Consider a finite number of balls initially placed in $L$ bins. At each time step a ball is taken from each non-empty bin. Then all the balls are uniformly reassigned into bins. This finite Markov chain is called Repeated Balls-into-Bins process and is a discrete time interacting particle system with parallel updating. We prove that, starting from a suitable (chaotic) set of initial states, as $L\to +\infty $, the numbers of balls in each bin become independent from the rest of the system i.e. we have propagation of chaos. We furthermore study some equilibrium properties of the limiting nonlinear process.

We consider exclusion processes on a rooted $d$-regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For $d\geq 3$, we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process “seen from the tagged particle” has an ergodic invariant measure.

We prove the convergence of the Sasamoto-Spohn model in equilibrium to the energy solution of the stochastic Burgers equation on the whole line. The proof, which relies on the second order Boltzmann-Gibbs principle, follows the approach of [9] and does not use any spectral gap argument.

Let $\mu $ be a compactly supported probability measure on the positive half-line and let $\mu ^{\boxtimes t}$ be the free multiplicative convolution semigroup. We show that the support of $\mu ^{\boxtimes t}$ varies continuously as $t$ changes. We also obtain the asymptotic length of the support of these measures.

We establish heat kernel upper bounds for a continuous-time random walk under unbounded conductances satisfying an integrability assumption, where we correct and extend recent results in [3] to a general class of speed measures. The resulting heat kernel estimates are governed by the intrinsic metric induced by the speed measure. We also provide a comparison result of this metric with the usual graph distance, which is optimal in the context of the random conductance model with ergodic conductances.

The proof of Theorem 2.3 in our paper [3] is fully justified only under the additional assumption $q_i(n)=a_in+b_i,\, i=1,...,\ell $.

We give upper and lower asymptotic bounds for the left tail and for the right tail of the continuous limiting QuickSort density $f$ that are nearly matching in each tail. The bounds strengthen results from a paper of Svante Janson (2015) concerning the corresponding distribution function $F$. Furthermore, we obtain similar bounds on absolute values of derivatives of $f$ of each order.

We prove that the Dean-Kawasaki SPDE admits a solution only in integer parameter regimes, in which case the solution is given in terms of a system of non-interacting particles.

In [1], a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Itô integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in [1], by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1].

We consider critical percolation on Galton-Watson trees and prove quenched analogues of classical theorems of critical branching processes. We show that the probability critical percolation reaches depth $n$ is asymptotic to a tree-dependent constant times $n^{-1}$. Similarly, conditioned on critical percolation reaching depth $n$, the number of vertices at depth $n$ in the critical percolation cluster almost surely converges in distribution to an exponential random variable with mean depending only on the offspring distribution. The incipient infinite cluster (IIC) is constructed for a.e. Galton-Watson tree and we prove a limit law for the number of vertices in the IIC at depth $n$, again depending only on the offspring distribution. Provided the offspring distribution used to generate these Galton-Watson trees has all finite moments, each of these results holds almost-surely.

For the Labouchere system with winning probability $p$ at each coup, we prove that the expectation of the largest bet size under any initial list is finite if $p>\frac{1} {2}$, and is infinite if $p\le \frac{1} {2}$, solving the open conjecture in [6]. The same result holds for a general family of betting systems, and the proof builds upon a recursive representation of the optimal betting system in the larger family.

We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequality in Kantorovich-Wasserstein distance inspired from the work of Chafaï, Hardy and Maïda on the Euclidean space. Their proof involves large deviation techniques together with an energy-distance comparison and a regularization procedure based on the superharmonicity of the Green function. This last ingredient is not available on a manifold. We solve this problem by using the heat kernel and its short-time asymptotic behavior.

We study the obtainment of closed-form formulas for the distribution of the jumps of a doubly-stochastic Poisson process. The problem is approached in two ways. On the one hand, we translate the problem to the computation of multiple derivatives of the Hazard process cumulant generating function; this leads to a closed-form formula written in terms of Bell polynomials. On the other hand, for Hazard processes driven by Lévy processes, we use Malliavin calculus in order to express the aforementioned distributions in an appealing recursive manner. We outline the potential application of these results in credit risk.

