Probability Theory and Related Fields最新文献

We investigate the large-scale behaviour of the Self-Repelling Brownian Polymer (SRBP) in the critical dimension $d=2$ . The SRBP is a model of self-repelling motion, which is formally given by the solution to a stochastic differential equation driven by a standard Brownian motion and with a drift given by the negative gradient of its own local time. As with its discrete counterpart, the "true" self-avoiding walk (TSAW) of Amit et al. (Phys Rev B 27(3):1635-1645, 1983. 10.1103/PhysRevB.27.1635), it is conjectured to be logarithmically superdiffusive, i.e. to be such that its mean-square displacement grows as $t{(logt)}^{\beta }$ for t large and some currently unknown $\beta \in (0,1)$ . The main result of the paper is an invariance principle for the SRBP under the weak coupling scaling, which corresponds to scaling the SRBP diffusively and simultaneously tuning down the strength of the self-interaction in a scale-dependent way. The diffusivity for the limiting Brownian motion is explicit and its expression provides compelling evidence that the $\beta$ above should be 1/2. Further, we derive the scaling limit of the so-called environment seen by the particle process, which formally solves a non-linear singular stochastic PDE of transport-type, and prove this is given by the solution of a stochastic linear transport equation with enhanced diffusivity.

We analyse the mixing profile of a random walk on a dynamic random permutation, focusing on the regime where the walk evolves much faster than the permutation. Two types of dynamics generated by random transpositions are considered: one allows for coagulation of permutation cycles only, the other allows for both coagulation and fragmentation. We show that for both types, after scaling time by the length of the permutation and letting this length tend to infinity, the total variation distance between the current distribution and the uniform distribution converges to a limit process that drops down in a single jump. This jump is similar to a one-sided cut-off, occurs after a random time whose law we identify, and goes from the value 1 to a value that is a strictly decreasing and deterministic function of the time of the jump, related to the size of the largest component in Erdős-Rényi random graphs. After the jump, the total variation distance follows this function down to 0.

We study the equilibrium fluctuations of an interacting particle system evolving on the discrete ring with $N\in N$ points, denoted by $T_N$ , and with three species of particles that we name A, B and C, but such that at each site there is only one particle. We prove that proper choices of density fluctuation fields (that match those from nonlinear fluctuating hydrodynamics theory) associated to the (two) conserved quantities converge, in the limit $N\to \infty$ , to a system of stochastic partial differential equations, that can either be the Ornstein-Uhlenbeck equation or the Stochastic Burgers equation. To understand the cross interaction between the two conserved quantities, we derive a general version of the Riemann-Lebesgue lemma which is of independent interest.

For a finite graph $G=(V,E)$ let $G^{\ast }$ be obtained by considering a random perfect matching of V and adding the corresponding edges to G with weight $\epsilon$ , while assigning weight 1 to the original edges of G. We consider whether for a sequence $\left(G_n\right)$ of graphs with bounded degrees and corresponding weights $\left({\epsilon }_n\right)$ , the (weighted) random walk on $\left(G_n^{\ast }\right)$ has cutoff. For graphs with polynomial growth we show that $log\left(\frac{1}{{\epsilon }_n}\right)\ll log\left|V_n\right|$ is a sufficient condition for cutoff. Under the additional assumption of vertex-transitivity we establish that this condition is also necessary. For graphs where the entropy of the simple random walk grows linearly up to some time of order $log|V_n|$ we show that $\frac{1}{{\epsilon }_n}\ll log\left|V_n\right|$ is sufficient for cutoff. In the special case of expander graphs we also provide a complete picture for the complementary regime $\frac{1}{{\epsilon }_n}\gtrsim log\left|V_n\right|$ .

Supplementary information: The online version contains supplementary material available at 10.1007/s00440-024-01342-9.

We define a fragmentation process which involves rectangles breaking up into progressively smaller pieces at rates that depend on their shape. Long, thin rectangles are more likely to break quickly, whereas squares break more slowly. Each rectangle is also more likely to split along its longest side. We are interested in how the system evolves over time: how many fragments are there of different shapes and sizes, and how did they reach that state? Using a standard transformation this fragmentation process with shape-dependent rates is equivalent to a two-dimensional branching random walk in continuous time in which the branching rate and the direction of each jump depend on the particles' position. Our main theorem gives an almost sure growth rate along paths for the number of particles in the branching random walk, which in turn gives the number of fragments with a fixed shape as the solution to an optimisation problem. This is a result of interest in the context of spatial branching systems and provides an example of a multitype branching process with a continuum of types.

