Geometry of Gaussian free field sign clusters and random interlacements

For a large class of amenable transient weighted graphs $G$, we prove that the sign clusters of the Gaussian free field on $G$ fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices like $\mathbb{Z}^d$, for $d \geqslant 3$, but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman in arXiv:0704.2560, and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from arXiv:1010.1490.Comment: 73 page

Développement physiologique des voies visuelles chez le rat normal et chez celui ayant subi des convulsions hyperthermiques

Les neurones des couches superficielles du collicule supérieur et du cortex visuel primaire du rat adulte sont sensibles à de basses fréquences spatiales de haut contraste défilant à des vitesses élevées. Entre les jours post-nataux 27-30 et l’âge adulte, les fréquences temporelles optimales des neurones du cortex visuel primaire augmentent, tandis que leurs seuils de contraste diminuent. Cependant, les fréquences spatiales optimales, les valeurs de résolution spatiale et les bandes passantes spatiales de ces neurones sont, dès l’ouverture des paupières, similaires à celles observées chez le rat adulte. Ces profils de réponse neuronale suggèrent que les projections rétino-colliculaires et rétino-géniculo-corticales sont essentiellement issues de neurones ganglionnaires rétinofuges magnocellulaires et koniocellulaires. Les neurones du cortex visuel primaire du rat ayant subi des convulsions hyperthermiques présentent, dès l’ouverture des paupières, de basses fréquences spatiales optimales, de larges bandes passantes directionnelles et temporelles ainsi que des seuils de contraste élevés par rapport aux neurones du rat normal. À l’âge adulte, de basses fréquences temporelles optimales et de larges bandes passantes spatiales sont également observées chez le rat ayant subi des convulsions hyperthermiques. L’altération des profils de réponse des neurones du cortex visuel primaire du rat ayant subi de convulsions hyperthermiques suggère un déséquilibre entre les mécanismes neuronaux excitateurs et inhibiteurs de cette aire corticale. Ces résultats suggèrent également qu’un épisode unique de convulsions fébriles infantiles suffit à altérer le développement des propriétés spatio-temporelles des champs récepteurs des neurones du cortex visuel primaire.Neurons in superficial layers of the rat superior colliculus and primary visual cortex are sensitive to highly contrasted low spatial frequencies drifting at fast speeds. Between post-natal days 27-30 and adulthood, the optimal temporal frequencies of neurons in the primary visual cortex increase, whereas their contrast thresholds decrease. However, the optimal spatial frequencies, spatial resolution values and spatial bandwidths of these neurons are, soon after eyelid opening, similar to those observed in the adult rat. These neuronal response profiles suggest that the retino-collicular and retino-geniculo-cortical projections are mainly innervated by magnocellular and koniocellular retinal ganglion cells. Neurons in the primary visual cortex of rats having experienced hyperthermic seizures are, soon after eyelid opening, sensitive to low optimal spatial frequencies and show broad directional and temporal bandwidths, as well as elevated contrast thresholds when compared to neurons of normal rats. At adulthood, low optimal temporal frequencies and broad spatial bandwidths are also observed in rats having experienced hyperthermic seizures. The alteration of response profiles of neurons in the primary visual cortex of rats having experienced hyperthermic seizures suggests an unbalance between excitatory and inhibitory mechanisms in this cortical structure. These results also suggest that a single episode of febrile seizures could be sufficient to impede the development of the spatio-temporal receptive field properties of neurons in the primary visual cortex

Arm exponent for the Gaussian free field on metric graphs in intermediate dimensions

We investigate the bond percolation model on transient weighted graphs ${G}$ induced by the excursion sets of the Gaussian free field on the corresponding metric graph. We assume that balls in ${G}$ have polynomial volume growth with growth exponent $\alpha$ and that the Green's function for the random walk on ${G}$ exhibits a power law decay with exponent $\nu$, in the regime $1\leq \nu \leq \frac{\alpha}{2}$. In particular, this includes the cases of ${G}=\mathbb Z^3$, for which $\nu=1$, and ${G}= \mathbb Z^4$, for which $\nu=\frac{\alpha}{2}=2$. For all such graphs, we determine the leading-order asymptotic behavior for the critical one-arm probability, which we prove decays with distance $R$ like $R^{-\frac{\nu}{2}+o(1)}$. Our results are in fact more precise and yield logarithmic corrections when $\nu>1$ as well as corrections of order $\log \log R$ when $\nu=1$. We further obtain very sharp upper bounds on truncated two-point functions close to criticality, which are new when $\nu>1$ and essentially optimal when $\nu=1$. This extends previous results from arXiv:2101.05801.Comment: 21 page

Phase transition for the late points of random walk

Let $X$ be a random walk on the torus of side length $N$ in dimension $d\geq 3$ with uniform starting point, and $t_{\text{cov}}$ be the expected value of its cover time, which is the first time that $X$ has visited every vertex of the torus at least once. For $\alpha > 0$, the set $\mathcal{L}^{\alpha}$ of $\alpha$-late points consists of those points not visited by $X$ at time $\alpha t_{\text{cov}}$. We prove the existence of a value $\alpha_* \in (\frac12,1)$ across which $\mathcal{L}^{\alpha}$ trivialises as follows: for all $\alpha > \alpha_*$ and $\epsilon\geq N^{-c}$ there exists a coupling of $\mathcal{L}^\alpha$ and two occupation sets $\mathcal{B}^{\alpha_\pm}$ of i.i.d. Bernoulli fields having the same density as $\mathcal{L}^{\alpha\pm \epsilon}$, which is asymptotic to $N^{-(\alpha\pm\epsilon)d}$, with the property that the inclusion $\mathcal{B}^{\alpha_+} \subseteq \mathcal{L}^{\alpha} \subseteq \mathcal{B}^{\alpha_-}$ holds with high probability as $N \to \infty$. On the contrary, when $\alpha \leq \alpha_*$ there is no such coupling. Corresponding results also hold for the vacant set of random interlacements at high intensities. The transition at $\alpha_*$ corresponds to the (dis-)appearance of `double-points' (i.e. neighboring pairs of points) in $\mathcal{L}^\alpha$. We further describe the law of $\mathcal{L}^{\alpha}$ for $\alpha>\frac12$ by adding independent patterns to $\mathcal{B}^{\alpha_{\pm}}$. In dimensions $d \geq 4$ these are exactly all two-point sets. When $d=3$ one must also include all connected three-point sets, but no other.Comment: 68 pages, 2 figure