We prove deviation bounds for the random variable $\sum _{i=1}^{n} f_i(Y_i)$ in which $\{Y_i\}_{i=1}^{\infty }$ is a Markov chain with stationary distribution and state space $[N]$, and $f_i: [N] \rightarrow [-a_i, a_i]$. Our bound improves upon previously known bounds in that the dependence is on $\sqrt{a_1^2+\cdots +a_n^2} $ rather than $\max _{i}\{a_i\}\sqrt{n} .$ We also prove deviation bounds for certain types of sums of vector–valued random variables obtained from a Markov chain in a similar manner. One application includes bounding the expected value of the Schatten $\infty $-norm of a random matrix whose entries are obtained from a Markov chain.

We consider a population of $N$ individuals. Each individual has a type belonging to some at most countable type space $K$. At each time step each individual of type $k\in K$ mutates to type $l\in K$ independently of the other individuals with probability $m_{k,l}$. It is shown that the associated empirical measure process is Markovian. For the two-type case $K=\{0,1\}$ we derive an explicit spectral decomposition for the transition matrix $P$ of the Markov chain $Y=(Y_n)_{n\ge 0}$, where $Y_n$ denotes the number of individuals of type $1$ at time $n$. The result in particular shows that $P$ has eigenvalues $(1-m_{0,1}-m_{1,0})^i$, $i\in \{0,\ldots ,N\}$. Applications to mean first passage times are provided.

This paper gives the Kolmogorov and Wasserstein bounds in normal approximation for the squared-length of total spin in the mean field classical $N$-vector models. The Kolmogorov bound is new while the Wasserstein bound improves a result obtained recently by Kirkpatrick and Nawaz [Journal of Statistical Physics, 165 (2016), no. 6, 1114–1140]. The proof is based on Stein’s method for exchangeable pairs.

Let $X(t)$ be a Gaussian random field $\mathbb R^d\to \mathbb R$. Using the notion of $(d-1)$-integral geometric measures, we establish a relation between (a) the volume of level sets, and (b) the number of crossings of the restriction of the random field to a line. Using this relation we prove the equivalence between the finiteness of the expectation and the finiteness of the second spectral moment matrix. Sufficient conditions for finiteness of higher moments are also established.

We study how the Gaussian multiplicative chaos (GMC) measures $\mu ^\gamma $ corresponding to the 2D Gaussian free field change when $\gamma $ approaches the critical parameter $2$. In particular, we show that as $\gamma \to 2^{-}$, $(2-\gamma )^{-1}\mu ^\gamma $ converges in probability to $2\mu '$, where $\mu '$ is the critical GMC measure.

We prove the subsequential tightness of centered maxima of two-dimensional Ginzburg-Landau fields with bounded elliptic contrast.

In this note we discuss “vacant set level set” percolation on a transient weighted graph. It interpolates between the percolation of the vacant set of random interlacements and the level set percolation of the Gaussian free field. We employ coupling and derive a stochastic domination from which we deduce in a rather general set-up a certain monotonicity property of the percolation function. In the case of regular trees this stochastic domination leads to a strict inequality between some eigenvalues related to Ornstein-Uhlenbeck semi-groups for which we have no direct analytical proof. It underpins a certain strict monotonicity property that has significant consequences for the percolation diagram. It is presently open whether a similar looking diagram holds in the case of ${\mathbb Z}^d$, $d \ge 3$.

We prove that reflected Brownian motion with normal reflections in a convex domain satisfies a dimension free Talagrand type transportation cost-information inequality. The result is generalized to other reflected diffusion processes with suitable drift and diffusion coefficients. We apply this to get such an inequality for interacting Brownian particles with rank-based drift and diffusion coefficients such as the infinite Atlas model. This is an improvement over earlier dimension-dependent results.

Mirror coupled reflected Brownian motions can simultaneously hit opposite sides of a wedge at different distances from the origin.

We consider the general version of Pólya urns recently studied by Bandyopadhyay and Thacker (2016+) and Mailler and Marckert (2017), with the space of colours being any Borel space $S$ and the state of the urn being a finite measure on $S$. We consider urns with random replacements, and show that these can be regarded as urns with deterministic replacements using the colour space $S\times [0,1]$.

We show that solutions to multidimensional SDEs with Lipschitz coefficients and driven by Brownian motion never reach the set where all coefficients vanish unless the initial position belongs to that set.

