random stepped surfaces
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random stepped surfaces Richard Kenyon (Yale) <latexit - PowerPoint PPT Presentation

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  1. <latexit sha1_base64="nk+5qLntFDRdai6LVmbqo7DdQus=">AB/nicbVA9SwNBEN3zM8avU7GyWQxCbMJdLSgI1gE8V8SBLC3t4kWbK3e+zuCeEI+FdsLBSx9XfY+W/cS1Jo4oOBx3szMwLYs608bxvZ2l5ZXVtPbeR39za3tl19/brWiaKQo1KLlUzIBo4E1AzHBoxgpIFHBoBMOrzG8gtJMinsziqETkb5gPUaJsVLXPbxjdEBUiG9AjKTAxQfC4bTrFrySNwFeJP6MFNAM1a71Q4lTSIQhnKidcv3YtNJiTKMchjn24mGmNAh6UPLUkEi0J10cv4Yn1glxD2pbAmDJ+rviZREWo+iwHZGxAz0vJeJ/3mtxPQuOikTcWJA0OmiXsKxkTjLAodMATV8ZAmhitlbcRYGocYmlrch+PMvL5J6ueSflcq35ULlchZHDh2hY1REPjpHFXSNqiGKErRM3pFb86T8+K8Ox/T1iVnNnOA/sD5/AHpKZTI</latexit> the hidden geometry of random stepped surfaces Richard Kenyon (Yale)

