Within the Kardar–Parisi–Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work [27] of Dauvergne, Ortmann, and Virág, this object was constructed and, upon a parabolic correction, shown to be the limit of one such model: Brownian last passage percolation. The limit object without parabolic correction, called the directed landscape, admits geodesic paths between any two space-time points $(x,s)$ and $(y,t)$ with $s<t$. In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations $x_1$ and $x_2$, and consider geodesics traveling $(x_1,0)\to (y,1)$ and $(x_2,0)\to (y,1)$. We prove that the set of $y\in \mathbb{R}$ for which these geodesics coalesce only at time 1 has Hausdorff dimension one-half. Second, we consider endpoints $(x,0)$ and $(y,1)$ between which there exist two geodesics intersecting only at times 0 and 1. We prove that the set of such $(x,y)\in {\mathbb{R}}^2$ also has Hausdorff dimension one-half. The proofs require several inputs of independent interest, including (i) connections to the so-called difference weight profile studied in [10]; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in [40].

For fixed functions $G,H:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$, consider the rotationally invariant probability density on ${\mathbb{R}}^n$ of the form

$${\mathrm{\mu }}^n(\mathrm{d}s)=\frac{1}{Z_n}G(\Vert s{\Vert }_2)e^{-nH(\Vert s{\Vert }_2)}\mathrm{d}s.$$

We show that when n is large, the Euclidean norm $\Vert Y^n{\Vert }_2$ of a random vector $Y^n$ distributed according to ${\mathrm{\mu }}^n$ satisfies a thin-shell property, in that its distribution is highly likely to concentrate around a value $s_0$ minimizing a certain variational problem. Moreover, we show that the fluctuations of this modulus away from $s_0$ have the order $1∕\sqrt{n}$ and are approximately Gaussian when n is large.

We apply these observations to rotationally invariant random simplices: the simplex whose vertices consist of the origin as well as independent random vectors $Y_1^n,\dots ,Y_p^n$ distributed according to ${\mathrm{\mu }}^n$, ultimately showing that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior. Our class of measures includes the Gaussian distribution, the beta distribution and the beta prime distribution on ${\mathbb{R}}^n$, provided a generalizing and unifying setting for the objects considered in Grote-Kabluchko-Thäle [Limit theorems for random simplices in high dimensions, ALEA 16, 141–177 (2019)].

Finally, the volumes of random simplices may be related to the determinants of random matrices, and we use our methods with this correspondence to show that if $A^n$ is an $n\times n$ random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants $c_0,c_1\in (0,\mathrm{\infty })$ and an absolute constant $C\in (0,\mathrm{\infty })$ such that

$$\underset{s\in \mathbb{R}}{sup}\left|\mathbb{P}\left[\frac{log\det (A^n)-log(n-1)!-c_0}{\sqrt{\frac{1}{2}logn+c_1}}<s\right]-{\int }_{-\mathrm{\infty }}^s\frac{e^{-u^2∕2}\mathrm{d}u}{\sqrt{2\mathrm{\pi }}}\right|<\frac{C}{{log}^{3∕2}n},$$

sharpening the $1∕{log}^{1∕3+o(1)}n$ bound in Nguyen and Vu [Random matrices: Law of the determinant, Ann. Probab. 42(1) (2014), 146–167].

We develop a stochastic calculus that makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The framework offers a unified treatment of real-valued and complex-valued semimartingales. The proposed calculus is a blueprint for the derivation of new relationships among stochastic processes with specific examples provided below.

In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller–Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman equation may have multiple solutions and we give a probabilistic interpretation for the minimal and the maximal one. Also, we show how to approximate them using finite time horizon problems. The analysis, covering both discrete and continuous time case, is supported with illustrative examples.

Let O be chosen uniformly at random from the group of $(N+L)\times (N+L)$ orthogonal matrices. Denote by $\tilde{O}$ the upper-left $N\times N$ corner of O, which we refer to as a truncation of O. In this paper we prove two conjectures of Forrester, Ipsen and Kumar (2020) on the number of real eigenvalues $N_{\mathbb{R}}^{(m)}$ of the product matrix ${\tilde{O}}_1\dots {\tilde{O}}_m$, where the matrices ${\{{\tilde{O}}_j\}}_{j=1}^m$ are independent copies of $\tilde{O}$. When L grows in proportion to N, we prove that

$$\mathbb{E}(N_{\mathbb{R}}^{(m)})=\sqrt{\frac{2mL}{\mathrm{\pi }}}\mathrm{arctanh}\left(\sqrt{\frac{N}{N+L}}\right)+O(1),N\to \mathrm{\infty }.$$

We also prove the conjectured form of the limiting real eigenvalue distribution of the product matrix. Finally, we consider the opposite regime where L is fixed with respect to N, known as the regime of weak non-orthogonality. In this case each matrix in the product is very close to an orthogonal matrix. We show that $\mathbb{E}(N_{\mathbb{R}}^{(m)})\sim c_{L,m}log(N)$ as $N\to \mathrm{\infty }$ and compute the constant $c_{L,m}$ explicitly. These results generalise the known results in the one matrix case due to Khoruzhenko, Sommers and Życzkowski (2010).

We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process. This model can be viewed as a special case of the random walk in a balanced random environment, for which the weak quenched limit is constructed as a function of the invariant measure of the environment viewed from the walk. We bypass the need to show the existence of this invariant measure. Instead, we find the limit of the quadratic variation of the walk and give an explicit formula for it.

The non-linear sewing lemma constructs flows of rough differential equations from a broad class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential equations, in the spirit of the backward error analysis. Mixing algebra and analysis, a Taylor formula with remainder and a composition formula are central in the expansion analysis. With a suitable algebraic structure on the non-smooth vector fields to be integrated, we recover in a single framework several results regarding high-order expansions for various kinds of driving paths. We also extend the notion of driving rough path. We introduce as an example a new family of branched rough paths, called aromatic rough paths modeled after aromatic Butcher series.

We study dynamic random conductance models on ${\mathbb{Z}}^2$ in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walks on dynamical percolation.

