In [ ], we have studied the Boltzmann random triangulation of the disk coupled to an Ising model with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the in nite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform in nite half-planar triangulation (UIHPT) decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of nite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning. In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in [ ] to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of quantum surfaces (a.k.a. Liouville quantum gravity). their roots in the physics literature [ ], and was revisited and popularized by Angel in [ ]. The peeling process proves to be a valuable tool for understanding the geometry of random planar maps without Ising model, see [ ] for a review of recent developments.In a previous paper [ ], we extended some enumeration results of Bernardi and Bousquet-Mélou [ ] to study the Ising-decorated random triangulations with Dobrushin boundary condition at its critical temperature. We used the peeling process to construct the local limit of the model, and to obtain several scaling limit results concerning the lengths of some Ising interfaces. In this paper, we extend similar results to the model at any temperature, and show how the large scale geometry of Ising-decorated random triangulations changes qualitatively at the critical temperature. In particular, our results con rm the physical intuition that, at large scale, Ising-decorated random maps at non-critical temperatures behave like non-decorated random maps.A similar model of Ising-decorated triangulations has been studied in a recent work of Albenque, Ménard and Schae er [ ]. They followed an approach reminiscent of Angel and Schramm in [ ] to show that the model has a local limit at any temperature, and obtained several properties of the limit such as one-endedness and recurrence for some temperatures. However they studied the model without boundary, and hence did not encounter the geometric consequences of the phase transition.We start by recalling some essential de nitions from [ ].Planar maps. Recall that a ( nite) planar map is a proper embedding of a nite connected graph into the sphere S 2 , viewed up to o...
Traditional Chinese medicine (TCM) has been practiced for thousands of years for treating human diseases. In comparison to modern medicine, one of the advantages of TCM is the principle of herb compatibility, known as TCM formulae. A TCM formula usually consists of multiple herbs to achieve the maximum treatment effects, where their interactions are believed to elicit the therapeutic effects. Despite being a fundamental component of TCM, the rationale of combining specific herb combinations remains unclear. In this study, we proposed a network-based method to quantify the interactions in herb pairs. We constructed a protein–protein interaction network for a given herb pair by retrieving the associated ingredients and protein targets, and determined multiple network-based distances including the closest, shortest, center, kernel, and separation, both at the ingredient and at the target levels. We found that the frequently used herb pairs tend to have shorter distances compared to random herb pairs, suggesting that a therapeutic herb pair is more likely to affect neighboring proteins in the human interactome. Furthermore, we found that the center distance determined at the ingredient level improves the discrimination of top-frequent herb pairs from random herb pairs, suggesting the rationale of considering the topologically important ingredients for inferring the mechanisms of action of TCM. Taken together, we have provided a network pharmacology framework to quantify the degree of herb interactions, which shall help explore the space of herb combinations more effectively to identify the synergistic compound interactions based on network topology.
In this paper we investigate the critical Fortuin-Kasteleyn (cFK) random map model. For each q ∈ [0, ∞] and integer n ≥ 1, this model chooses a planar map of n edges with a probability proportional to the partition function of critical q-Potts model on that map. Sheffield introduced the hamburger-cheeseburger bijection which maps the cFK random maps to a family of random words, and remarked that one can construct infinite cFK random maps using this bijection. We make this idea precise by a detailed proof of the local convergence. When q = 1, this provides an alternative construction of the UIPQ. In addition, we show that the limit is almost surely one-ended and recurrent for the simple random walk for any q, and mutually singular in distribution for different values of q.
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