  2. <latexit sha1_base64="FkXoPQ5IFkbwrfrxmUDmS+mhXlY=">ACBnicbVDLSgMxFM3UV62vUZciBIvgqsy0C8WFty4rGIf0BlLJk3b0EwyJBmhDF258VfcuFDEreAfuPNvzEy7qK0HAifn3Mu9wQRo0o7zo+VW1peWV3Lrxc2Nre2d+zdvYSscSkjgUTshUgRjlpK6pZqQVSYLCgJFmMLxK/eYDkYoKfqdHEfFD1Oe0RzHSRurYh4PzxAuRHgQBvB3flz0tZv4du+iUnAxwkbhTUrz8qmSodexvrytwHBKuMUNKtV0n0n6CpKaYkXHBixWJEB6iPmkbylFIlJ9kZ4zhsVG6sCekeVzDTJ3tSFCo1CgMTGW6oZr3UvE/rx3r3pmfUB7FmnA8GdSLGdQCpnALpUEazYyBGFJza4QD5BEWJvkCiYEd/7kRdIol9xKqXzjFKsXYI8OABH4AS4BRUwTWogTrA4BE8g1fwZj1ZL9a79TEpzVnTn3wB9bnL7dlmtY=</latexit> <latexit sha1_base64="eovm0ZYyi/sq4F9zpAzio2X0IQI=">AB63icbVBNS8NAEJ3Urxq/qh69LBbBU0jqQU9S8OKxgv2ANpTNdtMu3d2E3Y1Qv+CFw+KePUPefPfuGlz0NYHA4/3ZpiZF6WcaeP7305lY3Nre6e6+7tHxwe1Y5POjrJFKFtkvBE9SKsKWeStg0znPZSRbGIO1G07vC7z5RpVkiH80spaHAY8liRrApJNfzvGt7nv+AmidBCWpQ4nWsPY1GCUkE1QawrHW/cBPTZhjZRjhdO4OMk1TKZ4TPuWSiyoDvPFrXN0YZURihNlSxq0UH9P5FhoPROR7RTYTPSqV4j/ef3MxDdhzmSaGSrJclGcWQSVDyORkxRYvjMEkwUs7ciMsEKE2PjcW0IwerL6T8IrL3ho1Ju3ZRxVOINzuIQArqEJ9CNhCYwDO8wpsjnBfn3flYtlacuYU/sD5/AF/Oo0z</latexit> <latexit sha1_base64="xI2UNgHmi4Qa9AtpXIwWkmig8=">ACHicbVDLTgIxFO3gC/GFunTCauyAwsdGVI3LhEI48EkHTKHWhoO5O2Y4ITPsSNv+LGhca4cWHi31hgFgqepMnJOfem5x4/4kwb1/12Miura+sb2c3c1vbO7l5+/6Chw1hRqNOQh6rlEw2cSagbZji0IgVE+Bya/uhy6jfvQWkWylszjqAryECygFirNTLVyKiADij2AxgQLJpkgHOtYBYQCZhIXk4gZuj7+GZyVymWevmCW3JnwMvES0kBpaj18p+dfkhjAdJQTrRue25kuglRhlEOk1wn1hAROiIDaFsqbSDdTWbHTfCJVfo4CJV90uCZ+nsjIULrsfDt5DSlXvSm4n9eOzbBeTdhMoNSDr/KIg5NiGeNoX7TAE1fGwJoYrZrJgObVvUlqVztgRv8eRl0iXvEqpfF0uVC/SOrLoCB2jU+ShM1RFV6iG6oiR/SMXtGb8+S8O/Ox3w046Q7h+gPnK8fqiWhBg=</latexit> <latexit sha1_base64="izy/JBD+GJHR3RdsQHqxbKt0pTk=">AB+HicbVDLSgMxFL3js9ZHR126CRbBVZmpC8WFNy4ESrYB7RDyaSZNjTJDElGaId+iRsXirj1U9z5N6btLT1QOBwzj3cmxMmnGnjed/O2vrG5tZ2Yae4u7d/UHIPj5o6ThWhDRLzWLVDrClnkjYM5y2E0WxCDlthaPbmd96okqzWD6acUIDgQeSRYxgY6WeWyreM8kEm1CEbe65a9ijcHWiV+TsqQo95zv7r9mKSCSkM41rje4kJMqwMI5xOi91U0wSTER7QjqUSC6qDbH74FJ1ZpY+iWNknDZqrvxMZFlqPRWgnBTZDvezNxP+8TmqiqyBjMkNlWSxKEo5MjGatYD6TFi+NgSTBSztyIyxAoTY7sq2hL85S+vkma14l9U/IdquXaT1GAEziFc/DhEmpwB3VoAIEUnuEV3pyJ8+K8Ox+L0TUnzxzDHzifP8Qhkn0=</latexit> <latexit