Consider a population evolving from year to year through three seasons: spring, summer and winter. Every spring starts with N dormant individuals waking up independently of each other according to a given distribution. Once an individual is awake, it starts reproducing at a constant rate. By the end of spring, all individuals are awake and continue reproducing independently as Yule processes during the whole summer. In the winter, N individuals chosen uniformly at random go to sleep until the next spring, and the other individuals die. We show that because an individual that wakes up unusually early can have a large number of surviving descendants, for some choices of model parameters the genealogy of the population will be described by a Λ-coalescent. In particular, the beta coalescent can describe the genealogy when the rate at which individuals wake up increases exponentially over time. We also characterize the set of all Λ-coalescents that can arise in this framework.

We consider optimal stopping problems with finite-time horizon and state-dependent discounting. The underlying process is a one-dimensional linear diffusion and the gain function is time-homogeneous and difference of two convex functions. Under mild technical assumptions with local nature we prove fine regularity properties of the optimal stopping boundary including its continuity and strict monotonicity. The latter was never proven with probabilistic arguments. We also show that atoms in the signed measure associated with the second order spatial derivative of the gain function induce geometric properties of the continuation/stopping set that cannot be observed with smoother gain functions (we call them continuation bays and stopping spikes). The value function is continuously differentiable in time without any requirement on the smoothness of the gain function.

We consider the asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$ started from step initial data and obtain the exact Lyapunov exponents for $H_0(t)$, the integrated current of ASEP. As a corollary, we derive an explicit formula for the upper-tail large deviation rate function for $-H_0(t)$. Our result matches with the rate function for the integrated current of the totally asymmetric simple exclusion process (TASEP) obtained in [40].

A condition is identified that implies that solutions to the stochastic reaction-diffusion equation $\frac{\partial u}{\partial t}=\mathcal{A}u+f(u)+\mathit{\sigma }(u)\dot{W}$ on a bounded spatial domain never explode. We consider the case where σ grows polynomially and f is polynomially dissipative, meaning that f strongly forces solutions toward finite values. This result demonstrates the role that the deterministic forcing term f plays in preventing explosion.

We locate the critical threshold $p_c\sim 1∕\sqrt{3nlogn}$ at which it becomes likely that the complete graph $K_n$ can be obtained from the Erdős–Rényi graph ${\mathcal{G}}_{n,p}$ by iteratively completing copies of $K_4$ minus an edge. This refines work of Balogh, Bollobás and Morris that bounds the threshold up to multiplicative constants.

We show that nontrivial bi-infinite polymer Gibbs measures do not exist in typical environments in the inverse-gamma (or log-gamma) directed polymer model on the planar square lattice. The precise technical result is that, except for measures supported on straight-line paths, such Gibbs measures do not exist in almost every environment when the weights are independent and identically distributed inverse-gamma random variables. The proof proceeds by showing that when two endpoints of a point-to-point polymer distribution are taken to infinity in opposite directions but not parallel to lattice directions, the midpoint of the polymer path escapes. The proof is based on couplings, planar comparison arguments, and a recently discovered joint distribution of Busemann functions.

Many Gibbs measures with mean field interactions are known to be chaotic, in the sense that any collection of k particles in the n-particle system are asymptotically independent, as $n\to \mathrm{\infty }$ with k fixed or perhaps $k=o(n)$. This paper quantifies this notion for a class of continuous Gibbs measures on Euclidean space with pairwise interactions, with main examples being systems governed by convex interactions and uniformly convex confinement potentials. The distance between the marginal law of k particles and its limiting product measure is shown to be $O({(k∕n)}^{c\wedge 2})$, with c proportional to the squared temperature. In the high temperature case, this improves upon prior results based on subadditivity of entropy, which yield $O(k∕n)$ at best. The bound $O({(k∕n)}^2)$ cannot be improved, as a Gaussian example demonstrates. The results are non-asymptotic, and distance is quantified via relative Fisher information, relative entropy, or the squared quadratic Wasserstein metric. The method relies on an a priori functional inequality for the limiting measure, used to derive an estimate for the k-particle distance in terms of the $(k+1)$-particle distance.

We consider a branching random walk (BRW) taking its values in the $\mathtt{b}$-ary rooted tree ${\mathbb{W}}_{\mathtt{b}}$ (i.e. the set of finite words written in the alphabet $\{1,\dots ,\mathtt{b}\}$, with $\mathtt{b}\ge 2$). The BRW is indexed by a critical Galton–Watson tree conditioned to have n vertices; its offspring distribution is aperiodic and is in the domain of attraction of a γ-stable law, $\mathit{\gamma }\in (1,2]$. The jumps of the BRW are those of a nearest-neighbour null-recurrent random walk on ${\mathbb{W}}_{\mathtt{b}}$ (reflection at the root of ${\mathbb{W}}_{\mathtt{b}}$ and otherwise: probability $1∕2$ to move closer to the root of ${\mathbb{W}}_{\mathtt{b}}$ and probability $1∕(2\mathtt{b})$ to move away from it to one of the $\mathtt{b}$ sites above). We denote by ${\mathcal{R}}_{\mathtt{b}}(n)$ the range of the BRW in ${\mathbb{W}}_{\mathtt{b}}$ which is the set of all sites in ${\mathbb{W}}_{\mathtt{b}}$ visited by the BRW. We first prove a law of large numbers for $\mathrm{\#}{\mathcal{R}}_{\mathtt{b}}(n)$ and we also prove that if we equip ${\mathcal{R}}_{\mathtt{b}}(n)$ (which is a random subtree of ${\mathbb{W}}_{\mathtt{b}}$) with its graph-distance $d_{\mathtt{gr}}$, then there exists a scaling sequence ${(a_n)}_{n\in \mathbb{N}}$ satisfying $a_n\to \mathrm{\infty }$ such that the metric space $({\mathcal{R}}_{\mathtt{b}}(n),a_n^{-1}{d}_{\mathtt{gr}})$, equipped with its normalised empirical measure, converges to the reflected Brownian cactus with γ-stable branching mechanism: namely, a random compact real tree that is a variant of the Brownian cactus introduced by N. Curien, J-F. Le Gall and G. Miermont in [7].