We classify all subsets $S$ of the projective Hilbert space with the following property: for every point $\pm s_{0}\in S$, the spherical projection of $S\backslash \{\pm s_{0}\}$ on the hyperplane orthogonal to $\pm s_{0}$ is isometric to $S\backslash \{\pm s_{0}\}$. In probabilistic terms, this means that we characterize all zero-mean Gaussian processes $Z=(Z(t))_{t\in T}$ with the property that for every $s_{0}\in T$ the conditional distribution of $(Z(t))_{t\in T}$ given that $Z(s_{0})=0$ coincides with the distribution of $(\varphi (t; s_{0}) Z(t))_{t\in T}$ for some function $\varphi (t;s_{0})$. A basic example of such process is the stationary zero-mean Gaussian process $(X(t))_{t\in \mathbb{R} }$ with covariance function $\mathbb E [X(s) X(t)] = 1/\cosh (t-s)$. We show that, in general, the process $Z$ can be decomposed into a union of mutually independent processes of two types: (i) processes of the form $(a(t) X(\psi (t)))_{t\in T}$, with $a: T\to \mathbb{R} $, $\psi (t): T\to \mathbb{R} $, and (ii) certain exceptional Gaussian processes defined on four-point index sets. The above problem is reduced to the classification of metric spaces in which in every triangle the largest side equals the sum of the remaining two sides.

We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a $d$-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nested sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. Additionally, we describe a coupling between frog models on trees for which the degree of the smaller tree divides that of the larger one. This implies that the critical drift has a limit as $d$ tends to infinity along certain subsequences.

Bullets are fired from the origin of the positive real line, one per second, with independent speeds sampled uniformly from a discrete set. Collisions result in mutual annihilation. We show that a bullet with the second largest speed survives with positive probability, while a bullet with the smallest speed does not. This also holds for exponential spacings between firing times. Our results imply that the middle-velocity particle survives with positive probability in a two-sided version of the bullet process with three speeds known to physicists as ballistic annihilation.

We show that the local time of one-dimensional super-Brownian motion is locally $\gamma $-Hölder continuous near the boundary if $0<\gamma <3$ and fails to be locally $\gamma $-Hölder continuous if $\gamma >3$.

The frog model is an interacting particle system on a graph. Active particles perform independent simple random walks, while sleeping particles remain inert until visited by an active particle. Some number of sleeping particles are placed at each site sampled independently from a certain distribution, and then one particle is activated to begin the process. We show that the recurrence or transience of the model is sensitive not just to the expectation but to the entire distribution. This is in contrast to closely related models like branching random walk and activated random walk.

In this note we consider a certain class of Gaussian entire functions, characterized by some asymptotic properties of their covariance kernels, which we call admissible (as defined by Hayman). A notable example is the Gaussian Entire Function, whose zero set is well-known to be invariant with respect to the isometries of the complex plane.

We explore the rigidity of the zero set of Gaussian Taylor series, a phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire Function. In particular, we find that for a function of infinite order of growth, and having an admissible kernel, the zero set is “fully rigid”. This means that if we know the location of the zeros in the complement of any given compact set, then the number and location of the zeros inside that set can be determined uniquely. As far as we are aware, this is the first explicit construction in a natural class of random point processes with full rigidity.

We show that Internal Diffusion Limited Aggregation (IDLA )on $\mathbb{Z} ^{d}$ has near optimal Cheeger constant when the growing cluster is large enough. This implies, through a heat kernel lower bound derived previously in [11], that simple random walk evolving independently on growing in time IDLA cluster is recurrent when $d\ge 3$.

In this paper, we study the probability of visiting a distant point $a\in \mathbb{Z} ^{4}$ by a critical branching random walk starting at the origin. We prove that this probability is bounded by $1/(|a|^{2}\log |a|)$ up to a constant factor.

We consider a class of fractional stochastic volatility models (including the so-called rough Bergomi model), where the volatility is a superlinear function of a fractional Gaussian process. We show that the stock price is a true martingale if and only if the correlation $\rho $ between the driving Brownian motions of the stock and the volatility is nonpositive. We also show that for each $\rho <0$ and $m> \frac{1} {{1-\rho ^{2}}}$, the $m$-th moment of the stock price is infinite at each positive time.

We derive explicit Berry-Esseen bounds in the total variation distance for the Breuer-Major central limit theorem, in the case of a subordinating function $\varphi $ satisfying minimal regularity assumptions. Our approach is based on the combination of the Malliavin-Stein approach for normal approximations with Gebelein’s inequality, bounding the covariance of functionals of Gaussian fields in terms of maximal correlation coefficients.

