arXiv subjects

Loïc Gassmann

Publications and source records attributed to Loïc Gassmann.

3 recordsLinked to original sources

arXiv2026

Uniform analyticity of local observables in FK-percolation and analyticity of the Ising spontaneous magnetisation

We prove that, in the FK-percolation model, the probabilities of local events are uniformly analytic in the percolation parameter $p$ under suitable mixing assumptions on the measure, and satisfy a uniform exponential growth bound. This result allows us to prove that the magnetisation of the Potts model is analytic in a suitable range of parameters, including the Ising case in all dimensions $d \geq 3$ in the whole supercritical regime. We also provide a proof of the analyticity of the susceptibility of the Potts model with $q$ colours, for any $q \geq 2$ in the whole subcritical interval. Finally, we prove the analyticity of various quantities in the FK-percolation measure, including the multi-point and truncated multi-point connectivity probabilities.

math.PR↗

arXiv2025

Comparison of arm exponents in planar FK-percolation

By the FKG inequality for FK-percolation, the probability of the alternating two-arm event is smaller than the product of the probabilities of having a primal arm and a dual arm, respectively. In this paper, we improve this inequality by a polynomial factor for critical planar FK-percolation in the continuous phase transition regime ($1 \leq q \leq 4$). In particular, we prove that if the alternating two-arm exponent $α_{01}$ and the one-arm exponents $α_0$ and $α_1$ exist, then they satisfy the strict inequality $α_{01} > α_0 + α_1$. The question was formulated by Garban and Steif in the context of exceptional times and was brought to our attention by Radhakrishnan and Tassion, who obtained the same result for planar Bernoulli percolation through different methods.

math.PR↗

arXiv2025

Ordering and Convergence of Large Degrees in Random Hyperbolic Graphs

We describe the asymptotic behaviour of large degrees in random hyperbolic graphs, for all values of the curvature parameter $ α$. We prove that, with high probability, the node degrees satisfy the following ordering property: the ranking of the nodes by decreasing degree coincides with the ranking of the nodes by increasing distance to the centre, at least up to any constant rank. In the scale-free regime $ α>1/2$, the rank at which these two rankings cease to coincide is $n^{1/(1+8 α)+o(1)}$. We also provide a quantitative description of the large degrees by proving the convergence in distribution of the normalised degree process towards a Poisson point process. In particular, this establishes the convergence in distribution of the normalised maximum degree of the graph. A transition occurs at $ α= 1/2$, which corresponds to the connectivity threshold of the model. For $ α< 1/2$, the maximum degree is of order $n - O(n^{ α+ 1/2})$, whereas for $ α\geq 1/2$, the maximum degree is of order $n^{1/(2 α)}$. In the cases $ α< 1/2$ and $ α> 1/2$, the limit distribution of the maximum degree belongs to the class of extreme value distributions (Weibull for $ α< 1/2$ and Fréchet for $ α> 1/2$). This refines previous estimates on the maximum degree for $ α> 1/2$ and extends the study of large degrees to the dense regime $ α\leq 1/2$.

math.PR↗