We study dynamic allocation problems for discrete time multi-armed bandits under uncertainty, based on the the theory of nonlinear expectations. We show that, under independence assumption on the bandits and with some relaxation in the definition of optimality, a Gittins allocation index gives optimal choices. This involves studying the interaction of our uncertainty with controls which determine the filtration. We also run a simple numerical example which illustrates the interaction between the willingness to explore and uncertainty aversion of the agent when making decisions.

We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a structured seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. As geographic space labeling the colonies we consider a countable Abelian group $\mathbb{G}$ endowed with the discrete topology. The key example of interest is the Euclidean lattice $\mathbb{G}={\mathbb{Z}}^d$, $d\in \mathbb{N}$. Our goal is to classify the long-time behaviour of the system in terms of the underlying model parameters. In particular, we want to understand in what way the seed-bank enhances genetic diversity. We introduce three models of increasing generality, namely, individuals become dormant: (1) in the seed-bank of their colony; (2) in the seed-bank of their colony while adopting a random colour that determines their wake-up time; (3) in the seed-bank of a random colony while adopting a random colour. The extension in (2) allows us to model wake-up times with fat tails while preserving the Markov property of the evolution. The extension in (3) allows us to place individuals in different colony when they become dormant. For each of the three models we show that the system of continuum stochastic differential equations, describing the population in the large-colony-size limit, has a unique strong solution. We also show that the system converges to a unique equilibrium depending on a single density parameter that is determined by the initial state, and exhibits a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium) depending on the parameters controlling the migration and the seed-bank. The seed-bank slows down the loss of genetic diversity. In model (1), the dichotomy between clustering and coexistence is determined by migration only. In particular, clustering occurs for recurrent migration and coexistence occurs for transient migration, as for the system without seed-bank. In models (2) and (3), an interesting interplay between migration and seed-bank occurs. In particular, the dichotomy is affected by the seed-bank when the wake-up time has infinite mean. For instance, for critically recurrent migration the system exhibits clustering for finite mean wake-up time and coexistence for infinite mean wake-up time. Hence, at the critical dimension for the system without seed-bank, new universality classes appear when the seed-bank is added. If the wake-up time has a sufficiently fat tail, then the seed-bank determines the dichotomy and migration has no effect at all. The presence of the seed-bank makes the proof of convergence to a unique equilibrium a conceptually delicate issue. By combining duality arguments with coupling techniques, we show that our results also hold when we replace the Fisher-Wright diffusion function by a more general diffusion function, drawn from an appropriate class.

In this paper, we study the maximal edge-traversal time on optimal paths in First-passage percolation on the lattice ${\mathbb{Z}}^d$ for several edge distributions, including the Pareto and Weibull distributions. It is known to be unbounded when the edge distribution has unbounded support [J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 56-80, 1993]. We determine the order of the growth depending on the tail of the edge distribution.

In this paper, we study the $extended$ mean field control problem, which is a class of McKean–Vlasov stochastic control problem where the state dynamics and the reward functions depend upon the joint (conditional) distribution of the controlled state and the control process. By considering an appropriate controlled Fokker–Planck equation, we can formulate an optimization problem over a space of measure–valued processes and, under suitable assumptions, prove the equivalence between this optimization problem and the $extended$ mean–field control problem. Moreover, with the help of this new optimization problem, we establish the associated limit theory i.e. the $extended$ mean field control problem is the limit of a large population control problem where the interactions are achieved via the empirical distribution of state and control processes.

We give a short analytic proof of local large deviations for i.i.d. random variables in the domain of a multivariate α-stable law, $\mathit{\alpha }\in (0,1)\cup (1,2]$. Our method simultaneously covers lattice and nonlattice distributions (and mixtures thereof), bypassing aperiodicity considerations. The proof applies also to the dynamical setting.

In this paper we study the asymptotic behavior, as $t\downarrow 0$, of the spectral heat content $Q_D^{(\mathit{\alpha })}(t)$ for isotropic α-stable processes, $\mathit{\alpha }\in [1,2)$, in bounded $C^{1,1}$ open sets $D\subset {\mathbb{R}}^d$, $d\ge 2$. Together with the results from [2] for $d=1$ and [7] for $\mathit{\alpha }\in (0,1)$, the main theorem of this paper establishes the asymptotic behavior of the spectral heat content up to the second term for all $\mathit{\alpha }\in (0,2)$ and $d\ge 1$, and resolves the conjecture raised in [2].

We prove the hydrodynamic limit of a totally asymmetric zero range process on a torus with two lanes and randomly oriented edges. The asymmetry implies that the model is non-reversible. The random orientation of the edges is constructed in a bistochastic fashion which keeps the usual product distribution stationary for the quenched zero range model. It is also arranged to have no overall drift along the $\mathbb{Z}$ direction, which suggests diffusive scaling despite the asymmetry present in the dynamics. Indeed, using the relative entropy method, we prove the quenched hydrodynamic limit to be the heat equation with a diffusion coefficient depending on ergodic properties of the orientation of the edges.

The zero range process on this graph turns out to be non-gradient. Our main novelty is the introduction of a local equilibrium measure which decomposes the vertices of the graph into components of constant density. A clever choice of these components eliminates the non-gradient problems that normally arise during the hydrodynamic limit procedure.

This paper adapts the recently developed rigorous application of the supersymmetric transfer matrix approach for the Hermitian 1d band matrices to the case of the orthogonal symmetry. We consider $N\times N$ block band matrices consisting of $W\times W$ random Gaussian blocks (parametrized by $j,k\in \mathrm{\Lambda }=[1,n]\cap \mathbb{Z}$, $N=nW$) with a fixed entry’s variance $J_{jk}=W^{-1}({\mathit{\delta }}_{j,k}+\mathit{\beta }{\mathrm{\Delta }}_{j,k})$ in each block. Considering the limit $W,n\to \mathrm{\infty }$, we prove that the behaviour of the second correlation function of characteristic polynomials of such matrices in the bulk of the spectrum exhibit a crossover near the threshold $W\sim \sqrt{N}$.