We prove that the long term distribution of the queue length process in an ergodic generalised Jackson network obeys the Large Deviation Principle with a deviation function given by the quasipotential. The latter is related to the unique long term idempotent distribution, which is also a stationary idempotent distribution, of the large deviation limit of the queue length process. The proof draws on developments in queueing network stability and idempotent probability.

Let $X(\cdot )$ be a non-degenerate, positive recurrent one-dimensional diffusion process on $\mathbb{R} $ with invariant probability density $\mu (x)$, and let $\tau _{y}=\inf \{t\ge 0: X(t)=y\}$ denote the first hitting time of $y$. Let $\mathcal{X} $ be a random variable independent of the diffusion process $X(\cdot )$ and distributed according to the process’s invariant probability measure $\mu (x)dx$. Denote by $\mathcal{E} ^{\mu }$ the expectation with respect to $\mathcal{X} $. Consider the expression \[ \mathcal{E} ^{\mu }E_{x}\tau _{\mathcal{X} }=\int _{-\infty }^{\infty }(E_{x}\tau _{y})\mu (y)dy, \ x\in \mathbb{R} . \] In words, this expression is the expected hitting time of the diffusion starting from $x$ of a point chosen randomly according to the diffusion’s invariant distribution. We show that this expression is constant in $x$, and that it is finite if and only if $\pm \infty $ are entrance boundaries for the diffusion. This result generalizes to diffusion processes the corresponding result in the setting of finite Markov chains, where the constant value is known as Kemeny’s constant.

An infinite sequence of real random variables $(\xi _{1}, \xi _{2}, \dots )$ is said to be rotatable if every finite subsequence $(\xi _{1}, \dots , \xi _{n})$ has a spherically symmetric distribution. A celebrated theorem of Freedman states that $(\xi _{1}, \xi _{2}, \dots )$ is rotatable if and only if $\xi _{j} = \tau \eta _{j}$ for all $j$, where $(\eta _{1}, \eta _{2}, \dots )$ is a sequence of independent standard Gaussian random variables and $\tau $ is an independent nonnegative random variable. Freedman’s theorem is equivalent to a classical result of Schoenberg which says that a continuous function $\phi : \mathbb{R} _{+} \to \mathbb{C} $ with $\phi (0) = 1$ is completely monotone if and only if $\phi _{n}: \mathbb{R} ^{n} \to \mathbb{R} $ given by $\phi _{n}(x_{1}, \ldots , x_{n}) = \phi (x_{1}^{2} + \cdots + x_{n}^{2})$ is nonnegative definite for all $n \in \mathbb{N} $. We establish the analogue of Freedman’s theorem for sequences of random variables taking values in local fields using probabilistic methods and then use it to establish a local field analogue of Schoenberg’s result. Along the way, we obtain a local field counterpart of an observation variously attributed to Maxwell, Poincaré, and Borel which says that if $(\zeta _{1}, \ldots , \zeta _{n})$ is uniformly distributed on the sphere of radius $\sqrt{n} $ in $\mathbb{R} ^{n}$, then, for fixed $k \in \mathbb{N} $, the distribution of $(\zeta _{1}, \ldots , \zeta _{k})$ converges to that of a vector of $k$ independent standard Gaussian random variables as $n \to \infty $.

We prove a combinatorial statement about the distribution of directed currents in a complex “loop soup” and use it to give a new proof of the isomorphism, which relates loop measures and complex Gaussian free fields.

We study the behaviour of a class of edge-reinforced random walks on $\mathbb{Z} _{+}$, with heterogeneous initial weights, where each edge weight can be updated only when the edge is traversed from left to right. We provide a description for different behaviours of this process and describe phase transitions that arise as trade-offs between the strength of the reinforcement and that of the initial weights. Our result aims to complete the ones given by Davis [3, 4], Takeshima [9, 10] and Vervoort [11].

We investigate the relation to random configurations and combinatorial maps of the Eulerian networks defined by Poissonian emsembles of Markov loops.

We consider a discrete-time branching random walk in a stationary and ergodic environment $\xi =(\xi _{n})$ indexed by time $n\in \mathbb{N} $. Let $W_{n}(z)$ ($z\in \mathbb{C} ^{d}$) be the natural complex martingale of the process. We show sufficient conditions for its almost sure and quenched $L^{\alpha }$ convergence, as well as the existence of quenched moments and weighted moments of its limit, and also describe the exponential convergence rate.