sha1_base64="wxXfGoIhvdAleK9rAtgbGH8UMI=">ACFHicbVDLSsNAFJ34tr6iLt0MFkGolKSCuhLBjcsKVgtNDZPpxA5OJnHmRhpCd/6AGz/DbRe6UMStC3f+jdPHQq0H7nA4517u3BMkgmtwnC9rYnJqemZ2br6wsLi0vGKvrp3rOFWU1WgsYlUPiGaCS1YDoLVE8VIFAh2EVwf9/2LW6Y0j+UZAlrRuRK8pBTAkby7VLBi7j02x7nEvwa9vSNgtwtf3OZcW82Wl6+20OqYy3y46ZWcAPE7cESke7T317sLSY9W3P71WTNOISaCaN1wnQSaOVHAqWDdgpdqlhB6Ta5Yw1BJIqab+eCoLt4ySguHsTIlAQ/UnxM5ibTOosB0RgTa+q/XF/zGimEB82cyQFJulwUZgKDHuJ4RbXDEKIjOEUMXNXzFtE0UomBwLJgT378nj5LxSdnfL7qlJ4xANMYc20CbaRi7aR0foBFVRDVF0j3roBb1aD9az9Wa9D1snrNHMOvoF6+MbRDqhYQ=</latexit> <latexit sha1_base64="+AHCS9uGeUF2GUbjUlJ7Xvh60=">ACGnicbVA9SwNBFNzM8avqKXNYhQUJNzFQisRbCwsIhgNxCDvNu+Sxb29Y/edcAR/h41/xcZCETux8d+4F1NodGBhmJn3ljdhqQl3/0JianpmdmS3Pl+YXFpeXKyuqFTIjsCkSlZhWCBaV1NgkSQpbqUGIQ4WX4c1x4V/eorEy0eUp9iJoadlJAWQk64rwSkS34x2e5s8RL4NJpRkwOQ7HDSonKTgUaZFEba7nProZqp+zR+C/yXBiFTZCI3ryvtVNxFZjJqEAmvbgZ9SZwDGLVd4V7KLKYgbqCHbUc1xGg7g+Fpd3zLKV0eJcY9TXyo/pwYQGxtHocuGQP17bhXiP957Yyig85A6jQj1OL7oyhTnBJe9MS70qAglTsCwsiCNEHA4Jcm2VXQjB+8l9yUa8Fe7X6Wb16dDiqo8TW2QbZgHbZ0fshDVYkwl2zx7ZM3vxHrwn79V7+45OeKOZNfYL3scXCf6fqg=</latexit> <latexit sha1_base64="hbIWrsfLsami1o1nr8JwVmMHyEY=">ACKXicbVDLSgMxFM34tr6qLt0EW0FRykxd6EYpuHGpYFVoa82kd9pg5mHuHaEM/R03/obBUXd+iOmtQutHgczrk3yTl+ohWS6747Y+MTk1PTM7O5ufmFxaX8so5xqmRUJWxjs2lLxC0iqBKijRcJgZE6Gu48G+O+v7FHRhUcXRG3QaoWhHKlBSkJWa+coxGODFOqp2KDZxh7b4Aa/jraHM28ar8jZdlXvFnEJOHeDX15iaQEjgBFH/zvVSM19wS+4A/C/xhqTAhjhp5p/rVimIUQktUCseW5CjUwYUlJDL1dPERIhb0QbapZGIgRsZIOkPb5hlRYPYmNPRHyg/tzIRIjYDX07GQrq4KjXF/zaikF+41MRUlqk8nvh4JUc4p5vzbeUgYk6a4lQhpl/8plRxghyZabsyV4o5H/kvNydsteaflQuVwWMcMW2PrbJN5bI9V2DE7YVUm2T17ZC/s1Xlwnpw35+N7dMwZ7qyX3A+vwDVKSO</latexit> h : R 2 → R Minimize area: ZZ q 1 + h 2 x + h 2 y dx dy min h U 1 + s 2 + t 2 is the “surface tension”. √ Here σ ( s, t ) = ... Weierstrass-Enneper parameterization of minimal surfaces Let f, g be (arbitrary) analytic functions, then ✓Z ◆ Z Z f ( z )(1 − g ( z ) 2 ) dz, i f ( z )(1 + g ( z ) 2 ) dz, f ( z ) g ( z ) dz Re parameterizes a minimal surface in R 3 .