We study one-dimensional cellular automata whose rules are chosen at random from among r-neighbor rules with a large number n of states. Our main focus is the asymptotic behavior, as $n\to \mathrm{\infty }$, of the longest temporal period $X_{\mathit{\sigma },n}$ of a periodic solution with a given spatial period σ. We prove, when $\mathit{\sigma }\le r$, that this random variable is of order $n^{\mathit{\sigma }∕2}$, in that $X_{\mathit{\sigma },n}∕n^{\mathit{\sigma }∕2}$ converges to a nontrivial distribution. For the case $\mathit{\sigma }>r$, we present empirical evidence in support of the conjecture that the same result holds.

We consider a variant of simple random walk on a finite group. At each step, we choose an element, s, from a set of generators (“directions”) uniformly, and an integer, j, from a power law distribution (“speed”) associated with the chosen direction, and move from the current position, g, to $g{s}^j$. We show that if the finite group is nilpotent, the time it takes this walk to reach its uniform equilibrium is of the same order of magnitude as the diameter of a suitable pseudo-metric on the group, which is attached to the generators and speeds. Additionally, we give sharp bounds on the ${\ell }^2$-distance between the distribution of the position of the walker and the stationary distribution, and compute the relevant diameter for some examples.

This paper investigates spin distributions for a generic spherical p-spin model; we give a representation of spin distributions in terms of a stochastic process. In order to do this, we find a series of invariance principles analogous to the cavity equations used for the hypercube. The construction of these equations is a multistep process; we first change coordinates to write the sphere as a product space $S_{N-R^2}\times [-C,C]$ for some cutoff parameters R and C. We then construct an auxiliary Hamiltonian that writes the effect of the final spin as an independent Gaussian process, e.g., $H_N\approx H_{N-1}+Y_N+Z_N$ with $Y_N$ and $Z_N$ decoupled from $H_{N-1}$. The error rate in this approximation decays as $O(\frac{C^4}{R^2})$. The final step is to justify the double limit ${lim}_{C\to \mathrm{\infty }}{lim}_{R\to \mathrm{\infty }}$ to derive exact results; this double limit can be interpreted as a novel renormalization procedure.

We study the Markov chain on ${\mathbf{F}}_p$ obtained by applying a function f and adding $\pm \mathit{\gamma }$ with equal probability. When f is a linear function, this is the well-studied Chung–Diaconis–Graham process. We consider two cases: when f is the extension of a non-linear rational function which is bijective, and when $f(x)=x^2$. In the latter case, the stationary distribution is not uniform and we characterize it when $p=3(mod4)$. In both cases, we give an almost linear bound on the mixing time, showing that the deterministic function dramatically speeds up mixing. The proofs involve establishing bounds on exponential sums over the union of short intervals.

This paper focuses on the non-asymptotic concentration of the heteroskedastic Wishart-type matrices. Suppose Z is a $p_1$-by-$p_2$random matrix and $Z_{ij}\sim N(0,{\mathit{\sigma }}_{ij}^2)$ independently, we prove the expected spectral norm of Wishart matrix deviations (i.e., $\mathbb{E}‖Z{Z}^{\top }-\mathbb{E}Z{Z}^{\top }‖$) is upper bounded by

$$(1+\mathit{ϵ})\left\{2{\mathit{\sigma }}_C{\mathit{\sigma }}_R+{\mathit{\sigma }}_C^2+C{\mathit{\sigma }}_R{\mathit{\sigma }}_{\ast }\sqrt{log(p_1\wedge p_2)}+C{\mathit{\sigma }}_{\ast }^{2}log(p_1\wedge p_2)\right\},$$

where ${\mathit{\sigma }}_C^2:={max}_j{\sum }_{i=1}^{p_1}{\mathit{\sigma }}_{ij}^2$, ${\mathit{\sigma }}_R^2:={max}_i{\sum }_{j=1}^{p_2}{\mathit{\sigma }}_{ij}^2$ and ${\mathit{\sigma }}_{\ast }^2:={max}_{i,j}{\mathit{\sigma }}_{ij}^2$. A minimax lower bound is developed that matches this upper bound. Then, we derive the concentration inequalities, moments, and tail bounds for the heteroskedastic Wishart-type matrix under more general distributions, such as sub-Gaussian and heavy-tailed distributions. Next, we consider the cases where Z has homoskedastic columns or rows (i.e., ${\mathit{\sigma }}_{ij}\approx {\mathit{\sigma }}_i$ or ${\mathit{\sigma }}_{ij}\approx {\mathit{\sigma }}_j$) and derive the rate-optimal Wishart-type concentration bounds. Finally, we apply the developed tools to identify the sharp signal-to-noise ratio threshold for consistent clustering in the heteroskedastic clustering problem.

We consider large random planar maps and study the first-passage percolation distance obtained by assigning independent identically distributed lengths to the edges. We consider the cases of quadrangulations and of general planar maps. In both cases, the first-passage percolation distance is shown to behave in large scales like a constant times the usual graph distance. We apply our method to the metric properties of the classical Tutte bijection between quadrangulations with n faces and general planar maps with n edges. We prove that the respective graph distances on the quadrangulation and on the associated general planar map are in large scales equivalent when $n\to \mathrm{\infty }$.

We study the norm of the two-dimensional Brownian motion conditioned to stay outside the unit disk at all times. By conditioning the process is changed from barely recurrent to slightly transient. We obtain sharp results on the rate of escape to infinity of the process of future minima:

For this, we introduce a renewal structure attached to record times and values. Additional results are given for the long time behavior of the norm.

Let f be a Rademacher or a Steinhaus random multiplicative function. Let $\mathit{\epsilon }>0$ small. We prove that, as $x\to +\mathrm{\infty }$, we almost surely have

$$|\sum _{\begin{array}{c}n\le x\\ P(n)>\sqrt{x}\end{array}}f(n)|\le \sqrt{x}{(loglogx)}^{1∕4+\mathit{\epsilon }},$$

where $P(n)$ stands for the largest prime factor of n. This gives an indication of the almost sure size of the largest fluctuations of f.