We consider a supercritical branching process $Z_{n}$ in a stationary and ergodic random environment $\xi =(\xi _{n})_{n\ge 0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_{n}=Z_{n}/(\mathbb{E} [Z_{n}|\xi ])$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $\xi $ the law of $W$ conditioned on the environment $\xi $ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of [10], and of course it covers the well-known case of the martingale limit of a Galton-Watson process. Our proof combines analytical arguments with the recursive description of $W$.

Let $X$ and $Y$ denote two independent squared Bessel processes of dimension $m$ and $n-m$, respectively, with $n\geq 2$ and $m \in [0, n)$, making $X+Y$ a squared Bessel process of dimension $n$. For appropriately chosen function $s$, the process $s (X+Y)$ is a local martingale. We study the representation and the dynamics of $s(X+Y)$, projected on the filtration generated by $X$. This projection is a strict supermartingale if, and only if, $m<2$. The finite-variation term in its Doob-Meyer decomposition only charges the support of the Markov local time of $X$ at zero.

We consider the membrane model on a box $V_{N}\subset \mathbb{Z} ^{n}$ of size $(2N+1)^{n}$ with zero boundary condition in the subcritical dimensions $n=2$ and $n=3$. We show optimal estimates for the probability that the field is positive in a subset $D_{N}$ of $V_{N}$. In particular we obtain for $D_{N}=V_{N}$ that the probability to be positive on the entire domain is exponentially small and the rate is of the order of the surface area $N^{n-1}$.

This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof of a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.

Let $\mathcal{P} $ be the collection of Borel probability measures on $\mathbb{R} $, equipped with the weak topology, and let $\mu :[0,1]\rightarrow \mathcal{P} $ be a continuous map. Say that $\mu $ is presentable if $X_{t}\sim \mu _{t}$, $t\in [0,1]$, for some real process $X$ with continuous paths. It may be that $\mu $ fails to be presentable. Hence, firstly, conditions for presentability are given. For instance, $\mu $ is presentable if $\mu _{t}$ is supported by an interval (possibly, by a singleton) for all but countably many $t$. Secondly, assuming $\mu $ presentable, we investigate whether the quantile process $Q$ induced by $\mu $ has continuous paths. The latter is defined, on the probability space $((0,1),\mathcal{B} (0,1),\mbox{Lebesgue measure} )$, by \[ Q_{t}(\alpha )=\inf \, \bigl \{x\in \mathbb{R} :\mu _{t}(-\infty ,x]\ge \alpha \bigl \} \quad \quad \mbox{for all } t\in [0,1]\mbox{ and } \alpha \in (0,1). \] A few open problems are discussed as well.

We study Markov chains on a lattice in a codimension-one stratified independent random environment, exploiting results previously established in [2]. The random walk is first shown to be transient in dimension at least three. Focusing on dimension two, we provide sharp sufficient conditions for either recurrence or transience. We determine the critical scale of the local drift along the strata, corresponding to the frontier between the two regimes.

We show that bilinear variational estimates of Do, Muscalu, and Thiele [7] remain valid for a pair of general martingales with respect to the same filtration. Our result can also be viewed as an off-diagonal generalization of the Burkholder–Davis–Gundy inequality for martingale rough paths by Chevyrev and Friz [4].

In [8], the existence of the solution is proved for a scalar linearly growingbackward stochastic differential equation (BSDE) when the terminal value is$L\exp (\mu \sqrt{2\log (1+L)} )$-integrable for a positive parameter $\mu >\mu _{0}$ with a critical value $\mu _{0}$, and a counterexample is provided to show that the preceding integrability for $\mu <\mu _{0}$ is not sufficient to guarantee the existence of the solution. Afterwards, the uniqueness result (with $\mu >\mu _{0}$) is also given in [3] for the preceding BSDE under the uniformly Lipschitz condition of the generator. In this note, we prove that these two results still hold for the critical case: $\mu =\mu _{0}$.

We solve optimal stopping problems for an oscillating Brownian motion, i.e. a diffusion with positive piecewise constant volatility changing at the point $x=0$. Let $\sigma _{1}$ and $\sigma _{2}$ denote the volatilities on the negative and positive half-lines, respectively. Our main result is that continuation region of the optimal stopping problem with reward $((1+x)^{+})^{2}$ can be disconnected for some values of the discount rate when $2\sigma _{1}^{2}<\sigma _{2}^{2}$. Based on the fact that the skew Brownian motion in natural scale is an oscillating Brownian motion, the obtained results are translated into corresponding results for the skew Brownian motion.