We establish universal Gaussian fluctuations for the mesoscopic linear eigenvalue statistics in the vicinity of the cusp-like singularities of the limiting spectral density for Wigner-type random matrices. Prior to this work, the linear eigenvalue statistics at the cusp-like singularities were not studied in any ensemble. Our analysis covers not only the exact cusps but the entire transitionary regime from the square-root singularity at a regular edge through the sharp cusp to the bulk. We identify a new one-parameter family of functionals that govern the limiting bias and variance, continuously interpolating between the previously known formulas in the bulk and at a regular edge. Since cusps are the only possible singularities besides the regular edges, our result gives a complete description of the linear eigenvalue statistics in all regimes.

We consider the Glauber dynamics of a ferromagnetic Ising-Kac model on a three-dimensional periodic lattice of size ${(2N+1)}^3$ , in which the flipping rate of each spin depends on an average field in a large neighborhood of radius ${\gamma }^{-1}<<N$ . We study the random fluctuations of a suitably rescaled coarse-grained spin field as $N\to \infty$ and $\gamma \to 0$ ; we show that near the mean-field value of the critical temperature, the process converges in distribution to the solution of the dynamical ${\Phi }_3^4$ model on a torus. Our result settles a conjecture from Giacomin et al. (1999). The dynamical ${\Phi }_3^4$ model is given by a non-linear stochastic partial differential equation (SPDE) which is driven by an additive space-time white noise and which requires renormalisation of the non-linearity. A rigorous notion of solution for this SPDE and its renormalisation is provided by the framework of regularity structures (Hairer in Invent Math 198(2):269-504, 2014. 10.1007/s00222-014-0505-4). As in the two-dimensional case (Mourrat and Weber in Commun Pure Appl Math 70(4):717-812, 2017), the renormalisation corresponds to a small shift of the inverse temperature of the discrete system away from its mean-field value.

The purpose of this work is to provide an explicit construction of a strong Feller semigroup on the space of probability measures over the real line that additionally maps bounded measurable functions into Lipschitz continuous functions, with a Lipschitz constant that blows up in an integrable manner in small time. Our construction relies on a rearranged version of the stochastic heat equation on the circle driven by a coloured noise. Formally, this stochastic equation writes as a reflected equation in infinite dimension. Under the action of the rearrangement, the solution is forced to live in a space of quantile functions that is isometric to the space of probability measures on the real line. We prove the equation to be solvable by means of an Euler scheme in which we alternate flat dynamics in the space of random variables on the circle with a rearrangement operation that projects back the random variables onto the subset of quantile functions. A first challenge is to prove that this scheme is tight. A second one is to provide a consistent theory for the limiting reflected equation and in particular to interpret in a relevant manner the reflection term. The last step in our work is to establish the aforementioned Lipschitz property of the semigroup by adapting earlier ideas from the Bismut-Elworthy-Li formula.

We introduce a theory of non-commutative $L^p$ spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic integrals with respect to Itô-Grassmann processes, Girsanov's formula and a weak formulation of Grassmann SDEs. We apply this new setting to the construction of several unbounded random variables including a Grassmann analog of the  ${\Phi }_2^4$ Euclidean QFT in a bounded region and weak solution to singular SPDEs in the spirit of the early work of Jona-Lasinio and Mitter on the stochastic quantisation of  ${\Phi }_2^4$ .

In this paper we study the convergence of nonlinear Dirichlet problems for systems of variational elliptic PDEs defined on randomly perforated domains of (mathbb {R}^n). Under the assumption that the perforations are small balls whose centres and radii are generated by a stationary short-range marked point process, we obtain in the critical-scaling limit an averaged nonlinear analogue of the extra term obtained in the classical work of Cioranescu and Murat (Res Notes Math III, 1982). In analogy to the random setting recently introduced by Giunti, Höfer and Velázquez (Commun Part Differ Equ 43(9):1377–1412, 2018) to study the Poisson equation, we only require that the random radii have finite ((n-q))-moment, where (1<q<n) is the growth-exponent of the associated energy functionals. This assumption on the one hand ensures that the expectation of the nonlinear q-capacity of the spherical holes is finite, and hence that the limit problem is well defined. On the other hand, it does not exclude the presence of balls with large radii, that can cluster up. We show however that the critical rescaling of the perforations is sufficient to ensure that no percolating-like structures appear in the limit.