Let $\mathrm{\kappa }\in {\mathbb{N}}_{+}^{\ell }$ satisfy ${\mathrm{\kappa }}_1+\cdots +{\mathrm{\kappa }}_{\ell }=n$, and let ${\mathcal{U}}_{\mathrm{\kappa }}$ denote the multislice of all strings $u\in {[\ell ]}^n$ having exactly ${\mathrm{\kappa }}_i$ coordinates equal to i, for all $i\in [\ell ]$. Consider the Markov chain on ${\mathcal{U}}_{\mathrm{\kappa }}$ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant ${\mathit{\varrho }}_{\mathrm{\kappa }}$ for the chain satisfies

$${\mathit{\varrho }}_{\mathrm{\kappa }}^{-1}\le n\cdot \sum _{i=1}^{\ell }\frac{1}{2}{log}_2(4n∕{\mathrm{\kappa }}_i),$$

which is sharp up to constants whenever ℓ is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal–Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan–Szegedy Theorem.

This paper studies the local structure of continuous random fields on ${\mathbb{R}}^d$ taking values in a complete separable linear metric space $\mathbb{V}$. Extending seminal work of Falconer, we show that the generalized $(1+k)$-th order increment tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron. These results motivate the further study of the structure of $\mathbb{V}$-valued intrinsic random functions of order k (IRF${}_k$, $k=0,1,\dots$). To this end, we focus on the special case where $\mathbb{V}$ is a Hilbert space. Building on the work of Sasvari and Berschneider, we establish the spectral characterization of all second order $\mathbb{V}$-valued IRF${}_k$’s, extending the classical Matheron theory. Using these results, we further characterize the class of Gaussian, operator self-similar $\mathbb{V}$-valued IRF${}_k$’s, generalizing results of Dobrushin and Didier, Meerschaert and Pipiras, among others. These processes are the Hilbert-space-valued versions of the general k-th order operator fractional Brownian fields and are characterized by their self-similarity operator exponent as well as a finite trace class operator valued spectral measure. We conclude with several examples motivating future applications to probability and statistics.

To highlight certain similarities in graphical representations of several well known two-dimensional models of statistical mechanics, we introduce and study a new family of models which specializes to these cases after a proper tuning of the parameters.

To be precise, our model consists of two independent standard Potts models, with possibly different numbers of spins and different coupling constants (the four parameters of the model), defined jointly on a graph embedded in a surface and its dual graph, and conditioned on the event that the primal and dual interfaces between spins of different value do not intersect. We also introduce naturally coupled height function and bond percolation models, and we discuss fundamental properties of the resulting joint coupling.

As special cases we recover the standard Potts and random cluster model, the six-vertex model and loop $O(n)$ model, the random current, double random current and XOR-Ising model.

The paper deals with a random connection model, a random graph whose vertices are given by a homogeneous Poisson point process on ${\mathbb{R}}^d$, and edges are independently drawn with probability depending on the locations of the two end points. We establish central limit theorems (CLT) for general functionals on this graph under minimal assumptions that are a combination of the weak stabilization for the add-one cost and a $(2+\mathit{\delta })$-moment condition. As a consequence, CLTs for isomorphic subgraph counts, isomorphic component counts, the number of connected components are then derived. In addition, CLTs for Betti numbers and the size of the largest component are also proved for the first time.

We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [17]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the equation, as well as almost sure convergence of the associated particle system, for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions.

We consider the Ising model on the hexagonal lattice evolving according to Metropolis dynamics. We study its metastable behavior in the limit of vanishing temperature when the system is immersed in a small external magnetic field. We determine the asymptotic properties of the transition time from the metastable to the stable state up to a multiplicative factor and study the mixing time and the spectral gap of the Markov process. We give a geometrical description of the critical configurations and show how not only their size but their shape varies depending on the thermodynamical parameters. Finally we provide some results concerning polyiamonds of maximal area and minimal perimeter.

We study the shape of the normalized stable Lévy tree $\mathcal{T}$ near its root. We show that, when zooming in at the root at the proper speed with a scaling depending on the index of stability, we get the unnormalized Kesten tree. In particular the limit is described by a tree-valued Poisson point process which does not depend on the initial normalization. We apply this to study the asymptotic behavior of additive functionals of the form

$${\mathbf{Z}}_{\mathit{\alpha },\mathit{\beta }}={\int }_{\mathcal{T}}\mathrm{\mu }(\mathrm{d}x){\int }_0^{H(x)}{\mathit{\sigma }}_{r,x}^{\mathit{\alpha }}{\mathfrak{h}}_{r,x}^{\mathit{\beta }}\mathrm{d}r$$

as $max(\mathit{\alpha },\mathit{\beta })\to \mathrm{\infty }$, where μ is the mass measure on $\mathcal{T}$, $H(x)$ is the height of x and ${\mathit{\sigma }}_{r,x}$ (resp. ${\mathfrak{h}}_{r,x}$) is the mass (resp. height) of the subtree of $\mathcal{T}$ above level r containing x. Such functionals arise as scaling limits of additive functionals of the size and height on conditioned Bienaymé-Galton-Watson trees.

We consider the asymmetric exclusion process with a driven tagged particle on $\mathbb{Z}$ which has different jump rates from other particles. When the non-tagged particles have non-nearest-neighbor jump rates , we show that the tagged particle can have a speed which has a different sign from the mean derived from its jump rates. We also show the existence of some non-trivial invariant measures for the environment process viewed from the tagged particle. Our arguments are based on coupling, martingale methods, and analyzing currents through fixed bonds.

The contact process is a particular case of birth-and-death processes on infinite particle configurations. We consider the contact processes on locally compact separable metric spaces. We prove the existence of a one-parameter set of invariant measures in the critical regime under the condition imposed on the associated Markov jump process. This condition means that any pair of independent trajectories of this jump process run away from each other. The general scheme can be applied to the contact process on the lattice in a heterogeneous and random environments as well as to the contact process on graphs and on manifolds.

Consider the following one player game. A deck containing m copies of n different card types is shuffled uniformly at random. Each round the player tries to guess the next card in the deck, and then the card is revealed and discarded. It was shown by Diaconis, Graham, He, and Spiro that if m is fixed, then the maximum expected number of correct guesses that the player can achieve is asymptotic to $H_{m}logn$, where $H_m$ is the mth harmonic number.