We prove that, in the coupon collector’s problem, the point processes given by the times of $r^{th}$ arrivals for coupons of each type, centered and normalized in a proper way, converge toward a non-homogeneous Poisson point process. This result is then used to derive some generalizations and infinite-dimensional extensions of classical limit theorems on the topic.

We consider the linear stochastic wave equation driven by a Gaussian noise which is white in time and colored in space. We show that the solution satisfies a certain form of strong local nondeterminism and we use this property to derive the exact uniform modulus of continuity for the solution.

In this short note we study the asymptotic behaviour of the minima over compact intervals of Gaussian processes, whose paths are not necessarily smooth. We show that, beyond the logarithmic large deviation Gaussian estimates, this problem is closely related to the classical small-ball problem. Under certain conditions we estimate the term describing the correction to the large deviation behaviour. In addition, the asymptotic distribution of the location of the minimum, conditionally on the minimum exceeding a high threshold, is also studied.

Let $F(\sigma )$ be the random Dirichlet series $F(\sigma )=\sum _{p\in \mathcal{P} } \frac{X_{p}} {p^{\sigma }}$, where $\mathcal{P} $ is an increasing sequence of positive real numbers and $(X_{p})_{p\in \mathcal{P} }$ is a sequence of i.i.d. random variables with $\mathbb{P} (X_{1}=1)=\mathbb{P} (X_{1}=-1)=1/2$. We prove that, for certain conditions on $\mathcal{P} $, if $\sum _{p\in \mathcal{P} }\frac{1} {p}<\infty $ then with positive probability $F(\sigma )$ has no real zeros while if $\sum _{p\in \mathcal{P} }\frac{1} {p}=\infty $, almost surely $F(\sigma )$ has an infinite number of real zeros.

Kinetically constrained models (KCMs) are interacting particle systems on $\mathbb{Z} ^{d}$ with a continuous-time constrained Glauber dynamics, which were introduced by physicists to model the liquid-glass transition. One of the most well-known KCMs is the one-dimensional East model. Its generalization to higher dimension, the $d$-dimensional East model, is much less understood. Prior to this paper, convergence to equilibrium in the $d$-dimensional East model was proven to be at least stretched exponential, by Chleboun, Faggionato and Martinelli in 2015. We show that the $d$-dimensional East model exhibits exponential convergence to equilibrium in all settings for which convergence is possible.

For $\xi \geq 0$, Liouville first passage percolation (LFPP) is the random metric on $\varepsilon \mathbb{Z} ^{2}$ obtained by weighting each vertex by $\varepsilon e^{\xi h_{\varepsilon }(z)}$, where $h_{\varepsilon }(z)$ is the average of the whole-plane Gaussian free field $h$ over the circle $\partial B_{\varepsilon }(z)$. Ding and Gwynne (2018) showed that for $\gamma \in (0,2)$, LFPP with parameter $\xi = \gamma /d_{\gamma }$ is related to $\gamma $-Liouville quantum gravity (LQG), where $d_{\gamma }$ is the $\gamma $-LQG dimension exponent. For $\xi > 2/d_{2}$, LFPP is instead expected to be related to LQG with central charge greater than 1. We prove several estimates for LFPP distances for general $\xi \geq 0$. For $\xi \leq 2/d_{2}$, this leads to new bounds for $d_{\gamma }$ which improve on the best previously known upper (resp. lower) bounds for $d_{\gamma }$ in the case when $\gamma > \sqrt{8/3} $ (resp. $\gamma \in (0.4981, \sqrt{8/3} )$). These bounds are consistent with the Watabiki (1993) prediction for $d_{\gamma }$. However, for $\xi > 1/\sqrt{3} $ (or equivalently for LQG with central charge larger than 17) our bounds are inconsistent with the analytic continuation of Watabiki’s prediction to the $\xi >2/d_{2}$ regime. We also obtain an upper bound for the Euclidean dimension of LFPP geodesics.

We consider the empirical eigenvalue distribution of an $m\times m$ principle submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and Réffy identified the limiting spectral measure if $\frac{m} {n}\to \alpha $, as $n\to \infty $; under suitable scaling, the family $\{\mu _{\alpha }\}_{\alpha \in (0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $\alpha $) and uniform measure on the unit circle (as $\alpha \to 1$). In this note, we prove an explicit concentration inequality which shows that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $\mu _{\alpha }$ is typically of order $\sqrt{\frac {\log (m)}{m}} $ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new “Coulomb transport inequality” due to Chafaï, Hardy, and Maïda.

