{"id":"W2949901644","doi":"","title":"Non-amenable Cayley graphs of high girth have p_c < p_u and mean-field exponents","year":2012,"lang":"en","type":"preprint","venue":"arXiv (Cornell University)","topic":"Stochastic processes and statistical mechanics","field":"Mathematics","cited_by":2,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of British Columbia","funders":"","keywords":"Cayley graph; Mathematics; Percolation (cognitive psychology); Uniqueness; Critical exponent; Combinatorics; Field (mathematics); Discrete mathematics; Girth (graph theory); Pure mathematics; Graph; Mathematical analysis; Scaling; Geometry","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0003312448,0.0003600667,0.0004212412,0.001463676,0.001151477,0.001786906,0.000414092,0.0006593582,0.002425689],"category_scores_gemma":[0.003181732,0.0003076457,0.0003766777,0.0005578059,0.001648818,0.001512976,0.0008456325,0.001451638,0.0001816909],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0005316475,"about_ca_system_score_gemma":0.0003722342,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0006446053,"about_ca_topic_score_gemma":0.001108852,"domain_scores_codex":[0.9997503,0.00006465394,0.000008944698,0.00005439571,0.00005911202,0.00006253944],"domain_scores_gemma":[0.9959247,0.002413427,0.0006409934,0.0002085769,0.0002610673,0.0005512008],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.0001634782,0.00009509769,0.007961377,0.0002336347,0.00006841958,0.0006876403,0.0008039479,0.01813907,0.02920195,0.929593,0.002845345,0.01020714],"study_design_scores_gemma":[0.00004534306,0.0001025859,0.009736994,0.00003000915,0.00003339984,0.001107789,0.0004400637,0.0513199,0.009358485,0.9246291,0.003148506,0.00004784548],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9540316,0.0006065283,0.02732532,0.0007361866,0.00006432236,0.00003276991,0.0001074155,0.0001577196,0.01693809],"genre_scores_gemma":[0.9959347,0.0001619536,0.002923399,0.00006440725,0.00003364096,0.00001482916,0.00003684318,0.00001847146,0.0008117503],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.002425689,"threshold_uncertainty_score":0.008114755,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.08880060688306342,"score_gpt":0.2231094806132213,"score_spread":0.1343088737301578,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}