In this paper we consider an adversarial version of this game where a second player shuffles the deck according to some (possibly non-uniform) distribution. We prove that a certain greedy strategy for the shuffler is the unique optimal strategy in this game, and that the guesser can achieve at most $logn$expected correct guesses asymptotically for fixed m against this greedy strategy..

We show existence of an invariant probability measure for a class of functional McKean-Vlasov SDEs by applying Kakutani’s fixed point theorem to a suitable class of probability measures on a space of continuous functions. Unlike some previous works [1, 25], we do not assume a monotonicity condition to hold. Further, our conditions are even weaker than some results in the literature on invariant probability measures for functional SDEs without dependence on the law of the solution [12].

We derive local semicircle laws for random matrices with exchangeable entries which exhibit correlations that decay slowly in the dimension N of the matrix. To be precise, any ℓ-point correlation $𝔼[Y_1\cdots Y_{\ell }]$ between distinct matrix entries $Y_1,\dots ,Y_{\ell }$ may decay at a rate of only $N^{-\ell ∕2}$. We call our ensembles of (high temperature) Curie-Weiss type, and Curie-Weiss(β)-distributed entries directly fit within our framework in the high temperature regime $\mathit{\beta }\in [0,1]$. Using rank-one perturbations, we show that even in the low-temperature regime $\mathit{\beta }\in (1,\mathrm{\infty })$, where ℓ-point correlations survive in the limit, the local semicircle law still holds after rescaling the matrix entries with a constant which depends on β but not on N.

The purpose of this article is to investigate the continuum limit of a distributional identity for half-space directed polymers given in [BBC20]. The limiting identity turns out to relate the multiplicative-noise half-space stochastic heat equation with Dirichlet boundary condition to the same equation with Robin boundary condition. The identity is related to conjectured Gaussian fluctuation behavior of subcritical half-space KPZ.

We prove large deviation principles for the distribution of the empirical measure of the eigenvalues of Lax matrices following the Generalized Gibbs ensembles of the classical Toda chain introduced in [9]. We deduce the almost sure convergence of this empirical measure towards a limit which we describe in terms of the limiting empirical measure of Beta-ensembles. Our results apply to general smooth potentials.

We give general theorems on asymptotic normality for additive functionals of random tries generated by a sequence of independent strings. These theorems are applied to show asymptotic normality of the distribution of random fringe trees in a random trie. Formulas for asymptotic mean and variance are given. In particular, the proportion of fringe trees of size k (defined as number of keys) is asymptotically, ignoring oscillations, $c∕(k(k-1))$ for $k\ge 2$, where $c=1∕(1+H)$ with H the entropy of the letters. Another application gives asymptotic normality of the number of k-protected nodes in a random trie. For symmetric tries, it is shown that the asymptotic proportion of k-protected nodes (ignoring oscillations) decreases geometrically as $k\to \mathrm{\infty }$.

We analyze the scattering transform with the quadratic nonlinearity (STQN) of Gaussian processes without depth limitation. STQN is a nonlinear transform that involves a sequential interlacing convolution and nonlinear operators, which is motivated to model the deep convolutional neural network. We prove that with a proper normalization, the output of STQN converges to a chi-square process with one degree of freedom in the finite dimensional distribution sense, and we provide a total variation distance control of this convergence at each time that converges to zero at an exponential rate. To show these, we derive a recursive formula to represent the intricate nonlinearity of STQN by a linear combination of Wiener chaos, and then apply the Malliavin calculus and Stein’s method to achieve the goal.

We investigate the noise sensitivity of the top eigenvector of a sparse random symmetric matrix. Let v be the top eigenvector of an $N\times N$ sparse random symmetric matrix with an average of d non-zero centered entries per row. We resample k randomly chosen entries of the matrix and obtain another realization of the random matrix with top eigenvector $v^{[k]}$. Building on recent results on sparse random matrices and a noise sensitivity analysis previously developed for Wigner matrices, we prove that, if $d\ge N^{2∕9}$, with high probability, when $k\ll N^{5∕3}$, the vectors v and $v^{[k]}$ are almost collinear and, on the contrary, when $k\gg N^{5∕3}$, the vectors v and $v^{[k]}$ are almost orthogonal. A similar result holds for the eigenvector associated to the second largest eigenvalue of the adjacency matrix of an Erdős-Rényi random graph with average degree $d\ge N^{2∕9}$.

We consider the sample average of a centered random walk in ${\mathbb{R}}^d$ with regularly varying step size distribution. For the first exit time from a compact convex set A not containing the origin, we show that its tail is of lognormal type. Moreover, we show that the typical way for a large exit time to occur is by having a number of jumps growing logarithmically in the scaling parameter.

A random field $X={(X_v)}_{v\in G}$ on a quasi-transitive graph G is a factor of an i.i.d. process if it can be written as $X=\mathit{\phi }(Y)$ for some i.i.d. process $Y={(Y_v)}_{v\in G}$ and equivariant map φ. Such a map, also called a coding, is finitary if, for every vertex $v\in G$, there exists a finite (but random) set $U\subset G$ such that $X_v$ is determined by ${\{Y_u\}}_{u\in U}$. We construct a coding for the random-cluster model on G, and show that the coding is finitary whenever the free and wired measures coincide. This strengthens a result of Häggström–Jonasson–Lyons [18]. We also prove that the coding radius has exponential tails in the subcritical regime. As a corollary, we obtain a similar coding for the subcritical Potts model.

Our methods are probabilistic in nature, and at their heart lies the use of coupling-from-the-past for the Glauber dynamics. These methods apply to any monotone model satisfying mild technical (but natural) requirements. Beyond the random-cluster and Potts models, we describe two further applications – the loop $O(n)$ model and long-range Ising models. In the case of $G={\mathbb{Z}}^d$, we also construct finitary, translation-equivariant codings using a finite-valued i.i.d. process Y. To do this, we extend a mixing-time result of Martinelli–Olivieri [22] to infinite-range monotone models on quasi-transitive graphs of sub-exponential growth.

