Related papers: A derivative formula for the free energy function
Consider percolation on the triangular lattice. Let $\kappa(p)$ be the free energy at the zero field. We show that $$|\kappa'''(p)| \leq |p-p_c|^{-1/3+o(1)} \mbox{ if } p \neq p_c.$$ Furthermore, we show that there exists a sequence…
We calculate up to four loops the free energy of the two-dimensional (2D) O(n) nonlinear sigma-model regularized on the lattice with the 0-loop and 1-loop Symanzik improved actions. An effective coupling constant based on this calculation…
Sigma models arise frequently in particle physics and condensed-matter physics as low-energy effective theories. In this paper I compute the exact free energy at any temperature in two hierarchies of integrable sigma models in two…
We consider a free $p$-form gauge theory on a $d$-dimensional sphere of radius $R$ and calculate its free energy. We perform the calculation for generic values of $p$ and obtain the free energy as a function of $d, p$ and $R$. The result…
We calculate the 2-loop partition function of QCD on the lattice, using the Wilson formulation for gluons and the overlap-Dirac operator for fermions. Direct by-products of our result are the 2-loop free energy and average plaquette. Our…
We study the dependence of the free energy on the CP violating angle theta, in four-dimensional SU(N) gauge theories with N >= 3, and in the large-N limit. Using the Wilson lattice formulation for numerical simulations, we compute the first…
Consider an anisotropic independent bond percolation model on the $d$-dimensional hypercubic lattice, $d\geq 2$, with parameter $p$. We show that the two point connectivity function $P_{p}(\{(0,\dots,0)\leftrightarrow (n,0,\dots,0)\})$ is a…
We show that negative of the number of floppy modes behaves as a free energy for both connectivity and rigidity percolation, and we illustrate this result using Bethe lattices. The rigidity transition on Bethe lattices is found to be first…
We prove that for $\kappa\in(0,8)$, if $(\eta_1,\eta_2)$ is a $2$-SLE$_\kappa$ pair in a simply connected domain $D$ with an analytic boundary point $z_0$, then $\lim_{r\to 0^+}r^{-\alpha} \mathbb{P}[\mbox{dist}(z_0,\eta_j)<r,j=1,2]$…
We prove that the free energy of any spherical mixed $p$-spin model converges as the dimension $N$ tends to infinity. While the convergence is a consequence of the Parisi formula, the proof we give is independent of the formula and uses the…
This work addresses the question of whether it is possible to define simple pair-wise interaction terms to approximate free energies of proteins or polymers. Rather than ask how reliable a potential of mean force is, one can ask how…
We compute free energies as well as conformal anomalies associated with boundaries for a conformal free scalar field. To that matter, we introduce the family of spaces of the form $\mathbb{S}^a\times \mathbb{H}^b$, which are conformally…
We study the free energy for pure and mixed spherical $p$-spin models with i.i.d.\ disorder. In the mixed case, each $p$-interaction layer is assumed either to have regularly varying tails with exponent $\alpha_p$ or to satisfy a finite…
We calculate the perturbative value of the Free Energy in Lattice QCD in three dimensions, up to three loops. Our calculation is performed using the Wilson formulation for gluons in SU(N) gauge theories. The Free Energy is directly related…
For an odd prime $p$, we consider free actions of $(\mathbb{Z}/p)^2$ on $S^{2n-1}\times S^{2n-1}$ given by linear actions of $(\mathbb{Z}/p)^2$ on $\mathbb{R}^{4n}$. Simple examples include a lens space cross a lens space, but $k$-invariant…
We consider a model of p independent Ising spins on a dynamical planar phi-cubed graph. Truncating the free energy to two terms yields an exactly solvable model that has a third order phase transition from a pure gravity region (gamma=-1/2)…
We compute the complete topological expansion of the formal hermitian two-matrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1/ N expansion of the nonmixed correlation functions and give a new…
Reliably computing the free energy in a gauge theory like QCD is a challenging and resource-demanding endeavor. As an alternative, we explore here the possibility to obtain the associated thermodynamic anomaly by exploiting its relation to…
A $2$-SLE$_\kappa$ ($\kappa\in(0,8)$) is a pair of random curves $(\eta_1,\eta_2)$ in a simply connected domain $D$ connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLE$_\kappa$ curve in a…
The functional relation of the Riemann z\^eta function provides us with neither the nature nor the expression of z\^eta at positive odd numbers. From the function $F(z)=\frac{z^{-2n}}{e^z-1}$, we find a functional relation involving…