the independence preserving property. In particular, we found new classes of bijections having the independence preserving property. Our second main result is that these newly introduced bijections are fundamental in the class of (known) bijections with the independence preserving property, in the sense that many known bijections having the independence preserving property are derived from these maps by taking special parameters or performing some limiting procedure. This reveals that the independence preserving property, which has been investigated for specific functions individually, can be understood in a more unified manner.

We study a permuted random walk process on a graph G. Given a fixed sequence of permutations on the vertices of G, the permuted random walker alternates between taking random walk steps, and applying the next permutation in the sequence to their current position. Existing work on permuted random walks includes results on hitting times, mixing times, and asymptotic speed. The usual random walk on a regular tree, or generally any non-amenable graph, has positive speed, i.e. the distance from the origin grows linearly. Our focus is understanding whether permuted walks can be slower than the corresponding non-permuted walk, by carefully choosing the permutation sequence. We show that on regular trees (including the line), the permuted random walk is always stochastically faster. The proof relies on a majorization inequality for probability measures, plus an isoperimetric inequality for the tree. We also quantify how much slower the permuted random walk can possibly be when it is coupled with the corresponding non-permuted walk.

We look at random walks in Dirichlet environment. It was known that in dimension $d\ge 3$, if the walk is sub-ballistic, the displacement of the walk is polynomial of order κ for some explicit κ. We show that the walk, after renormalization, actually converges to a κ-stable completely asymmetric Lévy Process.

Place an active particle at the root of a d-ary tree and a single dormant particle at each non-root site. In discrete time, active particles move towards the root with probability p and, otherwise, away from the root to a uniformly sampled child vertex. When an active particle moves to a site containing a dormant particle, the dormant particle becomes active. The critical drift $p_d$ is the infimum over all p for which infinitely many particles visit the root almost surely. Guo, Tang, and Wei proved that ${sup}_{d\ge 3}{p}_d\le 1∕3$. We improve this bound to $5∕17$ with a shorter argument that generalizes to give bounds on ${sup}_{d\ge m}{p}_d$. We additionally prove that $lim\hspace{0.17em}sup{p}_d\le 1∕6$ by finding the limiting critical drift for a non-backtracking variant.

For a class of additive processes driven by the affine recursion $X_{n+1}=A_{n+1}{X}_n+B_{n+1}$, we develop a sample-path large deviations principle in the $M_1^{\prime }$ topology on $\mathbb{D}[0,1]$. We allow $B_n$ to have both signs and focus on the case where Kesten’s condition holds on $A_1$, leading to heavy-tailed distributions. The most likely paths in our large deviations results are step functions with both positive and negative jumps.

We construct a self-similar solution of the heat equation for the fractional Laplacian with Dirichlet boundary conditions in every fat cone. Furthermore, we give the entrance law from the vertex and the Yaglom limit for the corresponding killed isotropic stable Lévy process and precise large-time asymptotics for solutions of the Cauchy problem in the cone.

We first study a d-dimensional branching Brownian motion (BBM) among mild Poissonian obstacles, where a random trap field in ${\mathbb{R}}^d$ is created via a Poisson point process. The trap field consists of balls of fixed radius centered at the atoms of the Poisson point process. The mild obstacle rule is that when particles are inside traps, they branch at a lower rate, which is allowed to be zero, whereas when outside traps they branch at the normal rate. We prove upper bounds on the large-deviation probabilities for the total mass of BBM among mild obstacles, which we then use along with the Borel-Cantelli lemma to prove the corresponding strong law of large numbers. Our results are quenched, that is, they hold in almost every environment with respect to the Poisson point process. Our strong law improves on the existing corresponding weak law in [Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), 490–518]. We also study a d-dimensional BBM inside subdiffusively expanding balls, where the boundary of the ball is deactivating in the sense that once a particle of the BBM hits the moving boundary, it is instantly deactivated but will be reactivated at a later time provided its ancestral line is fully inside the expanding ball at that later time. We obtain a large-deviation result as time tends to infinity on the probability that the mass inside the ball is aytpically small. An essential ingredient in the proofs of the strong law of large numbers for BBM among mild obstacles turns out to be the large-deviation result on the mass of BBM inside expanding balls.

We prove, under mild conditions on fixed points and 2-cycles, the asymptotic normality of vincular pattern counts for a permutation chosen uniformly at random in a conjugacy class. Additionally, we prove that the limiting variance is always non-degenerate for classical pattern counts. The proof uses weighted dependency graphs.

While many questions in robust finance can be posed in the martingale optimal transport framework or its weak extension, others like the subreplication price of VIX futures, the robust pricing of American options or the construction of shadow couplings necessitate additional information to be incorporated into the optimization problem beyond that of the underlying asset. In the present paper, we take into account this extra information by introducing an additional parameter to the weak martingale optimal transport problem. We prove the stability of the resulting problem with respect to the risk neutral marginal distributions of the underlying asset, thus extending the results in [9]. A key step is the generalization of the main result in [7] to include the extra parameter into the setting. This result establishes that any martingale coupling can be approximated by a sequence of martingale couplings with specified marginals, provided that the marginals of this sequence converge to those of the original coupling. Finally, we deduce stability of the three previously mentioned motivating examples.

We introduce a technique for comparing the limit profile behavior of two reversible, commuting Markov chains on the same space, that share the same stationary distribution. We apply this technique to prove that the limit profile of star transpositions at time $t=nlogn+cn$ is equal to $d_{\text{T.V.}}(\text{Poiss}(1+e^{-c}),\text{Poiss}(1))$ by comparing to the limit profile of random transpositions, as studied in [29]. We also provide examples of important commuting Markov chains, whose limit profile behavior is unknown, which could give new directions for research.

A distributional equation as a criterion for invariant measures of Markov processes associated to Lévy-type operators is established. This is obtained via a characterization of infinitesimally invariant measures of the associated generators. Particular focus is put on the one-dimensional case where the distributional equation becomes a Volterra-Fredholm integral equation, and on solutions to Lévy-driven stochastic differential equations. The results are accompanied by various illustrative examples.