We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed p-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model’s covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constraint set is a compact metric space. This space is not related to the model’s spherical structure and can be carried over to other multi-species settings.

We introduce an algorithm that constructs a random graph with prescribed degree sequence together with a depth first exploration of it. In the so-called supercritical regime where the graph contains a giant component, we prove that the renormalized contour process of the Depth First Search Tree has a deterministic limiting profile that we identify. The proof goes through a detailed analysis of the evolution of the empirical degree distribution of unexplored vertices. This evolution is driven by an infinite system of differential equations which has a unique and explicit solution. As a byproduct, we deduce the existence of a macroscopic simple path and get a lower bound on its length.

We investigate the average minimum cost of a bipartite matching, with respect to the squared Euclidean distance, between two samples of n i.i.d. random points on a bounded Lipschitz domain in the Euclidean plane, whose common law is absolutely continuous with strictly positive Hölder continuous density. We confirm in particular the validity of a conjecture by D. Benedetto and E. Caglioti stating that the asymptotic cost as n grows is given by the logarithm of n multiplied by an explicit constant times the volume of the domain. Our proof relies on a reduction to the optimal transport problem between the associated empirical measures and a Whitney-type decomposition of the domain, together with suitable upper and lower bounds for local and global contributions, both ultimately based on PDE tools. We further show how to extend our results to more general settings, including Riemannian manifolds, and also give an application to the asymptotic cost of the random quadratic bipartite travelling salesperson problem.

This paper presents the existence and uniqueness of solutions to an abstract nonlinear equation driven by multiplicative noise of Lévy type. This equation covers many hydrodynamical models, including 2D Navier-Stokes equations, 2D MHD equations, the 2D Magnetic Bernard problem, and several Shell models of turbulence. In the literature on this topic, besides the classical Lipschitz and linear growth conditions, other atypical assumptionsare also required on the coefficients of the stochastic perturbations. The goal of this paper is to eliminate these atypical assumptions. Our assumption on the coefficients of stochastic perturbations is new even for the Wiener cases and, in one sense, is shown to be quite sharp. A new cutting off argument and energy estimation procedure play an important role in establishing the existence and uniqueness under this assumption.

Consider the long-range percolation model on the integer lattice ${\mathbb{Z}}^d$ in which all nearest-neighbour edges are present and otherwise x and y are connected with probability $q_{x,y}:=1-exp(-|x-y{|}^{-s})$, independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e. for $s>d$,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value $d=1$, $s=2$, where the spectral dimension is discontinuous. Towards this end, we present various on-diagonal heat kernel bounds, a number of which are new. In particular, the lower bounds are derived through the application of a general technique that utilises the translation invariance of the model. We highlight that, applying this general technique, we are able to partially extend our main result beyond the nearest-neighbour setting, and establish lower heat kernel bounds over the range of parameters $s\in (d,2d)$. We further note that our approach is applicable to short-range models as well.

The purpose of this paper is to highlight some hidden Markovian structure of the concave majorant of the Brownian motion. Several distributional identities are implied by the joint law of a standard one-dimensional Brownian motion B and its almost surely unique concave majorant K on $[0,\mathrm{\infty })$. In particular, the one-dimensional distribution of $2K_t-B_t$ is that of $R_5(t)$, where $R_5$ is a 5-dimensional Bessel process with $R_5(0)=0$. The process $2K-B$ shares a number of other properties with $R_5$, and we conjecture that it may have the distribution of $R_5$. We also describe the distribution of the convex minorant of a three-dimensional Bessel process with drift.

We consider a model of first passage percolation (FPP) where the nearest-neighbor edges of the standard two-dimensional Euclidean lattice are equipped with random variables. These variables are i.i.d. nonnegative, continuous, and have a finite moment generating function in a neighborhood of 0. We derive consequences about transverse increments of passage times, assuming the model satisfies certain properties. Approximately, the assumed properties are the following: We assume that the standard deviation of the passage time on scale r is of some order $\mathit{\sigma }(r)$, and $\left\{\mathit{\sigma }(r),r>0\right\}$ grows approximately as a power of r. Also, the tails of the passage time distributions for distance r satisfy an exponential bound on a scale $\mathit{\sigma }(r)$ uniformly over r. In addition, the boundary of the limit shape in a neighborhood of some fixed direction θ has a uniform quadratic curvature. By transverse increment we mean the difference between passage times from the origin to a pair of points which are approximately at the direction θ and the direction between the pair of points is the direction of the tangent to the boundary of the limit shape at the direction θ. The main consequence derived is the following. If $\mathit{\sigma }(r)$ varies as $r^{\mathit{\chi }}$ for some $\mathit{\chi }>0$, and ξ is such that $\mathit{\chi }=2\mathrm{\xi }-1$, then the fluctuation of the transverse increment of passage time between a pair of points situated at distance r from each other is of the order of $r^{\mathit{\chi }∕\mathrm{\xi }}$.

Consider the Langevin process, described by a vector (positions and momenta) in ${\mathbb{R}}^d\times {\mathbb{R}}^d$. Let $\mathcal{O}$ be a ${\mathcal{C}}^2$ open bounded and connected set of ${\mathbb{R}}^d$. Recent works showed the existence of a unique quasi-stationary distribution (QSD) of the Langevin process on the domain $D:=\mathcal{O}\times {\mathbb{R}}^d$. In this article, we study the overdamped limit of this QSD, i.e. when the friction coefficient goes to infinity. In particular, we show that the marginal law in position of the overdamped limit is the QSD of the overdamped Langevin process on the domain $\mathcal{O}$.