We prove the existence and uniqueness of a discrete nonnegative harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed when leaving a globally Lipschitz domain in $\mathbb{Z} ^{d}$. Our method is based on a systematic use of comparison arguments and discrete potential-theoretical techniques.

In N. V. Krylov, Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies, Electron. J. Probab., 4(2), 1999, it is proved under standard assumptions that the value functions of controlled diffusion processes can be approximated with order 1/6 error by those with controls which are constant on uniform time intervals. In this note we refine the proof and show that the provable rate can be improved to 1/4, which is optimal in our setting. Moreover, we demonstrate the improvements this implies for error estimates derived by similar techniques for approximation schemes, bringing these in line with the best available results from the PDE literature.

Let $X$ be an $\mu $-symmetric irreducible Markov process on $I$ with strong Feller property. In addition, we assume that $X$ possesses a tightness property. In this paper, we prove some conditional limiting theorems for the process $X$. The emphasis is on conditional ergodic theorem. These results are also discussed in the framework of one-dimensional diffusions.

Bi-log-concavity of probability measures is a univariate extension of the notion of log-concavity that has been recently proposed in a statistical literature. Among other things, it has the nice property from a modelisation perspective to admit some multimodal distributions, while preserving some nice features of log-concave measures. We compute the isoperimetric constant for a bi-log-concave measure, extending a property available for log-concave measures. This implies that bi-log-concave measures have exponentially decreasing tails. Then we show that the convolution of a bi-log-concave measure with a log-concave one is bi-log-concave. Consequently, infinitely differentiable, positive densities are dense in the set of bi-log-concave densities for $ L_{p}$-norms, $p\in \left [1,+\infty \right ]$. We also derive a necessary and sufficient condition for the convolution of two bi-log-concave measures to be bi-log-concave. We conclude this note by discussing a way of defining a multi-dimensional extension of the notion of bi-log-concavity.

In a fully general setting, we study the relation between martingale spaces under two locally absolutely continuous probabilities and prove that the martingale representation property (MRP) is always stable under locally absolutely continuous changes of probability. Our approach relies on minimal requirements, is constructive and, as shown by a simple example, enables us to study situations which cannot be covered by the existing theory.

We propose new concentration inequalities for self-normalized martingales. The main idea is to introduce a suitable weighted sum of the predictable quadratic variation and the total quadratic variation of the martingale. It offers much more flexibility and allows us to improve previous concentration inequalities. Statistical applications on autoregressive process, internal diffusion-limited aggregation process, and online statistical learning are also provided.

Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the $\gamma $-Liouville quantum gravity (LQG) metric, obtained by exponentiating the discrete Gaussian free field (GFF) and the circle average regularization of the continuum GFF respectively. We show that these two models can be coupled so that with high probability distances in these models agree up to $o(1)$ errors in the exponent, and thus have the same distance exponent.

Ding and Gwynne (2018) give a formula for the continuum LFPP distance exponent in terms of the $\gamma $-LQG dimension exponent $d_{\gamma }$. Using results of Ding and Li (2018) on the level set percolation of the discrete GFF, we bound the DLFPP distance exponent and hence obtain a new lower bound $d_{\gamma }\geq 2 + \frac{\gamma ^{2}} {2}$. This improves on previous lower bounds for $d_{\gamma }$ for the regime $\gamma \in (\gamma _{0}, 0.576)$, for some small nonexplicit $\gamma _{0} > 0$.

We study the non-random fluctuation in first passage percolation and show that it diverges. We also prove the divergence of non-random shape fluctuation, which was predicted in [Yu Zhang. The divergence of fluctuations for shape in first passage percolation. Probab. Theory. Related. Fields. 136(2) 298–320, 2006].

This paper is concerned with a new expression of the so-called Pennington-Worah distribution, characterizing the asymptotic empirical eigenvalue distribution of some non linear random matrix ensembles. More precisely consider $M= \frac{1} {m} YY^{*}$ with $Y=f(WX)$ where $W$ and $X$ are random rectangular matrices with i.i.d. centered entries. The function $f$ is applied pointwise and can be seen as an activation function in (random) neural networks. The asymptotic empirical distribution of this ensemble has been computed in [16] and [3]. Here it is related to the Marcenko-Pastur distribution and information plus noise matrices.