  3. <latexit sha1_base64="prc/w+MbpXs6SGiT/hNwQbGnME4=">AB+3icbVC7TsMwFHV4lvIKZWSxWiExVUkZYEKVWBgYikQfUhu1jnuTWnWcyHYQJeqvsDCAECs/wsbf4LYZoOVIVzo65173uMnCntON/W2vrG5tZ2Yae4u7d/cGgflVoqTiWFJo15LDs+UcCZgKZmkMnkUAin0PbH1/P/PYDSMVica8nCXgRCQULGCXaSH27NBjcxk8gQihjzcwroerbFafqzIFXiZuTCsrR6NtfvWFM0wiEpwo1XWdRHsZkZpRDtNiL1WQEDomIXQNFSQC5WXz3af41ChDHMTSlNB4rv6eyEik1CTyTWdE9EgtezPxP6+b6uDSy5hIUg2CLj4KUo51jGdB4CGTQDWfGEKoZGZXTEdEqpNXEUTgrt8ip1aruebV2V6vUr/I4CugEldEZctEFqMb1EBNRNEjekav6M2aWi/Wu/WxaF2z8plj9AfW5w+pIJQt</latexit> “Lozenge” tilings

  4. <latexit sha1_base64="ETFf3v2F7coyQGXgZl7fuhVdITg=">ACDXicbVC7SgNBFJ2NrxhfUubwShYhd1YaCUBG8sI5gFJCLOzd5Mhs7PLzF0hLvkBG3/FxkIRW3s7/8bJo9DEAxcO59w7c+/xEykMu63k1tZXVvfyG8WtrZ3dveK+wcNE6eaQ53HMtYtnxmQkEdBUpoJRpY5Eto+sPrid+8B21ErO5wlEA3Yn0lQsEZWqlXPJHxA6g+UBT2ib6hTAM1CEkCATWpDhkH0yuW3LI7BV0m3pyUyBy1XvGrE8Q8jUAhl8yYtucm2M2YRsEljAud1EDC+JD1oW2pYhGYbja9ZkxPrRLQMNa2FNKp+nsiY5Exo8i3nRHDgVn0JuJ/XjvF8LKbCZWkCIrPgpTSTGmk2hoIDRwlCNLGNfC7kr5gGnG0QZYsCF4iycvk0al7J2XK7eVUvVqHkeHJFjckY8ckGq5IbUSJ1w8kieySt5c56cF+fd+Zi15pz5zCH5A+fzB9qnAg=</latexit> <latexit sha1_base64="BewevTCbKpmUDwR2qkZiQIlA7g=">ACDHicbVC7TsNAEDzJrwClDQnIiSqyA4FVIBEQxk8pBChNaXdXzifLbu1kjBygfQ8Cs0FCBEywfQ8TdcHgWvqUYzs9rdCTMlLfn+pzczOze/sLi0XFpZXVvfKG9uNW2aG4ENkarUtEOwqKTGBklS2M4MQhIqbIU3ZyO/dYvGylRf0iDbgJ9LSMpgJx0Xa60YiAeQ5ahtjxKDVdg+sjtwBIm3Mo7PHEpv+qPwf+SYEoqbIr6dfnjqpeKPEFNQoG1ncDPqFuAISkUDktXucUMxA30seOohgRtxg/M+R7TumNL4lSTXysfp8oILF2kIQumQDF9rc3Ev/zOjlFR91C6iwn1GKyKMoVp5SPmuE9aVCQGjgCwkh3KxcxGBDk+iu5EoLfL/8lzVo1OKjWLmqV0+NpHUtsh+2yfRawQ3bKzlmdNZhg9+yRPbMX78F78l69t0l0xpvObLMf8N6/AEXAmxs=</latexit> lozenge tilings are stepped surfaces What happens for large system size?

  5. The Dimer Model Dimer cover of a graph: perfect matching of the vertices.

  6. <latexit sha1_base64="ItMjMLZMgSuLrlCO2U1brJeV3ys=">ACBHicbVC7SgNBFJ2NrxhfUcs0g0GwCrux0EoCWlhGMA9IQpid3CRD5rHMzArLksLGX7GxUMTWj7Dzb5wkW2jigQuHc+6dufeEWfG+v63l1tb39jcym8Xdnb39g+Kh0dNo2JNoUEV7odEgOcSWhYZjm0Iw1EhBxa4eR65rceQBum5L1NIugJMpJsyCixTuoXSzdMOBsricdKQkKVCDEn1jIK/WLZr/hz4FUSZKSMtT7xa/uQNFYgLSUE2M6gR/ZXkq0e43DtNCNDUSETsgIOo5KIsD0vkRU3zqlAEeKu1KWjxXf0+kRBiTiNB1CmLHZtmbif95ndgOL3spk1FsQdLFR8OY6vwLBE8YBqo5YkjhGrmdsV0TDSh1gVTcCEyevkma1EpxXqnfVcu0qiyOPSugEnaEAXaAaukV1EAUPaJn9IrevCfvxXv3PhatOS+bOUZ/4H3+AIjKmAI=</latexit> Dimers on honeycomb lattice

  7. <latexit sha1_base64="KRK7tE8CPWSOZ0uBWN3O6fnEZsg=">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</latexit> <latexit sha1_base64="3P0Ke3FwLXCuJxC+RJ75PBsUKU=">ACF3icbVDJSgNBEO1xjXGLevRSmBE8DTPjQVGUgAiCFwUThSEnk5FG3sWumvEPwL/6KFw+KeNWbf2NnObg9KHi8V0VvShT0pDvfzpj4xOTU9OFmeLs3PzCYmlpuWbSXAusilSl+iLiBpVMsEqSF5kGnkcKTyPrg/6/vkNaiPT5Iy6GTZjfpnIjhScrNQqeYe3PM4U7oB7I04AZhg2SMBoLQ3QW30UaCY4A9CH3Xa5XKvucPAH9JMCJlNsJq/TRaKcijzEhobgx9cDPqNnjmqRQeFds5AYzLq75JdYtTbjd3OwN/rqDdau0oZNqWwnBQP0+0eOxMd04sp0xpyvz2+uL/3n1nDrbzZ5MspwEcNFnVwBpdAPCdpSoyDVtYQLe2tIK645oJslEUbQvD75b+kFnrBphehuXK/iOAltla2yDBWyLVdgRO2FVJtg9e2TP7MV5cJ6cV+dt2DrmjGZW2A84718RuJrK</latexit> Thm[Kasteleyn 1965] For G a part of the honeycomb graph, let K be the bipartite adjacency matrix, ( 1 b ∼ w K wb = 0 else. Then | det K | is the number of dimer covers. Example: K is 12 × 12; det K = 20.

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