We investigate random graphs on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random mark and given marks and positions of the points we form an edge between two points independently with a probability depending via a kernel on the two marks and the distance of the points. Different kernels allow the mark to play different roles, like weight, radius or birth time of a vertex. The kernels depend on a parameter γ, which determines the power-law exponent of the degree distributions. A further independent parameter δ characterises the decay of the connection probabilities of vertices as their distance increases. We prove transience of the infinite cluster in the entire supercritical phase in regimes given by the parameters γ and δ, and complement these results by recurrence results if $d=2$. Our results are particularly interesting for the soft Boolean graph model discussed in the preprint [arXiv:2108:11252] and the age-dependent random connection model recently introduced by Gracar et al. [Queueing Syst. 93.3-4 (2019)]

In order to study the regularity of the density of a solution of a infinite activity jump driven stochastic differential equation we consider the following two-step approximation method. First, we use the solution of the moment problem in order to approximate the small jumps by another whose Lévy measure has finite support. In a second step we replace the approximation of the first two moments by a small noise Brownian motion based on the Assmussen-Rosiński approach. This approximation needs to satisfy certain properties in order to apply the “balance” method which allows the study of densities for the solution process based on Malliavin Calculus for the Brownian motion. Our results apply to situations where the Lévy measure is absolutely continuous with respect to the Lebesgue measure or purely atomic measures or combinations of them.

We study the weak convergence of β- and ${\mathit{\beta }}^{\prime }$-Delaunay tessellations in ${\mathbb{R}}^{d-1}$ that were introduced in part I of this paper, as $\mathit{\beta }\to \mathrm{\infty }$. The limiting stationary simplicial random tessellation, which is called the Gaussian-Delaunay tessellation, is characterized in terms of a space-time paraboloid hull process in ${\mathbb{R}}^{d-1}\times \mathbb{R}$. The latter object has previously appeared in the analysis of the number of shocks in the solution of the inviscid Burgers’ equation and the description of the local asymptotic geometry of Gaussian random polytopes. In this paper it is used to define a new stationary random simplicial tessellation in ${\mathbb{R}}^{d-1}$. As for the β- and ${\mathit{\beta }}^{\prime }$-Delaunay tessellation, the distribution of volume-power weighted typical cells in the Gaussian-Delaunay tessellation is explicitly identified, establishing thereby a new bridge to Gaussian random simplices. Also major geometric characteristics of these cells such as volume moments, expected angle sums and also the cell intensities of the Gaussian-Delaunay tessellation are investigated.

We prove that the Beta random walk, introduced in [BC17] 2017, has cubic fluctuations from the large deviation principle of the GUE Tracy-Widom type for arbitrary values $\mathit{\alpha }>0$ and $\mathit{\beta }>0$ of the parameters of the Beta distribution, removing previous restrictions on their values. Furthermore, we prove that the GUE Tracy-Widom fluctuations still hold in the intermediate disorder regime. We also show that any random walk in space-time random environment that matches certain moments with the Beta random walk also has GUE Tracy-Widom fluctuations in the intermediate disorder regime. As a corollary we show the emergence of GUE Tracy-Widom fluctuations from the large deviation principle for trajectories ending at boundary points for random walks in space (time-independent) i.i.d. Dirichlet random environment in dimension $d=2$ for a class of asymptotic behavior of the parameters.

A general framework for the study of regular variation (RV) is that of Polish star-shaped metric spaces, while recent developments in [41] have discussed RV with respect to a properly localised boundedness $\mathcal{B}$. Along the lines of the latter approach, we discuss the RV of Borel measures and random processes on a general Polish metric spaces $(\mathrm{D},d_{\mathrm{D}})$. Tail measures introduced in [47] appear naturally as limiting measures of regularly varying time series. We define tail measures on the measurable space $(\mathrm{D},\mathcal{D})$ indexed by $\mathcal{H}(\mathrm{D})$, a countable family of 1-homogeneous coordinate maps, and show some tractable instances for the investigation of RV when $\mathcal{B}$ is determined by $\mathcal{H}(\mathrm{D})$. This allows us to study the regular variation of càdlàg processes on $D({\mathbb{R}}^l,{\mathbb{R}}^d)$ retrieving in particular results obtained in [59] for RV of stationary càdlàg processes on the real line removing $l=1$ therein. Further, we discuss potential applications and open questions.

We propose a definition of directional multivariate subexponential and convolution equivalent densities and find a useful characterization of these notions for a class of integrable and almost radial decreasing functions. We apply this result to show that the density of the absolutely continuous part of the compound Poisson measure built on a given density f is directionally convolution equivalent and inherits its asymptotic behaviour from f if and only if f is directionally convolution equivalent. We also extend this characterization to the densities of more general infinitely divisible distributions on ${ℝ}^d$, $d\ge 1$, which are not pure compound Poisson.

We study the winding behavior of random walks on two oriented square lattices. One common feature of these walks is that they are bound to revolve clockwise. We also obtain quantitative results of transience/recurrence for each walk.

Given a weighted finite graph G, we consider a random partition of its vertex set induced by a measure on spanning rooted forests on G. The latter is a generalized parametric version of the classical Uniform Spanning Tree measure which can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter $q>0$. The related random trees—identifying the blocks of the partition—tend to cluster nodes visited by the random walk on time scale $1∕q$. We explore the emerging macroscopic structure by analyzing two-point correlations, as a function of the tuning parameter q. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block. This interaction potential can be seen as an affinity measure for “densely connected nodes” and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute this potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structure. For such geometries we show phase-transitions in the behavior of the random partition as a function of the tuning parameter and the edge weights. Moreover, as a corollary of our main results, we infer the right scaling of the parameters that give rise to the emergence of “giant” blocks.

We study biased random walk on the infinite connected component of supercritical percolation on the integer lattice ${\mathbb{Z}}^d$ for $d\ge 2$. For this model, Fribergh and Hammond showed the existence of an exponent γ such that: for $\mathit{\gamma }<1$, the random walk is sub-ballistic (i.e. has zero velocity asymptotically), with polynomial escape rate described by γ; whereas for $\mathit{\gamma }>1$, the random walk is ballistic, with non-zero speed in the direction of the bias. They moreover established, under the usual diffusive scaling about the mean distance travelled by the random walk in the direction of the bias, a central limit theorem when $\mathit{\gamma }>2$. In this article, we explain how Fribergh and Hammond’s percolation estimates further allow it to be established that for $\mathit{\gamma }\in (1,2)$ the fluctuations about the mean are of an anomalous polynomial order, with exponent given by ${\mathit{\gamma }}^{-1}$.