We prove that Schütz’s ASEP Markov duality functional is also a Markov duality functional for the stochastic six vertex model. We introduce a new method that uses induction on the number of particles to prove the Markov duality.

Start a planar Brownian motion and let it run until it hits some given barrier. We show that the barrier may be crafted so that the $x$ coordinate at the hitting time has any prescribed centered distribution with finite variance. This provides a new, complex-analytic proof of the Skorokhod embedding theorem. Our method is constructive and can give an explicit description of the barrier.

We give a new expression for the law of the eigenvalues of the discrete Anderson model on the finite interval $[1,N]$, in terms of two random processes starting at both ends of the interval. Using this formula, we deduce that the tail of the eigenvectors behaves approximately like $\exp (\sigma B_{|n-k|}-\gamma \frac{|n-k|} {4})$ where $\gamma , \sigma >0 $, $B_{s}$ is the Brownian motion and $k$ is uniformly chosen in $[1,N]$ independently of $B_{s}$. A similar result has recently been shown by B. Rifkind and B. Virag in the critical case, that is, when the random potential is multiplied by a factor $\frac{1} {\sqrt{N} }$.

We consider a random walk with catastrophes which was introduced to model population biology. It is known that this Markov chain gets eventually absorbed at $0$ for all parameter values. Recently, it has been shown that this chain exhibits a metastable behavior in the sense that it can persist for a very long time before getting absorbed. In this paper we study this metastable phase by making the parameters converge to extreme values. We obtain four different limits that we believe shed light on the metastable phase.

We consider the critical branching processes in correlated random environment which is positively associated and study the probability of survival up to the $n$-th generation. Moreover, when the environment is given by fractional Brownian motion, we estimate also the tail of progeny as well as the tail of width.

We prove that every amenable one-ended Cayley graph has an invariant one-ended spanning tree. More generally, for any one-ended amenable unimodular random graph we construct a factor of iid percolation (jointly unimodular subgraph) that is almost surely a one-ended spanning tree. In [2] and [1] similar claims were proved, but the resulting spanning tree had 1 or 2 ends, and one had no control of which of these two options would be the case.

Given a connected graph $G$ with some subset of its vertices excited and a fixed target vertex, in the geodesic-biased random walk on $G$, a random walker moves as follows: from an unexcited vertex, she moves to a uniformly random neighbour, whereas from an excited vertex, she takes one step along some fixed shortest path towards the target vertex. We show, perhaps counterintuitively, that the geodesic-bias can slow the random walker down exponentially: there exist connected, bounded-degree $n$-vertex graphs with excitations where the expected hitting time of a fixed target is at least $\exp (\sqrt [4]{n} / 100)$.

We obtain the distribution of the maximal average in a sequence of independent identically distributed exponential random variables. Surprisingly enough, it turns out that the inverse distribution admits a simple closed form. An application to ruin probability in a risk-theoretic model is also given.

We extend the random characteristics approach to Wigner matrices whose entries are not required to have a normal distribution. As an application, we give a simple and fully dynamical proof of the weak local semicircle law in the bulk.

We establish central limit theorems for general functionals on binomial point processes and their Poissonized version, which extends the results of Penrose–Yukich (Ann. Appl. Probab. 11(4), 1005–1041 (2001)) to the inhomogeneous case. Here functionals are required to be strongly stabilizing for add-one cost on homogeneous Poisson point processes and to satisfy some moments conditions. As an application, a central limit theorem for Betti numbers of random geometric complexes in the subcritical regime is derived.

Using X. Fernique’s results on the compactness of distributions of càdlàg random functions, we derive some embedding type càdlàguity moment estimates for stochastic processes with jumps.

We investigate concentration properties of spectral measures of Hermitian random matrices with partially dependent entries. More precisely, let $X_{n}$ be a Hermitian random matrix of the size $n\times n$ that can be split into independent blocks of the size at most $d_{n}=o(n^{2})$. We prove that under some mild conditions on the distribution of the entries of $X_{n}$, the empirical spectral measure of $X_{n}$ concentrates around its mean.

The main theorem is a strengthening of the recent result by Kemp and Zimmerman, where the size of the blocks grows as $o(\log n)$. As an application, we are able to upgrade the results of Schenker and Schulz on the convergence in expectation to the semicircle law of a class of random matrices with dependent entries to weak convergence in probability. Other applications include patterned random matrices, e.g. matrices of Toeplitz, Hankel or circulant type and matrices with heavy tailed entries in the domain of attraction of the Gaussian distribution.