We study a model of fully-packed dimer configurations (or perfect matchings) on a bipartite periodic graph that is two-dimensional but not planar. The graph is obtained from Z 2 via the addition of an extensive number of extra edges that break planarity (but not bipartiteness). We prove that, if the weight λ of the non-planar edges is small enough, the height function scales on large distances to the Gaussian Free Field with a λ-dependent amplitude, that coincides with the anomalous exponent of dimer-dimer correlations. Because of non-planarity, Kasteleyn's theory does not apply: the model is non-determinantal. Rather, we map the model to a system of interacting lattice fermions in the Luttinger universality class, that we then analyze via fermionic Renormalization Group methods.

Background: There is evidence of hypnosis being effective in reducing both emotional distress, anxiety, and depression. Hypnosis as a technique promotes enhanced mental mastery over the body. This may result in hypnosis being particularly salient in treating distress associated with somatic and psychological symptoms. Objective: This research aims at verifying the effectiveness of a group treatment, which provides for the use of hypnosis-related techniques in mixed-group outpatients with anxiety or depression. Methods: Participants. 31 outpatients (average age= 49.005; DS =12.1) including 13 with mild-moderate depression (average age= 49.17; DS=12.20) and 18 with mild-moderate anxiety (average age=48.84; DS=13.02). Procedure. The group treatment comprised 8 sessions, during which a hypnotic state was induced, aimed at generating a sensation of profound wellbeing in the participants. They could share their experience in a penultimate group session, and were provided with individualised recommendations on nutrition and lifestyle in the last individual session. Patients were prescribed to practice self-hypnosis every day at home with the support of a CD-ROM. Psychopathological symptomatology (SCL-90R), depression (BDI), and anxiety (STAI-Y1, SAS) were assessed at pre, post, and 3-month follow-up. Statistical analysis. Friedman, Kruskal-Wallis and Mann-Whitey tests were used. The Bonferroni’s correction was applied as needed. The effect size (Cohen’s d) was also measured. Results: For the total sample, for all tests, significant differences were observed in the phases. The effect size was found to vary from “small” at pre to ”medium” at post. A “large” effect size was observed when comparing pre and follow-up phases. An overall reduction in the symptoms of distress measured by the SCL-90 R – with the anxiety group showing better outcomes – alongside with an improvement in the symptoms of depression and anxiety were observed in all participants. Conclusion: The clinical impact appears to be relevant, as shown by the values for d. The treatment is cost-effective for highly prevalent disorders in outpatients. The outcomes of this study support the effectiveness of hypnotic group treatment.